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</div><h2>SL Paper 1</h2><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The Venn diagram below shows events <em>A</em> and <em>B</em> where \({\rm{P}}(A) = 0.3\) , \({\rm{P}}(A \cup B) = 0.6\) </span><span style="font-family: times new roman,times; font-size: medium;">and \({\rm{P}}(A \cap B) = 0.1\) . The values <em>m</em> , <em>n</em> , <em>p</em> and <em>q</em> are probabilities.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/beach.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Write down the value of <em>n</em> .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Find the value of <em>m</em> , of <em>p</em> , and of <em>q</em> .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \({\rm{P}}(B')\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) \(n = 0.1\) <em><strong>A1 N1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) \(m = 0.2\) , \(p = 0.3\) , \(q = 0.4\) <em><strong>A1A1A1 N3</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></strong></em></p>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">appropriate approach</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{P}}(B') = 1 - P(B)\) , \(m + q\) , \(1 - (n + p)\) <strong><em>(M1)</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(B') = 0.6\) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Most candidates were able to find the correct values for the Venn diagram. Unfortunately, however, there were many candidates who did not understand what each region of the diagram represents. For example, a very common error was thinking that \({\rm{P}}(B) = p\) , rather than the correct \({\rm{P}}(B) = p + n\) . </span></p>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Candidates seemed to understand the idea of the complement in part (b), but some were not able to find the correct answer because of confusion over the separation of the different regions in the diagram. </span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A box contains six red marbles and two blue marbles. Anna selects a marble from </span><span style="font-family: times new roman,times; font-size: medium;">the box. She replaces the marble and then selects a second marble.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the probability that the first marble Anna selects is red.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the probability that Anna selects two red marbles.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the probability that one marble is red and one marble is blue.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: In this question, method marks may be awarded for selecting without replacement, as noted in </span><span style="font-family: times new roman,times; font-size: medium;">the examples.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(R) = \frac{6}{8}\left( { = \frac{3}{4}} \right)\) <em><strong>A1 N1</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[1 mark]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to find \({\rm{P(Red)}} \times {\rm{P(Red)}}\) <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{P(}}R{\rm{)}} \times {\rm{P(}}R{\rm{)}}\) , \(\frac{3}{4} \times \frac{3}{4}\) , \(\frac{6}{8} \times \frac{5}{7}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(2R) = \frac{{36}}{{64}}\left( { = \frac{9}{{16}}} \right)\) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [2 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 1</span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to find \({\rm{P(Red)}} \times {\rm{P(Blue)}}\) <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{P(}}R{\rm{)}} \times {\rm{P(}}B{\rm{)}}\) , \(\frac{6}{8} \times \frac{2}{8}\) , \(\frac{6}{8} \times \frac{2}{7}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">recognizing two ways to get one red, one blue <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{P}}(RB) + {\rm{P}}(BR)\) , \(2\left( {\frac{{12}}{{64}}} \right)\) , \(\frac{6}{8} \times \frac{2}{7} + \frac{2}{8} \times \frac{6}{7}\)</span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\({\rm{P}}(1R,1B) = \frac{{24}}{{64}}\left( { = \frac{3}{8}} \right)\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1 N2</span></strong></em></p>
<p><em><strong> <span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<p><strong> <span style="font-family: times new roman,times; font-size: medium;">METHOD 2</span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">recognizing that \({\rm{P}}(1R,1B)\) is \(1 - {\rm{P}}(2B) - {\rm{P}}(2R)\) <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to find \({\rm{P}}(2R)\) and \({\rm{P}}(2B)\) <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{P}}(2R) = \frac{3}{4} \times \frac{3}{4}\) , \(\frac{6}{8} \times \frac{5}{7}\) ; \({\rm{P}}(2B) = \frac{1}{4} \times \frac{1}{4}\) , \(\frac{2}{8} \times \frac{1}{7}\)</span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\({\rm{P}}(1R,1B) = \frac{{24}}{{64}}\left( { = \frac{3}{8}} \right)\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1 N2</span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [3 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Candidates did very well on parts (a) and (b) of this probability question. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Candidates did very well on parts (a) and (b) of this probability question, and knew to multiply the probabilities of independent events in part (b). However, in part (c), very few candidates considered that there are two ways to draw one red and one blue marble, and therefore did not earn full marks on this question. There were also some candidates who tried to add, rather than multiply, the probabilities in parts (b) and (c). </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Candidates did very well on parts (a) and (b) of this probability question, and knew to multiply the probabilities of independent events in part (b). However, in part (c), very few candidates considered that there are two ways to draw one red and one blue marble, and therefore did not earn full marks on this question. There were also some candidates who tried to add, rather than multiply, the probabilities in parts (b) and (c). </span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Consider the events <em>A</em> and <em>B</em>, where \({\rm{P}}(A) = 0.5\) , \({\rm{P}}(B) = 0.7\) and \({\rm{P}}(A \cap B) = 0.3\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The Venn diagram below shows the events <em>A</em> and <em>B</em>, and the probabilities <em>p</em>, <em>q</em> and <em>r</em>.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/baba.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Write down the value of</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) <em>p</em> ;</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) <em>q</em> ;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iii) <em>r</em>.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a(i), (ii) and (iii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the value of \({\rm{P}}(A|B')\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Hence, or otherwise, show that the events <em>A</em> and <em>B</em> are <strong>not</strong> independent.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) \(p = 0.2\) <em><strong>A1 N1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \(q = 0.4\) <em><strong>A1 N1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iii) \(r = 0.1\) <em><strong>A1 N1 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">a(i), (ii) and (iii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(A|B') = \frac{2}{3}\) <em><strong>A2 N2</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>A1</strong></em> for an unfinished answer such as \(\frac{{0.2}}{{0.3}}\) . </span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">valid reason <em><strong>R1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{2}{3} \ne 0.5\) , \(0.35 \ne 0.3\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">thus, <em>A</em> and <em>B</em> are not independent <em><strong>AG N0</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[1 mark]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">As the definitions of <em>p</em> and <em>q</em> were not clear to candidates, both responses of <em>p</em> \(= 0.2\), <em>q</em> \(= 0.4\) </span><span style="font-family: times new roman,times; font-size: medium;">and <em>p</em> \(= 0.5\), <em>q</em> \(= 0.7\) were accepted for full marks. However, finding <em>r</em> eluded many.</span></p>
<div class="question_part_label">a(i), (ii) and (iii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Few </span><span style="font-family: times new roman,times; font-size: medium;">candidates answered the conditional probability correctly. Many attempted to use the formula </span><span style="font-family: times new roman,times; font-size: medium;">in the booklet without considering the complement, and there was little evidence of the Venn </span><span style="font-family: times new roman,times; font-size: medium;">diagram being utilized as a helpful aid.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">To show the events are not independent, many </span><span style="font-family: times new roman,times; font-size: medium;">correctly reasoned that \(0.3 \ne 0.35\) . A handful recognized that \({\rm{P}}(A|B') \ne {\rm{P}}(A)\) is an </span><span style="font-family: times new roman,times; font-size: medium;">alternative approach that uses the answer in part (b). Some candidates do not know the </span><span style="font-family: times new roman,times; font-size: medium;">difference between <strong>independent</strong> and <strong>mutually exclusive</strong>.</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = \frac{1}{2}{x^2} + kx + 8\) , where \(k \in \mathbb{Z}\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the values of <em>k</em> such that \(f(x) = 0\) has two equal roots.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Each value of <em>k</em> is equally likely for \( - 5 \le k \le 5\) . Find the probability that </span><span style="font-family: times new roman,times; font-size: medium;">\(f(x) = 0\) has no roots.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 1</span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of discriminant <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({b^2} - 4ac\) , discriminant = 0</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution into discriminant <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({k^2} - 4 \times \frac{1}{2} \times 8\) , \({k^2} - 16 = 0\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(k = \pm 4\) <em><strong>A1A1 N3</strong></em></span></p>
<p><strong> <span style="font-family: times new roman,times; font-size: medium;">METHOD 2</span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">recognizing that equal roots means perfect square <em><strong>(R1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. attempt to complete the square, \(\frac{1}{2}({x^2} + 2kx + 16)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct working</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{1}{2}{(x + k)^2}\) , \(\frac{1}{2}{k^2} = 8\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: times new roman,times; font-size: medium;">\(k = \pm 4\) <em><strong>A1A1 N3</strong></em></span></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [4 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of appropriate approach <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \({b^2} - 4ac < 0\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct working for <em>k</em> <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \( - 4 < k < 4\) , \({k^2} < 16\) , list all correct values of <em>k</em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(p = \frac{7}{{11}}\) <em><strong>A2 N3</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">A good number of candidates were successful in using the discriminant to find the correct values of <em>k</em> in part (a), however, there were many who tried to use the quadratic formula without recognizing the significance of the discriminant. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (b) was very poorly done by nearly all candidates. Common errors included finding the wrong values for <em>k</em>, and not realizing that there were 11 possible values for <em>k</em>. </span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let <em>X</em> be normally distributed with mean 100 cm and standard deviation 5 cm.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">On the diagram below, shade the region representing \({\rm{P}}(X > 105)\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/raymond.png" alt></span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Given that \({\rm{P}}(X < d) = {\rm{P}}(X > 105)\) , find the value of \(d\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Given that \({\rm{P}}(X > 105) = 0.16\) (correct to two significant figures), find \({\rm{P}}(d < X < 105)\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times;"><span style="font-size: medium;"><br><img src="images/daisy.png" alt></span><span style="font-size: medium;"> <em><strong>A1A1 N2</strong> </em></span></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>A1</strong></em> for vertical line to right of mean, <em><strong>A1</strong></em> for shading to right of <strong>their</strong> vertical line. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of recognizing symmetry <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times;"><span style="font-size: medium;">e.g. \(105\) is one standard deviation above the mean so \(d\) is one standard </span><span style="font-size: medium;">deviation below the mean, shading the corresponding part, </span><span style="font-size: medium;">\(105 - 100 = 100 - d\)</span></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(d = 95\) <em><strong>A1 N2</strong> </em></span></p>
<p><span style="font-family: times new roman,times;"><em><strong><span style="font-size: medium;">[2 marks]</span></strong></em></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of using complement <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(1 - 0.32\) , \(1 - p\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(d < X < 105) = 0.68\) <em><strong>A1 N2</strong> </em></span></p>
<p><span style="font-family: times new roman,times;"><em><strong><span style="font-size: medium;">[2 marks]</span></strong></em></span></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Most candidates did very well on part (a), shading the area under the normal curve. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Not all candidates realised that the problem could be solved by only using the symmetry of the normal distribution curve and the information given. Some of them saw the need to use tables and others just left it blank.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Candidates were only moderately successful on parts (b) and (c), which required understanding of the symmetry of the curve. Many candidates resorted to formulae or tables instead of reasoning through the question. </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Not all candidates realised that the problem could be solved by only using the symmetry of the normal distribution curve and the information given. Some of them saw the need to use tables and others just left it blank.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Candidates were only moderately successful on parts (b) and (c), which required understanding of the symmetry of the curve. Many candidates resorted to formulae or tables instead of reasoning through the question. </span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The ages of people attending a music concert are given in the table below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/chandler.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find <em>p</em> .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">The cumulative frequency diagram is given below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/joey.png" alt></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Use the diagram to estimate</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(i) the 80th percentile;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) the interquartile range.</span></p>
<p> </p>
<div class="marks">[5]</div>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of valid approach <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(92 + 52\) , line on graph at \(x = 31\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(p = 144\) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks] </span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) evidence of valid approach <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. line on graph, \(0.8 \times 160\) , using complement </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( = 29.5\) <em><strong>A1 N2</strong></em> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \({Q_1} = 23\) ; \({Q_3} = 29\) <em><strong> (A1)(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{IQR}} = 6\) (accept any notation that suggests an interval) <em><strong>A1 N3</strong></em> </span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[5 marks] </span></strong></em></p>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (a) was generally answered correctly, with most candidates showing a good grasp of cumulative frequency from a table. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">A surprising number of candidates had difficulty reading values off the cumulative frequency curve. A common incorrect answer for (b)(i) was 29, indicating carelessness with the given scale. Too many candidates gave 40 and 120 for the quartile values. </span></p>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The letters of the word PROBABILITY are written on 11 cards as shown below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/probability.png" alt></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Two cards are drawn at random without replacement.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Let <em>A</em> be the event the first card drawn is the letter A.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Let <em>B</em> be the event the second card drawn is the letter B.</span></p>
<p> </p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \({\rm{P}}(A)\) .</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \({\rm{P}}(B|A)\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \({\rm{P}}(A \cap B)\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(A) = \frac{1}{{11}}\) <em><strong>A1 N1</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[1 mark]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(B|A) = \frac{2}{{10}}\) <em><strong>A2 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">recognising that \({\rm{P}}(A \cap B) = {\rm{P}}(A) \times {\rm{P}}(B|A)\) <em><strong>(M1) </strong></em> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct values <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{P}}(A \cap B) = \frac{1}{{11}} \times \frac{2}{{10}}\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(A \cap B) = \frac{2}{{110}}\) <em><strong>A1 N3 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Most candidates answered part (a) correctly. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Few candidates used the concept of "B given A" to simply "write down" the answer of \(\frac{2}{{10}}\) . Instead, most reached for the formula in the booklet, with which few were successful. </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Few also made the connection that part (c) could be answered using both previous answers. Many found \({\rm{P}}(A \cap B)\) correctly even when answering part (b) incorrectly, although some candidates did not decrease the denominator for the second event. </span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The random variable \(X\) is normally distributed with a mean of 100. The following diagram shows the normal curve for \(X\).</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-11_om_16.32.27.png" alt="M17/5/MATME/SP1/ENG/TZ2/03"></p>
<p>Let \(R\) be the shaded region under the curve, to the right of 107. The area of \(R\) is 0.24.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down \({\text{P}}(X > 107)\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \({\text{P}}(100 < X < 107)\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \({\text{P}}(93 < X < 107)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>\({\text{P}}(X > 107) = 0.24\,\,\,\left( { = \frac{6}{{25}},{\text{ }}24\% } \right)\) <strong><em>A1</em></strong> <strong><em>N1</em></strong></p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid approach <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\({\text{P}}(X > 100) = 0.5,{\text{ P}}(X > 100) - {\text{P}}(X > 107)\)</p>
<p>correct working <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(0.5 - 0.24,{\text{ }}0.76 - 0.5\)</p>
<p>\({\text{P}}(100 < X < 107) = 0.26\,\,\,\left( { = \frac{{13}}{{50}},{\text{ }}26\% } \right)\) <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid approach <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(2 \times 0.26,{\text{ }}1 - 2(0.24),{\text{ P}}(93 < X < 100) = {\text{P}}(100 < X < 107)\)</p>
<p>\({\text{P}}(93 < X < 107) = 0.52\,\,\,\left( { = \frac{{13}}{{25}},{\text{ }}52\% } \right)\) <strong><em>A1 N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-size: medium; font-family: times new roman,times;">A <strong>four-sided</strong> die has three blue faces and one red face. The die is rolled.</span></p>
<p><span style="font-size: medium; font-family: times new roman,times;">Let <em>B</em> be the event a blue face lands down, and <em>R</em> be the event a red face lands down.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(i) P(<em>B</em>);</span><br><span style="font-family: times new roman,times; font-size: medium;">(ii) P(<em>R</em>).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">If the blue face lands down, the die is not rolled again. If the red face lands down, the die is rolled once again. This is represented by the following tree diagram, where <em>p</em>, <em>s</em>, <em>t</em> are probabilities.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><img src="data:image/png;base64,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" alt width="398" height="262"></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the value of <em>p</em>, of <em>s</em> and of <em>t</em>.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Guiseppi plays a game where he rolls the die. If a blue face lands down, he scores 2 and is finished. If the red face lands down, he scores 1 and rolls one more time. Let <em>X</em> be the total score obtained.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(i) Show that \({\text{P}}(X = 3) = \frac{3}{{16}}\) .</span><br><span style="font-family: times new roman,times; font-size: medium;">(ii) Find \({\text{P}}(X = 2)\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) Construct a probability distribution table for <em>X</em>.</span></p>
<p><span style="font-size: medium;"><span style="font-family: times new roman,times;">(ii) Calculate the expected value of</span> <span style="font-family: times new roman,times;"><em>X</em></span>.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">If the total score is 3, Guiseppi wins \(\$ 10\). If the total score is 2, Guiseppi gets nothing.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Guiseppi plays the game twice. Find the probability that he wins exactly \(\$ 10\).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) P(<em>B</em>) \( = \frac{3}{4}\) <em><strong>A1 N1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) P(<em>R</em>) \( = \frac{1}{4}\) <em><strong>A1 N1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em><strong>[2 marks]<br></strong></em></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\(p = \frac{3}{4}\) <em><strong>A1 N1</strong></em></span></p>
<p><span style="font-size: medium;"><span style="font-family: times new roman,times;">\(s = \frac{1}{4}\), \(t = \frac{3}{4}\) </span><span style="font-family: times new roman,times;"> <em><strong>A1 N1</strong></em></span></span></p>
<p><span style="font-size: medium;"><span style="font-family: times new roman,times;"><em><strong>[2 marks]<br></strong></em></span></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) \({\text{P}}(X = 3)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( = {\text{P (getting 1 and 2)}} = \frac{1}{4} \times \frac{3}{4}\) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( = \frac{3}{{16}}\) <em><strong>AG N0</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \({\text{P}}(X = 2) = \frac{1}{4} \times \frac{1}{4} + \frac{3}{4}{\text{ }}\left( {{\text{or }}1 - \frac{3}{{16}}} \right)\)<strong><em> </em></strong><em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( = \frac{{13}}{{16}}\) <em><strong>A1 N2</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em><strong>[3 marks]</strong></em></span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> <img src="data:image/png;base64,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" alt width="313" height="110"> <em><strong>A2 N2</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) evidence of using \({\text{E}}(X) = \sum{x{\text{P}}(X = x)} \) <em><strong>(M1)</strong></em></span><br><span style="font-family: times new roman,times; font-size: medium;">\({\text{E}}(X) = 2\left( {\frac{{13}}{{16}}} \right) + 3\left( {\frac{3}{{16}}} \right)\) <em><strong>(A1)</strong></em></span><br><span style="font-family: times new roman,times; font-size: medium;">\( = \frac{{35}}{{16}}{\text{ }}\left( { = 2\frac{3}{{16}}} \right)\) <em><strong>A1 N2</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em><strong>[5 marks]</strong></em></span></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">win \(\$ 10 \Rightarrow \) scores 3 one time, 2 other time <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{P}}(3) \times {\text{P}}(2) = \frac{{13}}{{16}} \times \frac{3}{{16}}\) (seen anywhere) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of recognising there are different ways of winning \(\$ 10\) <em><strong>(M1)</strong></em></span><br><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\text{P}}(3) \times {\text{P}}(2) + {\text{P}}(2) \times {\text{P}}(3)\) , \(2\left( {\frac{{13}}{{16}} \times \frac{3}{{16}}} \right)\) , \(\frac{{36}}{{256}} + \frac{3}{{256}} + \frac{{36}}{{256}} + \frac{3}{{256}}\)</span><br><span style="font-family: times new roman,times; font-size: medium;">\({\text{P(win }}\$ 10) = \frac{{78}}{{256}}{\text{ }}\left( { = \frac{{39}}{{128}}} \right)\) <em><strong>A1 N3</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em><strong>[4 marks]</strong></em></span></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-size: medium; font-family: times new roman,times;">This was the most difficult of the extended response questions for the candidates. Finding <em>s</em> and <em>t</em> correctly in part (b) was difficult, with many confused between writing appropriate probabilities on a single branch compared to at the final end of a multiple branch. Many candidates had no idea what to write for a probability distribution and those who did often had probabilities that did not sum to 1. Candidates who wrote a probability distribution often could correctly compute the expected value. The final part was the most challenging, but some good answers were seen. The most common error was not recognizing that there were two different ways of winning.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-size: medium; font-family: times new roman,times;">This was the most difficult of the extended response questions for the candidates. Finding <em>s</em> and <em>t</em> correctly in part (b) was difficult, with many confused between writing appropriate probabilities on a single branch compared to at the final end of a multiple branch. Many candidates had no idea what to write for a probability distribution and those who did often had probabilities that did not sum to 1. Candidates who wrote a probability distribution often could correctly compute the expected value. The final part was the most challenging, but some good answers were seen. The most common error was not recognizing that there were two different ways of winning.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-size: medium; font-family: times new roman,times;">This was the most difficult of the extended response questions for the candidates. Finding <em>s</em> and <em>t</em> correctly in part (b) was difficult, with many confused between writing appropriate probabilities on a single branch compared to at the final end of a multiple branch. Many candidates had no idea what to write for a probability distribution and those who did often had probabilities that did not sum to 1. Candidates who wrote a probability distribution often could correctly compute the expected value. The final part was the most challenging, but some good answers were seen. The most common error was not recognizing that there were two different ways of winning.</span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-size: medium; font-family: times new roman,times;">This was the most difficult of the extended response questions for the candidates. Finding <em>s</em> and <em>t</em> correctly in part (b) was difficult, with many confused between writing appropriate probabilities on a single branch compared to at the final end of a multiple branch. Many candidates had no idea what to write for a probability distribution and those who did often had probabilities that did not sum to 1. Candidates who wrote a probability distribution often could correctly compute the expected value. The final part was the most challenging, but some good answers were seen. The most common error was not recognizing that there were two different ways of winning.</span></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-size: medium; font-family: times new roman,times;">This was the most difficult of the extended response questions for the candidates. Finding <em>s</em> and <em>t</em> correctly in part (b) was difficult, with many confused between writing appropriate probabilities on a single branch compared to at the final end of a multiple branch. Many candidates had no idea what to write for a probability distribution and those who did often had probabilities that did not sum to 1. Candidates who wrote a probability distribution often could correctly compute the expected value. The final part was the most challenging, but some good answers were seen. The most common error was not recognizing that there were two different ways of winning.</span></p>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Jar A contains three red marbles and five green marbles. Two marbles are drawn </span><span style="font-family: times new roman,times; font-size: medium;">from the jar, one after the other, without replacement.</span></p>
</div>
<div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Jar B contains six red marbles and two green marbles. A fair six-sided die is tossed. </span><span style="font-family: times new roman,times; font-size: medium;">If the score is \(1\) or \(2\), a marble is drawn from jar A. Otherwise, a marble is drawn from </span><span style="font-family: times new roman,times; font-size: medium;">jar B.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Find the probability that</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (i) none of the marbles are green;</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (ii) exactly one marble is green.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the expected number of green marbles drawn from the jar.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Write down the probability that the marble is drawn from jar B.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) Given that the marble was drawn from jar B, write down the probability </span><span style="font-family: times new roman,times; font-size: medium;">that it is red.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Given that the marble is red, find the probability that it was drawn from jar A.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) attempt to find \({\rm{P(red)}} \times {\rm{P(red)}}\) <strong><em>(M1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg</em> \(\frac{3}{8} \times \frac{2}{7}\) , \(\frac{3}{8} \times \frac{3}{8}\) , \(\frac{3}{8} \times \frac{2}{8}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{P(none green)}} = \frac{6}{{56}}\) \(\left( { = \frac{3}{{28}}} \right)\) <em><strong>A1 N2</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) attempt to find \({\rm{P(red)}} \times {\rm{P(green)}}\) <strong><em>(M1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg</em> \(\frac{5}{8} \times \frac{3}{7}\) , \(\frac{3}{8} \times \frac{5}{8}\) , \(\frac{{15}}{{56}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">recognizing two ways to get one red, one green <strong><em>(M1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg</em> \(2{\rm{P}}(R) \times {\rm{P}}(G)\) , \(\frac{5}{8} \times \frac{3}{7} + \frac{3}{8} \times \frac{5}{7}\) , \(\frac{3}{8} \times \frac{5}{8} \times 2\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{P(exactly one green)}} = \frac{{30}}{{56}}\) \(\left( { = \frac{{15}}{{28}}} \right)\) <em><strong>A1 N2 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> </span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[5 marks] </span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{P(both green)}} = \frac{{20}}{{56}}\) (seen anywhere) <strong><em>(A1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution into formula for \({\rm{E}}(X)\) <strong><em>A1</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg</em> \(0 \times \frac{6}{{56}} + 1 \times \frac{{30}}{{56}} + 2 \times \frac{{20}}{{56}}\) , \(\frac{{30}}{{64}} + \frac{{50}}{{64}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">expected number of green marbles is \(\frac{{70}}{{56}}\) \(\left( { = \frac{5}{4}} \right)\) <strong><em>A1 N2 </em></strong></span></p>
<p><strong><em><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) \({\text{P(jar B)}} = \frac{4}{6}\) \(\left( { = \frac{2}{3}} \right)\) <em><strong>A1 N1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \({\text{P(red| jar B)}} = \frac{6}{8}\) \(\left( { = \frac{3}{4}} \right)\) <em><strong>A1 N1 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks] </span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">recognizing conditional probability <em><strong> (M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg</em> \({\rm{P}}(A|R)\) , \(\frac{{{\text{P(jar A and red)}}}}{{{\rm{P(red)}}}}\) , tree diagram </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to multiply along either branch (may be seen on diagram) <strong><em> (M1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg</em> \({\text{P(jar A and red)}} = \frac{1}{3} \times \frac{3}{8}\) \(\left( { = \frac{1}{8}} \right)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to multiply along <strong>other</strong> branch <em><strong> (M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \({\text{P(jar B and red)}} = \frac{2}{3} \times \frac{6}{8}\) \(\left( { = \frac{1}{2}} \right)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">adding the probabilities of two mutually exclusive paths <strong><em>(A1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg</em> \({\rm{P(red)}} = \frac{1}{3} \times \frac{3}{8} + \frac{2}{3} \times \frac{6}{8}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg</em> \({\text{P(jar A|red)}} = \frac{{\frac{1}{3} \times \frac{3}{8}}}{{\frac{1}{3} \times \frac{3}{8} + \frac{2}{3} \times \frac{6}{8}}}\) , \(\frac{{\frac{1}{8}}}{{\frac{5}{8}}}\) <strong><em>A1</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{P(jar A|red)}} = \frac{1}{5}\) <em><strong>A1 N3 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[6 marks] </span></strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Many candidates correctly found the probability of selecting no green marbles in two draws, although some candidates treated the second draw as if replacing the first. When finding the probability for exactly one green marble, candidates often failed to recognize two pathways for selecting one of each color.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Few candidates understood the concept of expected value in this context, often leaving this blank or treating as if a binomial experiment. Successful candidates often made a distribution table before making the final calculation.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Most candidates answered part (c) correctly. However, many overcomplicated (c)(ii) by using the conditional probability formula. Those with a clear understanding of the concept easily followed the “write down” instruction.</span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Only a handful of candidates correctly applied conditional probability to find \(P(A|R)\) in part (d). While some wrote down the formula, or drew a tree diagram, few correctly calculated \(P(red) = \frac{5}{8}\) . A common error was to combine the marbles in the two jars to give \(P(red) = \frac{9}{16}\) .</span></p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Adam travels to school by car (\(C\)) or by bicycle (\(B\)). On any particular day he is equally likely to travel by car or by bicycle.</p>
<p class="p1">The probability of being late (\(L\)) for school is \(\frac{1}{6}\) if he travels by car.</p>
<p class="p1">The probability of being late for school is \(\frac{1}{3}\) if he travels by bicycle.</p>
<p class="p1">This information is represented by the following tree diagram.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2015-12-14_om_05.16.45.png" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the value of \(p\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the probability that Adam will travel by car and be late for school.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the probability that Adam will be late for school.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Given that Adam is late for school, find the probability that he travelled by car.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Adam will go to school three times next week.</p>
<p class="p1">Find the probability that Adam will be late exactly once.</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">correct working <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;1 - \frac{1}{6}\)</p>
<p class="p1">\(p = \frac{5}{6}\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">multiplying along correct branches <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\frac{1}{2} \times \frac{1}{6}\)</p>
<p class="p1">\({\text{P}}(C \cap L) = \frac{1}{{12}}\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">multiplying along the other branch <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\frac{1}{2} \times \frac{1}{3}\)</p>
<p class="p1">adding probabilities of their \(2\) mutually exclusive paths <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\frac{1}{2} \times \frac{1}{6} + \frac{1}{2} \times \frac{1}{3}\)</p>
<p class="p1">correct working <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\frac{1}{{12}} + \frac{1}{6}\)</p>
<p class="p1">\({\text{P}}(L) = \frac{3}{{12}}\;\;\;\left( { = \frac{1}{4}} \right)\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N3</em></strong></p>
<p class="p1"><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">recognizing conditional probability (seen anywhere) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;{\text{P}}(C|L)\)</p>
<p class="p1">correct substitution of <strong>their </strong>values into formula <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\frac{{\frac{1}{{12}}}}{{\frac{3}{{12}}}}\)</p>
<p class="p1">\({\text{P}}(C|L) = \frac{1}{3}\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">valid approach <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg </em>\(X \sim B\) \(\left({3,{\text{ }}\frac{1}{4}} \right),{\text{ }}\left({\frac{1}{4}} \right){\left({\frac{3}{4}} \right)^2},{\text{ }}\left( {\begin{array}{*{20}{c}}3 \\1\end{array}} \right)\), three ways it could happen</p>
<p class="p1">correct substitution <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg </em>\(\;\;\;\left({\begin{array}{*{20}{c}}3\\1\end{array}}\right){\left({\frac{1}{4}}\right)^1}{\left({\frac{3}{4}} \right)^2},{\text{ }}\frac{1}{4}\times\frac{3}{4}\times\frac{3}{4} + \frac{3}{4} \times\frac{1}{4}\times\frac{3}{4} + \frac{3}{4}\times\frac{3}{4}\times\frac{1}{4}\)</p>
<p class="p1">correct working <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;3\left( {\frac{1}{4}} \right)\left( {\frac{9}{{16}}} \right),{\text{ }}\frac{9}{{64}} + \frac{9}{{64}} + \frac{9}{{64}}\)</p>
<p class="p1">\(\frac{{27}}{{64}}\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[4 marks]</em></strong></p>
<p class="p1"><strong><em>Total [15 marks]</em></strong></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Parts (a) and (b) of this question were answered correctly by nearly all candidates, and the majority earned full marks on part (c), as well. Unfortunately, there were a number of candidates who made arithmetic errors when multiplying or adding fractions.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Parts (a) and (b) of this question were answered correctly by nearly all candidates, and the majority earned full marks on part (c), as well. Unfortunately, there were a number of candidates who made arithmetic errors when multiplying or adding fractions. Candidates were not as successful in parts (d) and (e) of this question. Although many knew that conditional probability was necessary in part (d), many did not know to use their values from parts (b) and (c), and started from scratch with brand new, and often incorrect, calculations for the numerator and denominator. A majority of candidates did not recognize that binomial probability was needed in part (e), not realizing that there were three ways for Adam to be "late exactly once". A very common incorrect solution to part (e) was \(\frac{1}{4} \times \frac{3}{4} \times \frac{3}{4} = \frac{9}{{64}}\).</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Parts (a) and (b) of this question were answered correctly by nearly all candidates, and the majority earned full marks on part (c), as well. Unfortunately, there were a number of candidates who made arithmetic errors when multiplying or adding fractions. Candidates were not as successful in parts (d) and (e) of this question. Although many knew that conditional probability was necessary in part (d), many did not know to use their values from parts (b) and (c), and started from scratch with brand new, and often incorrect, calculations for the numerator and denominator. A majority of candidates did not recognize that binomial probability was needed in part (e), not realizing that there were three ways for Adam to be "late exactly once". A very common incorrect solution to part (e) was \(\frac{1}{4} \times \frac{3}{4} \times \frac{3}{4} = \frac{9}{{64}}\).</p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Parts (a) and (b) of this question were answered correctly by nearly all candidates, and the majority earned full marks on part (c), as well. Unfortunately, there were a number of candidates who made arithmetic errors when multiplying or adding fractions. Candidates were not as successful in parts (d) and (e) of this question. Although many knew that conditional probability was necessary in part (d), many did not know to use their values from parts (b) and (c), and started from scratch with brand new, and often incorrect, calculations for the numerator and denominator. A majority of candidates did not recognize that binomial probability was needed in part (e), not realizing that there were three ways for Adam to be "late exactly once". A very common incorrect solution to part (e) was \(\frac{1}{4} \times \frac{3}{4} \times \frac{3}{4} = \frac{9}{{64}}\).</p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Parts (a) and (b) of this question were answered correctly by nearly all candidates, and the majority earned full marks on part (c), as well. Unfortunately, there were a number of candidates who made arithmetic errors when multiplying or adding fractions. Candidates were not as successful in parts (d) and (e) of this question. Although many knew that conditional probability was necessary in part (d), many did not know to use their values from parts (b) and (c), and started from scratch with brand new, and often incorrect, calculations for the numerator and denominator. A majority of candidates did not recognize that binomial probability was needed in part (e), not realizing that there were three ways for Adam to be "late exactly once". A very common incorrect solution to part (e) was \(\frac{1}{4}\times\frac{3}{4} \times\frac{3}{4} = \frac{9}{{64}}\).</p>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Bag A contains three white balls and four red balls. Two balls are chosen at random </span><span style="font-family: times new roman,times; font-size: medium;">without replacement.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(i) Copy and complete the following tree diagram.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/M12P1TZ2Q9.jpg" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Find the probability that two white balls are chosen.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Bag A contains three white balls and four red balls. Two balls are chosen at random </span><span style="font-family: times new roman,times; font-size: medium;">without replacement.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Bag B contains four white balls and three red balls. When two balls are chosen at </span><span style="font-family: times new roman,times; font-size: medium;">random without replacement from bag B, the probability that they are both white is \(\frac{2}{7}\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A standard die is rolled. If 1 or 2 is obtained, two balls are chosen without replacement </span><span style="font-family: times new roman,times; font-size: medium;">from bag A, otherwise they are chosen from bag B.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the probability that the two balls are white.</span></p>
<p align="LEFT"> </p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Bag A contains three white balls and four red balls. Two balls are chosen at random without replacement.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Bag B contains four white balls and three red balls. When two balls are chosen at </span><span style="font-family: times new roman,times; font-size: medium;">random without replacement from bag B, the probability that they are both white is \(\frac{2}{7}\) . </span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A standard die is rolled. If 1 or 2 is obtained, two balls are chosen without replacement </span><span style="font-family: times new roman,times; font-size: medium;">from bag A, otherwise they are chosen from bag B.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Given that both balls are white, find the probability that they were chosen from bag A.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/M12P1TZ2Q9ams.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\frac{4}{6},\frac{3}{6}{\rm{and}}\frac{3}{6}\left( {\frac{2}{3},\frac{1}{2}{\rm{and}}\frac{1}{2}} \right)\) <em><strong>A1A1A1 N3</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) multiplying along the correct branches (may be seen on diagram) <em><strong> (A1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{3}{7} \times \frac{2}{6}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\frac{6}{{42}}\left( { = \frac{1}{7}} \right)\) <em><strong> A1 N2</strong> </em></span></p>
<p><em> <span style="font-family: times new roman,times; font-size: medium;"><strong>[5 marks]</strong> </span></em></p>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P(bag A) = }}\frac{2}{6}\left( { = \frac{1}{3}} \right)\) , \({\rm{P(bag B) = }}\frac{4}{6}\left( { = \frac{2}{3}} \right)\) (seen anywhere) <em><strong>(A1)(A1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">appropriate approach <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{P(}}WW \cap A) + {\rm{P}}(WW \cap B)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/M12P1TZ2Q9bms.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct calculation <em><strong>A1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{1}{3} \times \frac{1}{7} + \frac{2}{3} \times \frac{2}{7}\) , \(\frac{2}{{42}} + \frac{8}{{42}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(2W) = \frac{{60}}{{252}}\left( { = \frac{5}{{21}}} \right)\) <em><strong>A1 N3</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[5 marks] </span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">recognizing conditional probability <em><strong>(M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{{\rm{P}}(A \cap B)}}{{{\rm{P}}(B)}}\) , \({\rm{P}}(A|WW) = \frac{{{\rm{P}}(WW \cap A)}}{{{\rm{P}}(WW)}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct numerator <strong><em>(A1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{P}}(A \cap WW) = \frac{6}{{42}} \times \frac{2}{6},\frac{1}{{21}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct denominator <em><strong>(A1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{6}{{252}},\frac{5}{{21}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">probability \(\frac{{84}}{{420}}\left( { = \frac{1}{5}} \right)\) <em><strong>A1 N3</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks] </span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (a) of this question was answered correctly by the large majority of candidates. There were some who did not follow the instruction to copy and complete the tree diagram on their separate paper, and simply filled in the blanks on the exam paper. </span></p>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In part (b), many candidates struggled with finding the compound probability, and did not use the provided information in the appropriate manner. Quite a few candidates seemed to be confused about when they should add the probabilities or when they should multiply. </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In part (c), many recognized that the question dealt with conditional probability, and many tried to use the formula from the information booklet, but failed to realize that they had already found the required values for the numerator and denominator in their working for part (b). </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Throughout this question, it was discouraging to see the large number of candidates making arithmetic errors. There were a surprising number of candidates who multiplied fractions incorrectly, or found an incorrect value for simple multiplication such as \(2 \times 4 = 6\) or \(6 \times 7 = 43\) .</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The following is a cumulative frequency diagram for the time <em>t</em>, in minutes, taken by 80 students to complete a task.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><img src="images/maths_3.png" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the number of students who completed the task in less than 45 minutes.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the number of students who took between 35 and 45 minutes to complete the task.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Given that 50 students take less than <em>k </em>minutes to complete the task, find the value of \(k\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">attempt to find number who took less than 45 minutes <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> line on graph (vertical at approx 45, or horizontal at approx 70)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">70 students (accept 69) <strong><em>A1 N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">55 students completed task in less than 35 minutes <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">subtracting <strong>their </strong>values <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> 70 – 55</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">15 students <strong><em>A1 N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">correct approach <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> line from <em>y</em>-axis on 50</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(k = 33\) <strong><em>A1 N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let <em>A</em> and <em>B</em> be independent events, where \({\text{P}}(A) = 0.6\) and \({\text{P}}(B) = x\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down an expression for \({\text{P}}(A \cap B)\) .</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Given that \({\text{P}}(A \cup B) = 0.8\) ,</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) find <em>x</em> ;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) find \({\text{P}}(A \cap B)\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Hence</strong>, explain why <em>A</em> and <em>B</em> are <strong>not</strong> mutually exclusive.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{P}}(A \cap B) = {\text{P}}(A) \times {\text{P}}(B)( = 0.6x)\) <strong><em>A1 N1 </em></strong></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[1 mark]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) evidence of using \({\text{P}}(A \cup B) = {\text{P}}(A) + {\text{P}}(B) - {\text{P}}(A){\text{P}}(B)\) <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution <em><strong> A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(0.8 = 0.6 + x - 0.6x\) , \(0.2 = 0.4x\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(x = 0.5\) <em><strong>A1 N2</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \({\text{P}}(A \cap B) = 0.3\) <strong><em>A1 N1</em> </strong></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></strong></em></p>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">valid reason, with reference to \({\text{P(}}A \cap B)\) <strong><em>R1 N1</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\text{P(}}A \cap B) \ne 0\)</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[1 mark]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This question was well done by most candidates. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This question was well done by most candidates. When errors were made, candidates confused the terms "independent" and "mutually exclusive" and did not subtract the intersection when finding \({\text{P}}(A \cup B)\) . </span></p>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Candidates should also be aware of the command term "hence" used in part (c) where they were expected to provide a reason that involved \({\text{P}}(A \cap B)\) from their work in part (b). It seemed that many turned to the formula in the booklet instead of considering the conceptual meaning of the term. </span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">A discrete random variable \(X\) has the following probability distribution.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-13_om_05.16.18.png" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find \(p\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find \({\text{E}}(X)\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">summing probabilities to 1 <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>,\(\;\;\;\sum { = 1,{\text{ }}3 + 4 + 2 + x = 10} \)</p>
<p class="p1">correct working <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1">\(\frac{3}{{10}} + \frac{4}{{10}} + \frac{2}{{10}} + p = 1,{\text{ }}p = 1 - \frac{9}{{10}}\)</p>
<p class="p1">\(p = \frac{1}{{10}}\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N3</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>correct substitution into formula for \({\text{E}}(X)\) <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;0\left( {\frac{3}{{10}}} \right) + \ldots + 3(p)\)</p>
<p>correct working <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\frac{4}{{10}} + \frac{4}{{10}} + \frac{3}{{10}}\)</p>
<p>\({\text{E}}(X) = \frac{{11}}{{10}}\;\;\;(1.1)\) <strong><em>A1 N2</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<p><strong><em>Total [6 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Most candidates were able to find \(p\), however expectation emerged as surprisingly more difficult. Quite often \({\text{E}}(X)/4\) was found or candidates wrote the formula with no further work.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Most candidates were able to find \(p\), however expectation emerged as surprisingly more difficult. Quite often \({\text{E}}(X)/4\) was found or candidates wrote the formula with no further work.</p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">There are 20 students in a classroom. Each student plays only one sport. The table </span><span style="font-family: times new roman,times; font-size: medium;">below gives their sport and gender.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/tour.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">One student is selected at random.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Calculate the probability that the student is a male or is a tennis player.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) Given that the student selected is female, calculate the probability that the </span><span style="font-family: times new roman,times; font-size: medium;">student does not play football.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Two students are selected at random. Calculate the probability that neither </span><span style="font-family: times new roman,times; font-size: medium;">student plays football.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) correct calculation <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{9}{{20}} + \frac{5}{{20}} - \frac{2}{{20}}\) , \(\frac{{4 + 2 + 3 + 3}}{{20}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> \({\text{P(male or tennis)}} = \frac{{12}}{{20}}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1 N2</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) correct calculation <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{6}{{20}} \div \frac{{11}}{{20}}\) , \(\frac{{3 + 3}}{{11}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{P(not football|female)}} = \frac{6}{{11}}\) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong> <span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></strong></em></p>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;"><span style="color: #000000;">\({\text{P(first not football)}} = \frac{{11}}{{20}}\) , \({\text{P(second not football)}} = \frac{{10}}{{19}}\) </span></span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> \({\text{P(neither football)}} = \frac{{11}}{{20}} \times \frac{{10}}{{19}}\) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> \({\text{P(neither football)}} = \frac{{110}}{{380}}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1 N1</span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [3 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Many candidates had difficulty with this question, usually as a result of seeking to solve the </span><span style="font-family: times new roman,times; font-size: medium;">problem by formula instead of looking carefully at the table frequencies.</span></p>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A very common error </span><span style="font-family: times new roman,times; font-size: medium;">in part (b) was to assume identical probabilities for each selection instead of dependent </span><span style="font-family: times new roman,times; font-size: medium;">probabilities where there is no replacement.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-size: medium;"><span style="font-family: times new roman,times;">Let \(\boldsymbol{A} = \left( {\begin{array}{*{20}{c}}<br> 0&3 \\ <br> { - 2}&4 <br>\end{array}} \right)\)</span><span style="font-family: times new roman,times;"> and \(\boldsymbol{B} = \left( {\begin{array}{*{20}{c}}<br> { - 4}&0 \\ <br> 5&1 <br>\end{array}} \right)\)</span><span style="font-family: times new roman,times;">.</span></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find <strong><em>AB</em></strong> .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-size: medium;"><span style="font-family: TimesNewRomanPSMT;">Given that </span><span style="font-family: TimesNewRomanPS-BoldItalicMT;">\({\boldsymbol{X}} - 2{\boldsymbol{A}} = {\boldsymbol{B}}\)</span><span style="font-family: TimesNewRomanPSMT;">, find <em><strong>X</strong></em></span><span style="font-family: TimesNewRomanPSMT;">.</span></span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of multiplying <em><strong>(</strong></em><strong><em><span style="font-family: Times New Roman,Times New Roman;"><span style="font-family: Times New Roman,Times New Roman;">M1) </span></span></em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: Times New Roman,Times New Roman;"><span style="font-family: Times New Roman,Times New Roman;"><em>e.g. </em></span></span>one correct element, \((0 \times - 4) + (3 \times 5)\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\boldsymbol{AB}} = \left( {\begin{array}{*{20}{c}}<br>{15}&3\\<br>{28}&4<br>\end{array}} \right)\)<strong><em> <span style="font-family: times new roman,times;"><span style="font-family: Times New Roman,Times New Roman;"><span style="font-family: Times New Roman,Times New Roman;"><em><strong>A2 N3 </strong></em></span></span></span></em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: Times New Roman,Times New Roman;"><span style="font-family: Times New Roman,Times New Roman;"><strong>Note</strong></span></span>: Award <strong><em><span style="font-family: Times New Roman,Times New Roman;"><span style="font-family: Times New Roman,Times New Roman;">A1 </span></span></em></strong>for three correct elements. </span></p>
<p><span style="font-size: medium;"><em><strong><span style="font-family: times new roman,times;">[3 marks]</span></strong></em></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">finding \(2{\boldsymbol{A}} = \left( {\begin{array}{*{20}{c}}<br>0&6\\<br>{ - 4}&8<br>\end{array}} \right)\)<strong><em><span style="font-family: Times New Roman,Times New Roman;"><span style="font-family: Times New Roman,Times New Roman;"> (A1)</span></span></em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">adding 2\({\boldsymbol{A}}\) to both sides (may be seen first) <strong><em>(</em></strong><strong><em><span style="font-family: Times New Roman,Times New Roman;"><span style="font-family: Times New Roman,Times New Roman;">M1) </span></span></em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: Times New Roman,Times New Roman;"><span style="font-family: Times New Roman,Times New Roman;"><em>e.g. </em></span></span>\({\boldsymbol{X}} = {\boldsymbol{B}} +2{\boldsymbol{A}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: Times New Roman,Times New Roman;"><span style="font-family: Times New Roman,Times New Roman;"><em><strong> </strong></em></span></span>\({\boldsymbol{X}} = \left( {\begin{array}{*{20}{c}}<br>{ - 4}&6\\<br>1&9<br>\end{array}} \right)\)<strong><em><span style="font-family: Times New Roman,Times New Roman;"><span style="font-family: Times New Roman,Times New Roman;"> <span style="font-family: times new roman,times;"><span style="font-family: Times New Roman,Times New Roman;"><span style="font-family: Times New Roman,Times New Roman;"><em><strong>A1 N2</strong></em></span></span></span></span></span></em></strong></span></p>
<p><span style="font-size: medium;"><em><strong><span style="font-family: times new roman,times;">[3 marks]</span></strong></em></span></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">The large majority of candidates answered this question successfully. There were only a small number of candidates who seemed to have never worked with matrices before. Occasionally a candidate would incorrectly approach part (b) by finding an inverse of matrix <strong><em>A</em></strong>.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">The large majority of candidates answered this question successfully. There were only a small number of candidates who seemed to have never worked with matrices before. Occasionally a candidate would incorrectly approach part (b) by finding an inverse of matrix <strong><em>A</em></strong>.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The probability distribution of a discrete random variable <em>X</em> is given by \[{\rm{P}}(X = x) = \frac{{{x^2}}}{{14}}{\text{, }}x \in \left\{ {1{\text{, }}2{\text{, }}k} \right\}{\text{, where}} k > 0\] .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down \({\rm{P}}(X = 2)\) .</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Show that \(k = 3\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \({\rm{E}}(X)\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(X = 2) = \frac{4}{{14}}\) \(\left( { = \frac{2}{7}} \right)\) <em><strong>A1 N1</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[1 mark]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(X = 1) = \frac{1}{{14}}\) <em><strong> (A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(X = k) = \frac{{{k^2}}}{{14}}\) <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">setting the sum of probabilities \( = 1\) <em><strong>M1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{1}{{14}} + \frac{4}{{14}} + \frac{{{k^2}}}{{14}} = 1\) , \(5 + {k^2} = 14\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({k^2} = 9\) (accept \(\frac{{{k^2}}}{{14}} = \frac{9}{{14}}\) ) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(k = 3\) <em><strong>AG N0</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [4 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution into \({\rm{E}}(X) = \sum {x{\rm{P}}(X = x)} \) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(1\left( {\frac{1}{{14}}} \right) + 2\left( {\frac{4}{{14}}} \right) + 3\left( {\frac{9}{{14}}} \right)\)</span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\({\rm{E}}(X) = \frac{{36}}{{14}}\) \(\left( { = \frac{{18}}{7}} \right)\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1 N1</span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [2 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Although many candidates were successful in working with the probability function, students had difficulty following the "show that" instruction of this question. Many substituted \(k = 3\) and worked backwards to show that the sum of probabilities is 1. Some would argue that \(k = 4\) does not work, but were unable to give a complete justification for \(k = 3\) . </span><span style="font-family: times new roman,times; font-size: medium;">A good number of students seemed unprepared to find an expected value. Many candidates wrote a formula and did not know what to do with it, while others divided \({\rm{E}}(X)\) by 3 or by 6, which confuses the concept of a mean in a probability distribution with the more common understanding.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Although many candidates were successful in working with the probability function, students had difficulty following the "show that" instruction of this question. Many substituted \(k = 3\) and worked backwards to show that the sum of probabilities is 1. Some would argue that \(k = 4\) does not work, but were unable to give a complete justification for \(k = 3\) . </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">A good number of students seemed unprepared to find an expected value. Many candidates wrote a formula and did not know what to do with it, while others divided \({\rm{E}}(X)\) by 3 or by 6, which confuses the concept of a mean in a probability distribution with the more common understanding.</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-size: medium;"><span style="font-family: TimesNewRomanPSMT;">The following table shows the probability distribution of a discrete random variable </span><em><span style="font-family: TimesNewRomanPS-ItalicMT;">X </span></em><span style="font-family: TimesNewRomanPSMT;">.</span></span></p>
<p align="LEFT"><span style="font-size: medium;"><img src="data:image/png;base64,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" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-size: medium;"><span style="font-family: TimesNewRomanPSMT;">Find the value of </span><em><span style="font-family: TimesNewRomanPS-ItalicMT;">k </span></em><span style="font-family: TimesNewRomanPSMT;">.</span></span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-size: medium; font-family: times new roman,times;">Find \({\text{E}}(X)\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of summing to 1 <strong><em> (</em></strong><strong><em><span style="font-family: Times New Roman,Times New Roman;"><span style="font-family: Times New Roman,Times New Roman;">M1) </span></span></em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: Times New Roman,Times New Roman;"><span style="font-family: Times New Roman,Times New Roman;"><em>e.g.</em> </span></span><span style="font-family: Times New Roman;">\(\sum\limits_{}^{} {p = 1{\text{, }}0.3 + k + 2k + 0.1 = 1} \)</span></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct working <em><strong>(</strong></em><strong><em><span style="font-family: Times New Roman,Times New Roman;"><span style="font-family: Times New Roman,Times New Roman;">A1) </span></span></em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: Times New Roman,Times New Roman;"><span style="font-family: Times New Roman,Times New Roman;"><em>e.g.</em> </span></span><span style="font-family: Times New Roman;">\(0.4 + 3k{\text{, }}3k = 0.6\)</span></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: Times New Roman,Times New Roman;"><span style="font-family: Times New Roman,Times New Roman;">\(k = 0.2\) <em><strong>A1 N2</strong></em></span></span></span></p>
<p><span style="font-size: medium;"><em><strong><span style="font-family: times new roman,times;">[3 marks]</span></strong></em></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution into formula \({\text{E}}(X)\) <em><strong>(</strong><strong>A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em><span style="font-family: Times New Roman,Times New Roman;"><span style="font-family: Times New Roman,Times New Roman;">e.g. </span></span></em><span style="font-family: Times New Roman,Times New Roman;"><span style="font-family: Times New Roman,Times New Roman;">\(0(0.3) + 2(k) + 5(2k) + 9(0.1){\text{, }}12k + 0.9\)</span></span></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct working </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: Times New Roman,Times New Roman;"><span style="font-family: Times New Roman,Times New Roman;"><em>e.g.</em> </span></span>\(0(0.3) + 2(0.2) + 5(0.4) + 9(0.1){\text{, }}0.4 + 2.0 + 0.9\)<em> <strong>(</strong><strong>A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{E}}(X)\)<span style="font-family: Times New Roman,Times New Roman;"><span style="font-family: Times New Roman,Times New Roman;">= 3.3 <span style="font-family: times new roman,times;"><span style="font-family: Times New Roman,Times New Roman;"><span style="font-family: Times New Roman,Times New Roman;"><em><strong>A1 N2 </strong></em></span></span></span></span></span></span></p>
<p><span style="font-size: medium;"><em><strong><span style="font-family: times new roman,times;">[3 marks]</span></strong></em></span></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-size: medium; font-family: times new roman,times;">Overall, this question was very well done. A few candidates left this question blank, or used methods which would indicate they were unfamiliar with discrete random variables. In part (b), there were a good number of candidates who set up their work correctly, but then had trouble adding or multiplying decimals without a calculator. A common type of error for these candidates was \(5(0.4) = 0.2\) .</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-size: medium; font-family: times new roman,times;">Overall, this question was very well done. A few candidates left this question blank, or used methods which would indicate they were unfamiliar with discrete random variables. In part (b), there were a good number of candidates who set up their work correctly, but then had trouble adding or multiplying decimals without a calculator. A common type of error for these candidates was \(5(0.4) = 0.2\) .<br></span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>In a group of 20 girls, 13 take history and 8 take economics. Three girls take both history and economics, as shown in the following Venn diagram. The values \(p\) and \(q\) represent numbers of girls.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-11_om_08.39.12.png" alt="M17/5/MATME/SP1/ENG/TZ1/01"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of \(p\);</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of \(q\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>A girl is selected at random. Find the probability that she takes economics but not history.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>valid approach <em><strong>(M1)</strong></em></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(p + 3 = 13,{\text{ }}13 - 3\)</p>
<p>\(p = 10\) <em><strong>A1</strong></em> <em><strong>N2</strong></em></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid approach <em><strong>(M1)</strong></em></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(p + 3 + 5 + q = 20,{\text{ }}10 - 10 - 8\)</p>
<p>\(q = 2\) <em><strong>A1</strong></em> <em><strong>N2</strong></em></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid approach <em><strong>(M1)</strong></em></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(20 - p - q - 3,{\text{ }}1 - \frac{{15}}{{20}},{\text{ }}n(E \cap H') = 5\)</p>
<p>\(\frac{5}{{20}}\,\,\,\left( {\frac{1}{4}} \right)\) <em><strong>A1</strong></em> <em><strong>N2</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Ann and Bob play a game where they each have an eight-sided die. Ann’s die has three green faces and five red faces; Bob’s die has four green faces and four red faces. They take turns rolling their own die and note what colour faces up. The first player to roll green wins. Ann rolls first. Part of a tree diagram of the game is shown below.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-13_om_11.04.17.png" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the probability that Ann wins on her first roll.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>The probability that Ann wins on her third roll is \(\frac{5}{8} \times \frac{4}{8} \times p \times q\ \times \frac{3}{8}\).</p>
<p class="p1">Write down the value of \(p\) and of \(q\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>The probability that Ann wins on her tenth roll is \(\frac{3}{8}{r^k}\) where \(r \in \mathbb{Q},{\text{ }}k \in \mathbb{Z}\).</p>
<p class="p1">Find the value of \(r\) and of \(k\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the probability that Ann wins the game.</p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">recognizing Ann rolls green <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;{\text{P(G)}}\)</p>
<p class="p1">\(\frac{3}{8}\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>(i) \(p = \frac{4}{8},{\text{ }}q = \frac{5}{8}\) or \(q = \frac{4}{8},{\text{ }}p = \frac{5}{8}\) <strong><em>A1A1 N2</em></strong></p>
<p>(ii) recognizes Ann and Bob lose \(9\) times <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\(\overbrace {{A_L}{B_\(\overbrace {{A_L}{B_ \ldots \(\overbrace {{A_L}{B_{\text{ 9 times, }}\underbrace {\left( {\frac{5}{8} \times \frac{4}{8}} \right) \times \ldots \times \left( {\frac{5}{8} \times \frac{4}{8}} \right)}_{{\text{9 times}}}\)</p>
<p>\(k = 9\;\;\;\)(seen anywhere) <strong><em>A1 N2</em></strong></p>
<p>correct working <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;{\left( {\frac{5}{8} \times \frac{4}{8}} \right)^9} \times \frac{3}{8},{\text{ }}\left( {\frac{5}{8} \times \frac{4}{8}} \right) \times \ldots \times \left( {\frac{5}{8} \times \frac{4}{8}} \right) \times \frac{3}{8}\)</p>
<p>\(r = \frac{{20}}{{64}}\;\;\;\left( { = \frac{5}{{16}}} \right)\) <strong><em>A1 N2</em></strong></p>
<p><strong><em>[6 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">recognize the probability is an infinite sum <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\)Ann wins on her \({{\text{1}}^{{\text{st}}}}\) roll or \({{\text{2}}^{{\text{nd}}}}\) roll or \({{\text{3}}^{{\text{rd}}}}\) roll…, \({S_\infty }\)</p>
<p class="p1">recognizing GP <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1">\({u_1} = \frac{3}{8}\;\;\;\)(seen anywhere) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">\(r = \frac{{20}}{{64}}\;\;\;\)(seen anywhere) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">correct substitution into infinite sum of GP <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\frac{{\frac{3}{8}}}{{1 - \frac{5}{{16}}}},{\text{ }}\frac{3}{8}\left( {\frac{1}{{1 - \left( {\frac{5}{8} \times \frac{4}{8}} \right)}}} \right),{\text{ }}\frac{1}{{1 - \frac{5}{{16}}}}\)</p>
<p class="p1">correct working <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><span class="s1"><em>eg</em></span>\(\;\;\;\frac{{\frac{3}{8}}}{{\frac{{11}}{{16}}}},{\text{ }}\frac{3}{8} \times \frac{{16}}{{11}}\)</p>
<p class="p1">\({\text{P (Ann wins)}} = \frac{{48}}{{88}}\;\;\;\left( { = \frac{6}{{11}}} \right)\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N1</em></strong></p>
<p class="p1"><strong><em>[7 marks]</em></strong></p>
<p class="p1"><strong><em>Total [15 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Some teachers’ comments suggested that the word ‘loses’ in the diagram was misleading. But candidate scripts did not indicate any adverse effect.</p>
<p class="p1">a) <span class="Apple-converted-space"> </span>Very well answered.</p>
<p class="p1">b) <span class="Apple-converted-space"> </span>i) Probabilities \(p\) and \(q\) were typically found correctly. ii) Fewer candidates identified the common ratio and number of rolls correctly.</p>
<p class="p1">Few candidates recognized that this was an infinite geometric sum although some did see that a geometric progression was involved.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Some teachers’ comments suggested that the word ‘loses’ in the diagram was misleading, But candidate scripts did not indicate any adverse effect.</p>
<p class="p1">a) <span class="Apple-converted-space"> </span>Very well answered.</p>
<p class="p1">b) <span class="Apple-converted-space"> </span>i) Probabilities \(p\) and \(q\) were typically found correctly. ii) Fewer candidates identified the common ratio and number of rolls correctly.</p>
<p class="p1">Few candidates recognized that this was an infinite geometric sum although some did see that a geometric progression was involved.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Some teachers’ comments suggested that the word ‘loses’ in the diagram was misleading, But candidate scripts did not indicate any adverse effect.</p>
<p class="p1">a) <span class="Apple-converted-space"> </span>Very well answered.</p>
<p class="p1">b) <span class="Apple-converted-space"> </span>i) Probabilities \(p\) and \(q\) were typically found correctly. ii) Fewer candidates identified the common ratio and number of rolls correctly.</p>
<p class="p1">Few candidates recognized that this was an infinite geometric sum although some did see that a geometric progression was involved.</p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">A data set has a mean of 20 and a standard deviation of 6.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Each value in the data set has 10 added to it. Write down the value of</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) the new mean;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) the new standard deviation.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Each value in the original data set is multiplied by 10.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Write down the value of the new mean.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Find the value of the new variance.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) new mean is \(20 + 10 = 30\) <em><strong>A1 N1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) new sd is 6 <em><strong>A1 N1</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) new mean is \(20 \times 10 = 200\) <em><strong>A1 N1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) <strong>METHOD 1</strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">variance is 36 <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">new variance is \(36 \times 100 = 3600\) <em><strong> A1 N2</strong></em></span></p>
<p align="LEFT"><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 2</span></strong></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">new sd is 60 <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">new variance is \({60^2} = 3600\) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b(i) and (ii).</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The following box-and-whisker plot represents the examination scores of a group of students.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-22_om_06.17.41.png" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Write down the median score.</p>
<p class="p1">The range of the scores is 47 marks, and the interquartile range is 22 marks.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p2">Find the value of</p>
<p class="p2">(i) <span class="Apple-converted-space"> </span>\(c\);</p>
<p class="p2">(ii) <span class="Apple-converted-space"> </span>\(d\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1"><span class="s1">60 <span class="Apple-converted-space"> </span></span><strong><em>A1 <span class="Apple-converted-space"> </span>N1</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>(i) valid approach <strong><em>(M1)</em></strong></p>
<p><em>eg </em>\(\max - \min = {\rm{range}},c = 40 + 47\)</p>
<p>\(c = 87\) <strong><em>A1 N2</em></strong></p>
<p>(ii) valid approach <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;Q3 - Q1 = IQR,{\text{ }}74 - 22\)</p>
<p>\(d = 52\) <strong><em>A1 N2</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>A data set has <em>n</em> items. The sum of the items is 800 and the mean is 20.</p>
</div>
<div class="specification">
<p>The standard deviation of this data set is 3. Each value in the set is multiplied by 10.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <em>n</em>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the value of the new mean.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of the new variance.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>correct approach <em><strong>(A1)</strong></em></p>
<p><em>eg</em> \(\frac{{800}}{n} = 20\)</p>
<p>40 <em><strong>A1 N2</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>200 <em><strong>A1 N1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>recognizing variance = <em>σ </em><sup>2</sup> <em><strong>(M1)</strong></em></p>
<p><em>eg</em> 3<sup>2</sup> = 9</p>
<p>correct working to find new variance <em><strong>(A1)</strong></em></p>
<p><em>eg σ </em><sup>2 </sup>× 10<sup>2</sup>, 9 × 100</p>
<p>900 <em><strong> A1 N3</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>new standard deviation is 30 <em><strong> (A1)</strong></em></p>
<p>recognizing variance = <em>σ </em><sup>2</sup> <em><strong>(M1)</strong></em></p>
<p><em>e</em>g 3<sup>2</sup> = 9, 30<sup>2</sup></p>
<p>900 <em><strong> A1 N3</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">There are <span class="s1">10 </span>items in a data set. The sum of the items is <span class="s1">60</span>.</p>
</div>
<div class="specification">
<p class="p1">The variance of this data set is <span class="s1">3</span>. Each value in the set is multiplied by <span class="s1">4</span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the mean.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) Write down the value of the new mean.</p>
<p class="p1">(ii) Find the value of the new variance.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">correct approach <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(\frac{{60}}{{10}}\)</p>
<p class="p1"><span class="Apple-converted-space">\({\text{mean}} = 6\) </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>new mean \( = 24\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N1</em></strong></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>valid approach <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\({\text{variance}} \times {(4)^2},{\text{ }}3 \times 16\), new standard deviation \( = 4\sqrt 3 \)</p>
<p class="p1">new variance \( = 48\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">While most candidates were able to answer part (a) of this question correctly, they were not as successful in part (b). It seems that the item on the Maths SL syllabus dealing with the "effect of constant changes to the original data" was skipped over in many schools.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">While most candidates were able to answer part (a) of this question correctly, they were not as successful in part (b). It seems that the item on the Maths SL syllabus dealing with the "effect of constant changes to the original data" was skipped over in many schools.</p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">José travels to school on a bus. On any day, the probability that José will miss the bus is \(\frac{1}{3}\) </span><span style="font-family: times new roman,times; font-size: medium;">.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">If he misses his bus, the probability that he will be late for school is \(\frac{7}{8}\) </span><span style="font-family: times new roman,times; font-size: medium;">.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">If he does not miss his bus, the probability that he will be late is \(\frac{3}{8}\) </span><span style="font-family: times new roman,times; font-size: medium;">.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let <em>E</em> be the event “he misses his bus” and <em>F</em> the event “he is late for school”.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The information above is shown on the following tree diagram.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/party.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Find</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) \({\rm{P}}(E \cap F)\) ;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \({\rm{P}}(F)\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Find the probability that</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) José misses his bus and is not late for school;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) José missed his bus, given that he is late for school.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The cost for each day that José catches the bus is 3 euros. José goes to school on </span><span style="font-family: times new roman,times; font-size: medium;">Monday and Tuesday.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Copy and complete the probability distribution table.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><img src="images/bus.png" alt></span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The cost for each day that José catches the bus is 3 euros. José goes to school on </span><span style="font-family: times new roman,times; font-size: medium;">Monday and Tuesday.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the expected cost for José for both days.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) \(\frac{7}{{24}}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1 N1</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) evidence of <strong>multiplying</strong> along the branches <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{2}{3} \times \frac{5}{8}\) , \(\frac{1}{3} \times \frac{7}{8}\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>adding</strong> probabilities of two mutually exclusive paths <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\left( {\frac{1}{3} \times \frac{7}{8}} \right) + \left( {\frac{2}{3} \times \frac{3}{8}} \right)\) , \(\left( {\frac{1}{3} \times \frac{1}{8}} \right) + \left( {\frac{2}{3} \times \frac{5}{8}} \right)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(F) = \frac{{13}}{{24}}\) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [4 marks]</span></strong></em></p>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) \(\frac{1}{3} \times \frac{1}{8}\) </span><span style="font-family: times new roman,times; font-size: medium;"> <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(\frac{1}{{24}}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) recognizing this is \({\rm{P}}(E|F)\) <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{7}{{24}} \div \frac{{13}}{{24}}\)</span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(\frac{{168}}{{312}}\) \(\left( { = \frac{7}{{13}}} \right)\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A2 N3</span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [5 marks]</span></strong></em></p>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><br><img src="images/andy.png" alt><em><strong><span style="font-family: times new roman,times; font-size: medium;"> A2A1 N3</span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution into \({\rm{E}}(X)\) formula <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(0 \times \frac{1}{9} + 3 \times \frac{4}{9} + 6 \times \frac{4}{9}\) , \(\frac{{12}}{9} + \frac{{24}}{9}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{E}}(X) = 4\) (euros) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [2 marks]</span></strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Candidates generally handled some or all of parts (a) and (b) well. Errors included adding </span><span style="font-family: times new roman,times; font-size: medium;">probabilities along branches and trying to use the union formula from the information booklet.</span></p>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Candidates generally handled some or all of parts (a) and (b) well. Errors included adding probabilities along branches and trying to use the union formula from the information booklet. On part (b)(ii), many candidates knew that they were supposed to use some type of conditional probability but did not know how to find \({\rm{P}}(E|F)\) . Many candidates made errors working with fractions. Some candidates who missed part (a)(ii) were able to earn follow-through credit on part (b)(ii).</span></p>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Many candidates had difficulty completing the probability distribution table. While the </span><span style="font-family: times new roman,times; font-size: medium;">common error of finding the probability for \(x = 3\) as \(\frac{2}{9}\) was understandable as the </span><span style="font-family: times new roman,times; font-size: medium;">candidate did not appreciate that there were two ways of paying three euros, it was </span><span style="font-family: times new roman,times; font-size: medium;">disappointing that these candidates often correctly found \({\rm{P}}(X = 4)\) as \(\frac{4}{9}\) and did not note </span><span style="font-family: times new roman,times; font-size: medium;">that the probabilities failed to sum to one. These candidates could not earn full follow-through </span><span style="font-family: times new roman,times; font-size: medium;">marks on their expected value calculation in part (d). Some candidates did use the </span><span style="font-family: times new roman,times; font-size: medium;">probabilities summing to one with incorrect probabilities in part (c); these candidates often </span><span style="font-family: times new roman,times; font-size: medium;">earned full follow-through marks in part (d), as a majority of candidates knew the method for </span><span style="font-family: times new roman,times; font-size: medium;">finding expected value.</span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Many candidates had difficulty completing the probability distribution table. While the </span><span style="font-family: times new roman,times; font-size: medium;">common error of finding the probability for \(x = 3\) as \(\frac{2}{9}\) was understandable as the </span><span style="font-family: times new roman,times; font-size: medium;">candidate did not appreciate that there were two ways of paying three Euros, it was </span><span style="font-family: times new roman,times; font-size: medium;">disappointing that these candidates often correctly found \({\rm{P}}(X) = 4\) as \(\frac{4}{9}\) and did not note </span><span style="font-family: times new roman,times; font-size: medium;">that the probabilities failed to sum to one. These candidates could not earn full follow-through </span><span style="font-family: times new roman,times; font-size: medium;">marks on their expected value calculation in part (d). Some candidates did use the </span><span style="font-family: times new roman,times; font-size: medium;">probabilities summing to one with incorrect probabilities in part (c); these candidates often </span><span style="font-family: times new roman,times; font-size: medium;">earned full follow-through marks in part (d), as a majority of candidates knew the method for </span><span style="font-family: times new roman,times; font-size: medium;">finding expected value.</span></p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">In a group of 16 students, 12 take art and 8 take music. One student takes neither </span><span style="font-family: times new roman,times; font-size: medium;">art nor music. The Venn diagram below shows the events art and music. The values </span><span style="font-family: times new roman,times; font-size: medium;"><em>p</em> , <em>q</em> , <em>r</em> and <em>s</em> represent numbers of students.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/titanic.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Write down the value of <em>s</em> .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) Find the value of <em>q</em> .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iii) Write down the value of <em>p</em> and of <em>r</em> .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a(i), (ii) and (iii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) A student is selected at random. Given that the student takes music, </span><span style="font-family: times new roman,times; font-size: medium;">write down the probability the student takes art.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) <strong>Hence</strong>, show that taking music and taking art are <strong>not</strong> independent events.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Two students are selected at random, one after the other. Find the probability </span><span style="font-family: times new roman,times; font-size: medium;">that the first student takes <strong>only</strong> music and the second student takes <strong>only</strong> art.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) \(s = 1\) <em><strong>A1 N1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) evidence of appropriate approach <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(21 - 16\) , \(12 + 8 - q = 15\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(q = 5\) <em><strong>A1 N2</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(iii) \(p = 7\) , \(r = 3\) <em><strong>A1A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[5 marks]</span></strong></em></p>
<div class="question_part_label">a(i), (ii) and (iii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) \({\rm{P(art|music)}} = \frac{5}{8}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A2 N2</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) <strong>METHOD 1</strong></span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\({\rm{P(art)}} = \frac{{12}}{{16}}\) \(\left( { = \frac{3}{4}} \right)\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of correct reasoning <em><strong>R1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{3}{4} \ne \frac{5}{8}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">the events are not independent <em><strong>AG N0</strong></em></span></p>
<p><strong> <span style="font-family: times new roman,times; font-size: medium;">METHOD 2</span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> \({\rm{P(art)}} \times {\rm{P(music)}} = \frac{{96}}{{256}}\) \(\left( { = \frac{3}{8}} \right)\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of correct reasoning <em><strong>R1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{12}}{{16}} \times \frac{8}{{16}} \ne \frac{5}{{16}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">the events are not independent <em><strong>AG N0</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [4 marks]</span></strong></em></p>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\text{P(first takes only music)}} = \frac{3}{{16}}\)</span><span style="font-family: times new roman,times; font-size: medium;"> (seen anywhere) <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\text{P(second takes only art)}} = \frac{7}{{15}}\) </span><span style="font-family: times new roman,times; font-size: medium;">(seen anywhere) <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of valid approach <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{3}{{16}} \times \frac{7}{{15}}\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\text{P(music and art)}} = \frac{{21}}{{240}}\) \(\left( { = \frac{7}{{80}}} \right)\) </span><span style="font-family: times new roman,times; font-size: medium;"> </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1 N2</span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">A majority of candidates found the values in the Venn diagram easily. Common errors include giving \(s = 16\) , and also neglecting <strong><em>s</em></strong> in finding \(q = 4\) (e.g. \(12 + 8 - 16\) ) . Some interpreted the values as probabilities, despite the question explicitly stating that <em>p</em>, <em>q</em>, <em>r</em> and <em>s</em> represent numbers of students. Occasionally the values for <em>p</em> and <em>r</em> were misinterpreted as being inclusive of <em>q</em>. Follow-through marks were often earned in subsequent parts for such cases. </span></p>
<div class="question_part_label">a(i), (ii) and (iii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">For (b), rather than think of the situations conceptually, most candidates reached for the formula for conditional probability, with mixed results. Few candidates considered that independence means \({\rm{P}}(A|M) = {\rm{P}}(A)\) . Most applied \({\rm{P}}(A \cap B) = {\rm{P}}(A) \times {\rm{P}}(B)\) , with many giving incomplete or incorrect calculations. Some candidates compared the wrong things and showed, for example, that \(\frac{5}{8} \ne \frac{3}{8}\) , which incorrectly compares \({\rm{P}}(A|M)\) with \({\rm{P}}(A \cap M)\) . Others stated that because there is an intersection, the events are independent, which is an insufficient explanation. </span></p>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (c) was commonly answered as if there is replacement, with many candidates calculating \(\frac{3}{{16}} \times \frac{7}{{16}}\) . However, implicit in the phrasing "one after the other" is that there is no replacement. </span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A running club organizes a race to select girls to represent the club in a competition.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The times taken by the group of girls to complete the race are shown in the table below.</span></p>
<p><span style="font-family: TimesNewRomanPSMT;"><br><img style="display: block; margin-left: auto; margin-right: auto;" src="images/brit.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the value of \(p\) and of \(q\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A girl is chosen at random.</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (i) Find the probability that the time she takes is less than \(14\) minutes.</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (ii) Find the probability that the time she takes is at least \(26\) minutes.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A girl is selected for the competition if she takes less than \(x\) minutes to complete the </span><span style="font-family: times new roman,times; font-size: medium;">race.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Given that \(40\%\) of the girls are not selected,</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (i) find the number of girls who are not selected;</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (ii) find \(x\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Girls who are not selected, but took less than \(25\) minutes to complete the race, are </span><span style="font-family: times new roman,times; font-size: medium;">allowed another chance to be selected. The new times taken by these girls are shown </span><span style="font-family: times new roman,times; font-size: medium;">in the cumulative frequency diagram below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/crop.png" alt></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Write down the number of girls who were allowed another chance.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Find the percentage of the whole group who were selected.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to find \(p\) <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg</em> \(120 - 70\) , \(50 + 20 + x = 120\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(p = 50\) <em><strong>A1 N2</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to find \(q\) <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg</em> \(180 - 20\) , \(200 - 20 - 20\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(q = 160\) <em><strong>A1 N2 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks] </span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) \(\frac{{70}}{{200}}\) \(\left( { = \frac{7}{{20}}} \right)\) <strong><em>A1 N1</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) valid approach <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(20 + 20\) , \(200 - 160\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\frac{{40}}{{200}}\) \(\left( { = \frac{1}{5}} \right)\) <strong><em>A1 N2</em></strong></span></p>
<p><strong><em><span style="font-family: times new roman,times; font-size: medium;"> </span></em></strong></p>
<p><strong><em><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) attempt to find number of girls <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg</em> \(0.4\), \(\frac{{40}}{{100}} \times 200\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(80\) are not selected <em><strong>A1 N2</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \(120\) are selected <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(x = 20\) <em><strong>A1 N2 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em><strong> </strong></em></span></p>
<p><em><span style="font-family: times new roman,times; font-size: medium;"><strong>[4 marks]</strong> </span></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) \(30\) given second chance <em><strong>A1 N1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \(20\) took less than \(20\) minutes <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to find <strong>their</strong> selected total (may be seen in \(\%\) calculation) <strong><em>(M1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(120 + 20\) \(( = 140)\) , \(120 + \)<strong> their</strong> answer from (i) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(70\) (\(\%\)) <em><strong>A1 N3 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> </span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks] </span></strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="font-size: 13.28px; font-family: sans-serif; left: 430.973px; top: 924.133px; transform: scale(0.994531, 1); transform-origin: 0% 0% 0px;" dir="ltr" data-font-name="Helvetica" data-canvas-width="224.7640066984892"><span style="font-family: times new roman,times; font-size: medium;">Overall, candidates were very successful in parts (a), (b) and (c) of this question. Most of the errors in these parts had to do with candidates not understanding terms such as "at least" or "less than".</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Overall, candidates were very successful in parts (a), (b) and (c) of this question. Most of the errors in these parts had to do with candidates not understanding terms such as "at least" or "less than".</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Overall, candidates were very successful in parts (a), (b) and (c) of this question. Most of the errors in these parts had to do with candidates not understanding terms such as "at least" or "less than".</span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="font-size: 13.28px; font-family: sans-serif; left: 96.032px; top: 1004.16px; transform: scale(1.0304, 1); transform-origin: 0% 0% 0px;" dir="ltr" data-font-name="Helvetica" data-canvas-width="460.5902537266589"><span style="font-family: times new roman,times; font-size: medium;">Part (d) was quite challenging for candidates, who may not have read the question carefully and studied the values in the diagram. Many seemed confused by the idea that not all the girls who were given a second chance were selected. In part (d)(ii), many did not find the percentage of the whole group, but rather the percentage of the girls who were given a second chance.</span></p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The following Venn diagram shows the events \(A\) and \(B\)<span class="s1">, where \({\text{P}}(A) = 0.4,{\text{ P}}(A \cup B) = 0.8\) and \({\text{P}}(A \cap B) = 0.1\)</span>. The values \(p\) and \(q\) are probabilities.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-01-31_om_17.49.37.png" alt="M16/5/MATME/SP1/ENG/TZ1/02"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Write down the value of \(q\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Find the value of \(p\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find \({\text{P}}(B)\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> \(q = 0.1\)</span> <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N1</em></strong></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>appropriate approach <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\({\text{P}}(A) - q,{\text{ }}0.4 - 0.1\)</p>
<p class="p1"><span class="Apple-converted-space">\(p = 0.3\) </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">valid approach <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg </em>\(\,\,\,\,\,\)\({\text{P}}(A \cup B) = {\text{P}}(A) + {\text{P}}(B) - {\text{P}}(A \cap B),{\text{ P}}(A \cap B) + {\text{P}}(B \cap A')\)</p>
<p class="p1">correct values <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(0.8 = 0.4 + {\text{P}}(B) - 0.1,{\text{ }}0.1 + 0.4\)</p>
<p class="p1"><span class="Apple-converted-space">\({\text{P}}(B) = 0.5\) </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">This question was well done by most candidates. In part (b), the intersection \({\text{P}}(A \cap B)\) was sometimes overlooked, incorrectly using \({\text{P}}(A \cup B) = {\text{P}}(A) + {\text{P}}(B)\) instead.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>This question was well done by most candidates. In part (b), the intersection \({\text{P}}(A \cap B)\) was sometimes overlooked, incorrectly using \({\text{P}}(A \cup B) = {\text{P}}(A) + {\text{P}}(B)\) instead.</p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Events \(A\) and \(B\) are independent with \({\text{P}}(A \cap B) = 0.2\) and \({\text{P}}(A' \cap B) = 0.6\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \({\text{P}}(B)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \({\text{P}}(A \cup B)\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">valid interpretation (may be seen on a Venn diagram) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p2">eg\(\,\,\,\,\,\)\({\text{P}}(A \cap B) + {\text{P}}(A' \cap B),{\text{ }}0.2 + 0.6\)</p>
<p class="p3"><span class="s2">\({\text{P}}(B) = 0.8\) <span class="Apple-converted-space"> </span></span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p3"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">valid attempt to find \({\text{P}}(A)\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p1"><span class="s1"><em>eg</em></span>\(\,\,\,\,\,\)\({\text{P}}(A \cap B) = {\text{P}}(A) \times {\text{P}}(B),{\text{ }}0.8 \times A = 0.2\)</p>
<p class="p1">correct working for \({\text{P}}(A)\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p1"><span class="s1"><em>eg</em></span>\(\,\,\,\,\,\)\(0.25,{\text{ }}\frac{{0.2}}{{0.8}}\)</p>
<p class="p1">correct working for \({\text{P}}(A \cup B)\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p1"><span class="s1"><em>eg</em></span>\(\,\,\,\,\,\)\(0.25 + 0.8 - 0.2,{\text{ }}0.6 + 0.2 + 0.05\)</p>
<p class="p2"><span class="s2">\({\text{P}}(A \cup B) = 0.85\) <span class="Apple-converted-space"> </span></span><strong><em>A1 <span class="Apple-converted-space"> </span>N3</em></strong></p>
<p class="p2"><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The following diagram shows a board which is divided into three regions \(A\), \(B\) and \(C\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2015-12-13_om_16.53.51.png" alt></p>
<p class="p1">A game consists of a contestant throwing one dart at the board. The probability of hitting each region is given in the following table.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2015-12-13_om_16.54.43.png" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the probability that the dart does <strong>not </strong>hit the board.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">The contestant scores points as shown in the following table.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2015-12-13_om_17.03.42.png" alt></p>
<p class="p1">Given that the game is fair, find the value of \(q\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">evidence of summing probabilities to \(1\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\frac{5}{{20}} + \frac{4}{{20}} + \frac{1}{{20}} + p = 1,\;\;\;\sum { = 1} \)</p>
<p class="p1">correct working <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;p = 1 - \frac{{10}}{{20}}\)</p>
<p class="p1">\(p = \frac{{10}}{{20}}\;\;\;\left( { = \frac{1}{2}} \right)\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">correct substitution into \({\text{E}}(X)\) <strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\frac{4}{{20}}(q) + \frac{1}{{20}}(10) + \frac{{10}}{{20}}( - 3)\)</p>
<p class="p1">valid reasoning for fair game (seen anywhere, including equation) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;{\text{E}}(X) = 0\), points lost\( = \)points gained</p>
<p class="p1">correct working <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;4q + 10 - 30 = 0,\;\;\;\frac{4}{{20}}q + \frac{{10}}{{20}} = \frac{{30}}{{20}}\)</p>
<p class="p1">\(q = 5\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[4 marks]</em></strong></p>
<p class="p1"><strong><em>Total [7 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">The large majority of candidates answered part (a) of the question correctly by summing the probabilities to 1.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Part (b), however was not as well done. Many candidates seemed to be unfamiliar with the idea of a "fair game", despite this topic being listed in the syllabus. The most common error in part (b) was setting \(E(X) = 1\) rather than \(E(X) = 0\).</p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The cumulative frequency curve below represents the marks obtained by 100 students.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/M12P2TZ2Q1.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the median mark.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the interquartile range.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of median position <em><strong>(M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. 50, line on sketch </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">median is 56 <em><strong>A1 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks] </span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">lower quartile \(= 40\) , upper quartile \(= 70\) <em><strong> (A1)(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">interquartile range \(= 30\) <em><strong>A1 N3</strong> </em></span></p>
<p><span style="font-size: medium;"><em><strong><span style="font-family: times new roman,times;">[3 marks]</span> </strong></em></span></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Overall, this question was done well by candidates. In part (a), a surprising number of candidates found the median position (the cumulative frequency) on the <em>y</em>-axis, but did not find the median mark on the <em>x</em>-axis. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Overall, this question was done well by candidates. In part (a), a surprising number of candidates found the median position (the cumulative frequency) on the <em>y</em>-axis, but did not find the median mark on the <em>x</em>-axis. Similar misunderstanding was shown by some candidates in part (b), when attempting to find the interquartile range. </span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The following box-and-whisker plot shows the number of text messages sent by students in a school on a particular day.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of the interquartile range.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>One student sent<em> k</em> text messages, where <em>k</em> > 11 . Given that <em>k</em> is an outlier, find the least value of <em>k</em>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>recognizing <em>Q</em><sub>1</sub> or <em>Q</em><sub>3</sub> (seen anywhere) <em><strong>(M1)</strong></em></p>
<p>eg 4,11 , indicated on diagram</p>
<p>IQR = 7 <em><strong>A1 N2</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>recognizing the need to find 1.5 IQR <em><strong>(M1)</strong></em></p>
<p><em>eg </em> 1.5 × IQR, 1.5 × 7</p>
<p>valid approach to find <em>k </em> <em><strong> (M1)</strong></em></p>
<p><em>eg </em>10.5 + 11, 1.5 × IQR + <em>Q</em><sub>3</sub></p>
<p>21.5 <em><strong>(A1) </strong></em></p>
<p><em>k</em> = 22 <em><strong>A1 N3</strong></em></p>
<p><strong>Note:</strong> If no working shown, award <em><strong>N2</strong></em> for an answer of 21.5.</p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">The random variable X has the following probability distribution.<br></span></p>
<p><img src="data:image/png;base64,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" alt></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Given that \({\rm{E}}(X) = 1.7\) , find <em>q</em> .</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct substitution into \({\rm{E}}(X) = \sum {px} \) (seen anywhere) <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(1s + 2 \times 0.3 + 3q = 1.7\) , \(s + 3q = 1.1\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">recognizing \(\sum {p = 1} \) (seen anywhere) <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct substitution into \(\sum {p = 1} \) <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(s + 0.3 + q = 1\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">attempt to solve simultaneous equations <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct working <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(0.3 + 2q = 0.7\) , \(2s = 1\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(q = 0.2\) <em><strong>A1 N4</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[6 marks]</span></strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">Candidates generally earned either full marks or only one mark on this question. The most common error was where candidates only wrote the equation for \({\rm{E}}(X) = 1.7\) , and tried to rearrange that equation to solve for <em>q</em>. The candidates who also knew that the sum of the probabilities must be equal to 1 were very successful in solving the resulting system of equations. </span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">In a class of <span class="s1">21 </span>students, <span class="s1">12 </span>own a laptop, <span class="s1">10 </span>own a tablet, and <span class="s1">3 </span>own neither.</p>
<p class="p1">The following Venn diagram shows the events “own a laptop” and “own a tablet”.</p>
<p class="p1">The values \(p\), \(q\), \(r\) and \(s\) represent numbers of students.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-02-02_om_09.38.31.png" alt="M16/5/MATME/SP1/ENG/TZ2/08"></p>
</div>
<div class="specification">
<p class="p1">A student is selected at random from the class.</p>
</div>
<div class="specification">
<p class="p1">Two students are randomly selected from the class. Let \(L\) be the event a “student owns a laptop”.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Write down the value of \(p\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Find the value of \(q\).</p>
<p class="p1">(iii) <span class="Apple-converted-space"> </span>Write down the value of \(r\) and of \(s\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) Write down the probability that this student owns a laptop.</p>
<p class="p1">(ii) Find the probability that this student owns a laptop or a tablet but not both.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span><strong>Copy </strong>and complete the following tree diagram. (Do <strong>not </strong>write on this page.)</p>
<p class="p2" style="text-align: center;"><img src="images/Schermafbeelding_2017-02-02_om_13.55.21.png" alt="M16/5/MATME/SP1/ENG/TZ2/08.c"></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Write down the probability that the second student owns a laptop given that the first owns a laptop.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> \(p = 3\)</span> <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N1</em></strong></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>valid approach <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\((12 + 10 + 3) - 21,{\text{ }}22 - 18\)</p>
<p class="p1"><span class="Apple-converted-space">\(q = 4\) </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1">(iii) <span class="Apple-converted-space"> \(r = 8,{\text{ }}s = 6\)</span> <span class="Apple-converted-space"> </span><strong><em>A1A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> \(\frac{{12}}{{21}}{\text{ }}\left( { = \frac{4}{7}} \right)\)</span> <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A2 <span class="Apple-converted-space"> </span>N2</em></strong></span></p>
<p class="p2">(ii) <span class="Apple-converted-space"> </span>valid approach <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\(8 + 6,{\text{ }}r + s\)</p>
<p class="p2"><span class="Apple-converted-space">\(\frac{{14}}{{21}}{\text{ }}\left( { = \frac{2}{3}} \right)\) </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <img src="images/Schermafbeelding_2017-02-02_om_13.57.14.png" alt="M16/5/MATME/SP1/ENG/TZ2/08.c/M"> <strong><em>A1A1A1 N3</em></strong></p>
<p class="p1">(ii) <span class="Apple-converted-space"> \(\frac{{11}}{{20}}\)</span> <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1 <span class="Apple-converted-space"> </span>N1</em></strong></span></p>
<p class="p2"><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">On the whole, candidates were very successful on this question, with the majority of candidates earning most of the available marks.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">On the whole, candidates were very successful on this question, with the majority of candidates earning most of the available marks.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">On the whole, candidates were very successful on this question, with the majority of candidates earning most of the available marks. The most common error was seen in part (c)(ii), where many candidates did not earn the mark. It is also interesting to note that many of the candidates who answered this part correctly did so by using the formula for conditional probability, rather than recognizing that the required probability is given to them in the second branch of the tree diagram.</p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Two boxes contain numbered cards as shown below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/anne.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Two cards are drawn at random, one from each box.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Copy and complete the table below to show all nine equally likely outcomes.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/bt.png" alt></span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let <em>S</em> be the sum of the numbers on the two cards.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the probability of each value of <em>S</em>.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the expected value of <em>S</em>.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Anna plays a game where she wins \(\$ 50\) if <em>S</em> is even and loses \(\$ 30\) if <em>S</em> is odd.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Anna plays the game 36 times. Find the amount she expects to have at the end </span><span style="font-family: times new roman,times; font-size: medium;">of the 36 games.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/chef.png" alt></span><em><span style="font-family: times new roman,times; font-size: medium;"><strong> A2 N2</strong> </span></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks] </span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(12) = \frac{1}{9}\) , \({\rm{P}}(13) = \frac{3}{9}\) , \({\rm{P}}(14) = \frac{3}{9}\) , \({\rm{P}}(15) = \frac{2}{9}\) <em><strong>A2 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [2 marks]</span></strong></em></p>
<p> </p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution into formula for \({\text{E}}(X)\) <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{E}}(S) = 12 \times \frac{1}{9} + 13 \times \frac{3}{9} + 14 \times \frac{3}{9} + 15 \times \frac{2}{9}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{E}}(S) = \frac{{123}}{9}\) <em><strong>A2 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 1</span></strong></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct expression for expected gain E(<em>A</em>) for 1 game <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{4}{9} \times 50 - \frac{5}{9} \times 30\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\rm{E}}(A) = \frac{{50}}{9}\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">amount at end = expected gain for 1 game \( \times 36\) <em><strong> (M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">= 200 (dollars) <em><strong>A1 N2</strong></em></span></p>
<p align="LEFT"><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 2</span></strong></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">attempt to find expected number of wins and losses <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g.</span><span style="font-family: times new roman,times; font-size: medium;"> \(\frac{4}{9} \times 36\) , \(\frac{5}{9} \times 36\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">attempt to find expected gain E(<em>G</em>) <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(16 \times 50 - 30 \times 20\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\text{E}}(G) = 200\) (dollars) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Most candidates completed parts (a), (b) and (c) successfully. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Most candidates completed part (b) successfully. </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Many found the expected value correctly, while some showed difficulty with the arithmetic. </span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This was often left blank or only superficially attempted. Some found the expected value \(\frac{{50}}{9}\) but did not answer the question about the amount of money. </span></p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>Jim heated a liquid until it boiled. He measured the temperature of the liquid as it cooled. The following table shows its temperature, \(d\) degrees Celsius, \(t\) minutes after it boiled.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-11_om_08.45.05.png" alt="M17/5/MATME/SP1/ENG/TZ1/04"></p>
</div>
<div class="specification">
<p>Jim believes that the relationship between \(d\) and \(t\) can be modelled by a linear regression equation.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the independent variable.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the boiling temperature of the liquid.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Jim describes the correlation as <strong>very strong</strong>. Circle the value below which best represents the correlation coefficient.</p>
<p>\[0.992\quad \quad \quad 0.251\quad \quad \quad 0\quad \quad \quad - 0.251\quad \quad \quad - 0.992\]</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Jim’s model is \(d = - 2.24t + 105\), for \(0 \leqslant t \leqslant 20\). Use his model to predict the decrease in temperature for any 2 minute interval.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>\(t\) <em><strong>A1</strong></em> <em><strong>N1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>105 <em><strong>A1</strong></em> <em><strong>N1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\( - 0.992\) <strong><em>A2</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid approach <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\frac{{{\text{d}}d}}{{{\text{d}}t}} = - 2.24;{\text{ }}2 \times 2.24,{\text{ }}2 \times - 2.24,{\text{ }}d(2) = - 2 \times 2.24 \times 105,\)</p>
<p>finding \(d({t_2}) - d({t_1})\) where \({t_2} = {t_1} + 2\)</p>
<p>4.48 (degrees) <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p> </p>
<p><strong>Notes:</strong> Award no marks for answers that <strong>directly </strong>use the table to find the decrease in temperature for 2 minutes <em>eg</em> \(\frac{{105 - 98.4}}{2} = 3.3\).</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Bill and Andrea play two games of tennis. The probability that Bill wins the first game is \(\frac{4}{5}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">If Bill wins the first game, the probability that he wins the second game is \(\frac{5}{6}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">If Bill loses the first game, the probability that he wins the second game is \(\frac{2}{3}\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Copy</strong> and complete the following tree diagram. <em>(Do <strong>not</strong> write on this page.)</em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em> </em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em><br><img src="images/maths_9a_1.png" alt></em></span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the probability that Bill wins the first game and Andrea wins the second game.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the probability that Bill wins at least one game.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Given that Bill wins at least one game, find the probability that he wins both games.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: 'times new roman', times;"><span style="font-size: medium;"><br><img src="images/maths_9a.png" alt> <strong><em>A1A1A1 N3</em></strong></span></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Note:</em></strong> Award <em><strong>A1</strong></em> for each correct <strong>bold</strong> probability.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px 'Times New Roman';"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em><strong>[3 marks]</strong></em></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">multiplying along the branches (may be seen on diagram) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(\frac{4}{5} \times \frac{1}{6}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{4}{{30}}\left( {\frac{2}{{15}}} \right)\) <strong><em>A1 N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 1</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">multiplying along the branches (may be seen on diagram) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(\frac{4}{5} \times \frac{5}{6},{\text{ }}\frac{4}{5} \times \frac{1}{6},{\text{ }}\frac{1}{5} \times \frac{2}{3}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">adding their probabilities of three mutually exclusive paths <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(\frac{4}{5} \times \frac{5}{6} + \frac{4}{5} \times \frac{1}{6} + \frac{1}{5} \times \frac{2}{3},{\text{ }}\frac{4}{5} + \frac{1}{5} \times \frac{2}{3}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">correct simplification <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(\frac{{20}}{{30}} + \frac{4}{{30}} + \frac{2}{{15}},{\text{ }}\frac{2}{3} + \frac{2}{{15}} + \frac{2}{{15}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{28}}{{30}}{\text{ }}\left( { = \frac{{14}}{{15}}} \right)\) <strong><em>A1 N3</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 2</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">recognizing “Bill wins at least one” is complement of “Andrea wins 2” <strong><em>(R1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> finding P (Andrea wins 2)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">P (Andrea wins both) \( = \frac{1}{5} \times \frac{1}{3}\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">evidence of complement <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(1 - p,{\text{ }}1 - \frac{1}{{15}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{14}}{{15}}\) <strong><em>A1 N3</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">P (<em>B </em>wins both) \(\frac{4}{5} \times \frac{5}{6}{\text{ }}\left( { = \frac{2}{3}} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">evidence of recognizing conditional probability <strong><em>(R1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \({\text{P}}(A\left| B \right.),{\text{ P (Bill wins both}}\left| {{\text{Bill wins at least one), tree diagram}}} \right.\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">correct substitution <strong><em>(A2)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(\frac{{\frac{4}{5} \times \frac{5}{6}}}{{\frac{{14}}{{15}}}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{20}}{{28}}{\text{ }}\left( { = \frac{5}{7}} \right)\) <strong><em>A1 N3</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Two standard six-sided dice are tossed. A diagram representing the sample space is </span><span style="font-family: times new roman,times; font-size: medium;">shown below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/dice.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(X\) be the sum of the scores on the two dice.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Find \({\rm{P}}(X = 6)\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) Find \({\rm{P}}(X > 6)\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iii) Find \({\rm{P}}(X = 7|X > 6)\) .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">a(i), (ii) and (iii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Elena plays a game where she tosses two dice.</span></p>
<p style="margin-left: 60px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">If the sum is 6, she <strong>wins</strong> 3 points.</span></p>
<p style="margin-left: 60px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">If the sum is greater than 6, she <strong>wins</strong> 1 point.</span></p>
<p style="margin-left: 60px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">If the sum is less than 6, she <strong>loses</strong> <em>k </em>points.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the value of <em>k</em> for which the game is fair.</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) number of ways of getting \(X = 6\) is 5 <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(X = 6) = \frac{5}{{36}}\) </span><span style="font-family: times new roman,times; font-size: medium;"> <em><strong>A1 N2</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) number of ways of getting \(X > 6\) is 21 <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(X > 6) = \frac{{21}}{{36}}\left( { = \frac{7}{{12}}} \right)\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1 N2</span></strong></em></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(iii) \({\rm{P}}(X = 7|X > 6) = \frac{6}{{21}}\left( { = \frac{2}{7}} \right)\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A2 N2</span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[6 marks]</span></strong></em></p>
<div class="question_part_label">a(i), (ii) and (iii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to find \({\rm{P}}(X < 6)\) <em><strong>M1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(1 - \frac{5}{{36}} - \frac{{21}}{{36}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(X < 6) = \frac{{10}}{{36}}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">fair game if \({\rm{E}}(W) = 0\) (may be seen anywhere) <em><strong>R1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to substitute into \({\rm{E}}(X)\) formula <em><strong>M1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(3\left( {\frac{5}{{36}}} \right) + 1\left( {\frac{{21}}{{36}}} \right) - k\left( {\frac{{10}}{{36}}} \right)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution into \({\rm{E}}(W) = 0\) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(3\left( {\frac{5}{{36}}} \right) + 1\left( {\frac{{21}}{{36}}} \right) - k\left( {\frac{{10}}{{36}}} \right) = 0\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">work towards solving <em><strong>M1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(15 + 21 - 10k = 0\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(36 = 10k\) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(k = \frac{{36}}{{10}}( = 3.6)\) </span><span style="font-family: times new roman,times; font-size: medium;"> <em><strong>A1 N4</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [8 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a(i), (ii) and (iii).</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">There are nine books on a shelf. For each book, <em>x</em> is the number of pages, and <em>y</em> is the </span><span style="font-family: times new roman,times; font-size: medium;">selling price in pounds (£). Let <em>r</em> be the correlation coefficient.</span></p>
</div>
<div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><img src="images/diving.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the possible minimum and maximum values of <em>r</em> .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Given that \(r = 0.95\) , which of the following diagrams best represents the data.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/cow.png" alt></span></p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">For the data in diagram D , which <strong>two</strong> of the following expressions describe </span><span style="font-family: times new roman,times; font-size: medium;">the correlation between <em>x</em> and <em>y</em> ?</span></p>
<p style="text-align: center;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">perfect, zero, linear, strong positive, strong negative, </span><span style="font-family: times new roman,times; font-size: medium;">weak positive, weak negative</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">min value of <em>r</em> is \( - 1\), max value of <em>r</em> is 1 <em><strong>A1A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">C <em><strong>A1 N1</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[1 mark]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">linear, strong negative <em><strong>A1A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A box contains 100 cards. Each card has a number between one and six written on it. </span><span style="font-family: times new roman,times; font-size: medium;">The following table shows the frequencies for each number.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/take_that.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Calculate the value of <em>k</em>.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Find</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) the median;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) the interquartile range.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of using \(\sum {{f_i} = 100} \) <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(k = 4\) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) evidence of median position <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. 50th item, \(26 + 10 + 20 = 56\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\text{median}} = 3\) <em><strong>A1 N2</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) \({Q_1} = 1\) and \({Q_3} = 5\) <strong><em>(A1)(A1)</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\text{interquartile range}} = 4\) (accept 1 to 5 or \(5 - 1\) , etc.) <em><strong>A1 N3</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[5 marks]</span></strong></em></p>
<p align="LEFT"> </p>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Frequencies and median seemed well understood, but quartiles and inter-quartile range less </span><span style="font-family: times new roman,times; font-size: medium;">so.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Frequencies and median seemed well-understood, but quartiles and interquartile range less </span><span style="font-family: times new roman,times; font-size: medium;">so. A few students, probably based on past papers, drew cumulative frequency diagrams, </span><span style="font-family: times new roman,times; font-size: medium;">generating slightly different answers for median and quartiles.</span></p>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<br><hr><br><div class="specification">
<p>A city hired 160 employees to work at a festival. The following cumulative frequency curve shows the number of hours employees worked during the festival.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-14_om_07.01.49.png" alt="M17/5/MATME/SP1/ENG/TZ2/08.a.ii"></p>
</div>
<div class="specification">
<p>The city paid each of the employees £8 per hour for the first 40 hours worked, and £10 per hour for each hour they worked after the first 40 hours.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the median number of hours worked by the employees.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the number of employees who worked 50 hours or less.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the amount of money an employee earned for working 40 hours;</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the amount of money an employee earned for working 43 hours.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the number of employees who earned £200 or less.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Only 10 employees earned more than £\(k\). Find the value of \(k\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>evidence of median position <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)80th employee</p>
<p>40 hours <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>130 employees <strong><em>A1</em></strong> <strong><em>N1</em></strong></p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>£320 <strong><em>A1</em></strong> <strong><em>N1</em></strong></p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>splitting into 40 and 3 <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)3 hours more, \(3 \times 10\)</p>
<p>correct working <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(320 + 3 \times 10\)</p>
<p>£350 <strong><em>A1</em></strong> <strong><em>N3</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid approach <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)200 is less than 320 so 8 pounds/hour, \(200 \div 8,{\text{ }}25,{\text{ }}\frac{{200}}{{320}} = \frac{x}{{40}}\),</p>
<p>18 employees <strong><em>A2</em></strong> <strong><em>N3</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid approach <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(160 - 10\)</p>
<p>60 hours worked <strong><em>(A1)</em></strong></p>
<p>correct working <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(40(8) + 20(10),{\text{ }}320 + 200\)</p>
<p>\(k = 520\) <strong><em>A1</em></strong> <strong><em>N3</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Events <em>A</em> and <em>B</em> are such that \({\rm{P}}(A) = 0.3\) , \({\rm{P}}(B) = 0.6\) and \({\rm{P}}(A \cup B) = 0.7\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/washing.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The values <em>q</em> , <em>r</em> , <em>s</em> and <em>t</em> represent probabilities.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the value of <em>t</em> .</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Show that \(r = 0.2\) . </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Write down the value of <em>q</em> and of <em>s</em> .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Write down \({\rm{P}}(B')\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Find \({\rm{P}}(A|B')\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c(i) and (ii).</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\(t = 0.3\) <em><strong>A1 N1</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[1 mark] </span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) correct values <em><strong>A1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(0.3 + 0.6 - 0.7\) , \(0.9 - 0.7\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(r = 0.2\) <em><strong>AG N0</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \(q = 0.1\) , \(s = 0.4\) <em><strong>A1A1 N2</strong></em> </span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) \(0.4\) <em><strong>A1 N1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \({\rm{P}}(A|B') = \frac{1}{4}\) <em><strong>A2 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<div class="question_part_label">c(i) and (ii).</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Parts (a), (b), and (c)(i) of this Venn diagram probability question were answered quite well with candidates consistently earning full marks. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Parts (a), (b), and (c)(i) of this Venn diagram probability question were answered quite well with candidates consistently earning full marks. Only a few candidates worked backwards from the \(r = 0.2\) given in the "show that" portion of part (b).</span></p>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Many candidates struggled on part (c)(ii), either not recognizing conditional probability or multiplying probabilities to find the numerator as if the events were independent. A number of candidates who successfully found the probability in part (c)(ii) left their incomplete answer of \(\frac{{0.1}}{{0.4}}\) . </span></p>
<div class="question_part_label">c(i) and (ii).</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">In a class of 100 boys, 55 boys play football and 75 boys play rugby. Each boy must play </span><span style="font-family: times new roman,times; font-size: medium;">at least one sport from football and rugby.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Find the number of boys who play both sports.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Write down the number of boys who play only rugby.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">One boy is selected at random.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Find the probability that he plays only one sport.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) Given that the boy selected plays only one sport, find the probability that he </span><span style="font-family: times new roman,times; font-size: medium;">plays rugby.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Let <em>A</em> be the event that a boy plays football and <em>B</em> be the event that a boy plays rugby.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Explain why <em>A</em> and <em>B</em> are <strong>not</strong> mutually exclusive.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Show that <em>A</em> and <em>B</em> are <strong>not</strong> independent.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) evidence of substituting into \(n(A \cup B) = n(A) + n(B) - n(A \cap B)\) <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(75 + 55 - 100\) , Venn diagram </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">30 <em><strong>A1 N2 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) 45 <em><strong>A1 N1</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i)<strong> METHOD 1</strong> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of using complement, Venn diagram <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(1 - p\) , \(100 - 30\)</span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(\frac{{70}}{{100}}\) \(\left( { = \frac{7}{{10}}} \right)\) </span><em><span style="font-family: times new roman,times; font-size: medium;"><strong>A1 N2</strong> </span></em></p>
<p><strong> <span style="font-family: times new roman,times; font-size: medium;">METHOD 2 </span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to find P(only one sport) , Venn diagram <em><strong>(M1) </strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{25}}{{100}} + \frac{{45}}{{100}}\)</span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(\frac{{70}}{{100}}\) \(\left( { = \frac{7}{{10}}} \right)\) </span><em><span style="font-family: times new roman,times; font-size: medium;"><strong>A1 N2</strong> </span></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \(\frac{{45}}{{70}}\) \(\left( { = \frac{9}{{14}}} \right)\) <em><strong>A2 N2</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em><strong> [4 marks]</strong></em> </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">valid reason in words or symbols <em><strong>(R1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{P}}(A \cap B) = 0\) if mutually exclusive, \({\rm{P}}(A \cap B) \ne 0\) if not mutually exclusive </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct statement in words or symbols <em><strong>A1 N2</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{P}}(A \cap B) = 0.3\) , \({\rm{P}}(A \cup B) \ne {\rm{P}}(A) + {\rm{P}}(B)\) , \({\rm{P}}(A) + {\rm{P}}(B) > 1\) , some students play both sports, sets intersect </span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks] </span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">valid reason for independence <em><strong>(R1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{P}}(A \cap B) = {\rm{P}}(A) \times {\rm{P}}(B)\) , \({\rm{P}}(B|A) = {\rm{P}}(B)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution <em><strong>A1A1 N3</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{30}}{{100}} \ne \frac{{75}}{{100}} \times \frac{{55}}{{100}}\) , \(\frac{{30}}{{55}} \ne \frac{{75}}{{100}}\)</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Overall, this question was very well done. There were some problems with the calculation of conditional probability, where a considerable amount of candidates tried to use a formula instead of using its concept and analysing the problem. It is the kind of question where it can be seen if the concept is not clear to candidates. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Overall, this question was very well done. There were some problems with the calculation of conditional probability, where a considerable amount of candidates tried to use a formula instead of using its concept and analysing the problem. It is the kind of question where it can be seen if the concept is not clear to candidates. </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In part (c), candidates were generally able to explain in words why events were mutually exclusive, though many gave the wrong values for P(<em>A</em>) and P(<em>B</em>). </span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">There was a great amount of confusion between the concepts of independent and mutually exclusive events. In part (d), the explanations often referred to mutually exclusive events. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">It was evident that candidates need more practice with questions like (c) and (d). </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Some students equated probabilities and number of elements, giving probabilities greater than 1. </span></p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The diagram below shows the probabilities for events <em>A</em> and <em>B</em> , with \({\rm{P}}(A') = p\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/tree.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the value of <em>p</em> .</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \({\rm{P}}(B)\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \({\rm{P}}(A'|B)\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\(p = \frac{4}{5}\) <em><strong>A1 N1</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[1 mark]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">multiplying along the branches <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{1}{5} \times \frac{1}{4}\) , \(\frac{{12}}{{40}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">adding products of probabilities of two mutually exclusive paths <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{1}{5} \times \frac{1}{4} + \frac{4}{5} \times \frac{3}{8}\) , \(\frac{1}{{20}} + \frac{{12}}{{40}}\)</span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\({\rm{P}}(B) = \frac{{14}}{{40}}\) \(\left( { = \frac{7}{{20}}} \right)\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1 N2</span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [3 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">appropriate approach which must include \({A'}\) (may be seen on diagram) <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{{\rm{P}}(A' \cap B)}}{{{\rm{P}}(B)}}\) (do not accept \(\frac{{{\rm{P}}(A \cap B)}}{{{\rm{P}}(B)}}\) )</span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\({\rm{P}}(A'|B) = \frac{{\frac{4}{5} \times \frac{3}{8}}}{{\frac{7}{{20}}}}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">(A1)</span></strong></em></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\({\rm{P}}(A'|B) = \frac{{12}}{{14}}\) \(\left( { = \frac{6}{7}} \right)\) </span><span style="font-family: times new roman,times; font-size: medium;"> <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [3 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">While nearly every candidate answered part (a) correctly, many had trouble with the other parts of this question. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In part (b), many candidates did not multiply along the branches of the tree diagram to find the required values, and many did not realize that there were two paths for P(<em>B</em>). There were also many candidates who understood what the question required, but then did not know how to multiply fractions correctly, and these calculation errors led to an incorrect answer. </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In part (c), most candidates attempted to use a formula for conditional probability found in the information booklet, but very few substituted the correct values. </span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Celeste wishes to hire a taxicab from a company which has a large number of taxicabs.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">The taxicabs are randomly assigned by the company.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"> The probability that a taxicab is yellow is 0.4.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"> The probability that a taxicab is a Fiat is 0.3.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"> The probability that a taxicab is yellow or a Fiat is 0.6.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><span style="font: 17.0px 'Times New Roman';"> </span>Find the probability that the taxicab hired by Celeste is <strong>not </strong>a yellow Fiat.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">recognize need for intersection of <em>Y </em>and <em>F</em> <strong><em>(R1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \({\text{P}}(Y \cap F){\text{, }}0.3 \times 0.4\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">valid approach to find \({\text{P}}(Y \cap F)\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \({\text{P}}(Y) + {\text{P}}(F) - {\text{P}}(Y \cup F)\), Venn diagram</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">correct working (may be seen in Venn diagram) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(0.4 + 0.3 - 0.6\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(Y \cap F) = 0.1\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">recognize need for complement of \(Y \cap F\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(1 - {\text{P}}(Y \cap F){\text{, }}1 - 0.1\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}\left( {(Y \cap F)'} \right) = 0.9\) <strong><em>A1 N3</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>The following table shows the probability distribution of a discrete random variable \(A\), in terms of an angle \(\theta \).</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-11_om_09.10.36.png" alt="M17/5/MATME/SP1/ENG/TZ1/10"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \(\cos \theta = \frac{3}{4}\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that \(\tan \theta > 0\), find \(\tan \theta \).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Let \(y = \frac{1}{{\cos x}}\), for \(0 < x < \frac{\pi }{2}\). The graph of \(y\)between \(x = \theta \) and \(x = \frac{\pi }{4}\) is rotated 360° about the \(x\)-axis. Find the volume of the solid formed.</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>evidence of summing to 1 <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\sum {p = 1} \)</p>
<p>correct equation <strong><em>A1</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\cos \theta + 2\cos 2\theta = 1\)</p>
<p>correct equation in \(\cos \theta \) <strong><em>A1</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\cos \theta + 2(2{\cos ^2}\theta - 1) = 1,{\text{ }}4{\cos ^2}\theta + \cos \theta - 3 = 0\)</p>
<p>evidence of valid approach to solve quadratic <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)factorizing equation set equal to \(0,{\text{ }}\frac{{ - 1 \pm \sqrt {1 - 4 \times 4 \times ( - 3)} }}{8}\)</p>
<p>correct working, clearly leading to required answer <strong><em>A1</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\((4\cos \theta - 3)(\cos \theta + 1),{\text{ }}\frac{{ - 1 \pm 7}}{8}\)</p>
<p>correct reason for rejecting \(\cos \theta \ne - 1\) <strong><em>R1</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\cos \theta \) is a probability (value must lie between 0 and 1), \(\cos \theta > 0\)</p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>R0 </em></strong>for \(\cos \theta \ne - 1\) without a reason.</p>
<p> </p>
<p>\(\cos \theta = \frac{3}{4}\) <em><strong>AG N0</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid approach <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)sketch of right triangle with sides 3 and 4, \({\sin ^2}x + {\cos ^2}x = 1\)</p>
<p>correct working </p>
<p><strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)missing side \( = \sqrt 7 ,{\text{ }}\frac{{\frac{{\sqrt 7 }}{4}}}{{\frac{3}{4}}}\)</p>
<p>\(\tan \theta = \frac{{\sqrt 7 }}{3}\) <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to substitute either limits or the function into formula involving \({f^2}\) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\pi \int_\theta ^{\frac{\pi }{4}} {{f^2},{\text{ }}\int {{{\left( {\frac{1}{{\cos x}}} \right)}^2}} } \)</p>
<p>correct substitution of both limits and function <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\pi \int_\theta ^{\frac{\pi }{4}} {{{\left( {\frac{1}{{\cos x}}} \right)}^2}{\text{d}}x} \)</p>
<p>correct integration <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\tan x\)</p>
<p>substituting <strong>their </strong>limits into <strong>their </strong>integrated function and subtracting <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\tan \frac{\pi }{4} - \tan \theta \)</p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>M0 </em></strong>if they substitute into original or differentiated function.</p>
<p> </p>
<p>\(\tan \frac{\pi }{4} = 1\) <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(1 - \tan \theta \)</p>
<p>\(V = \pi - \frac{{\pi \sqrt 7 }}{3}\) <strong><em>A1</em></strong> <strong><em>N3</em></strong></p>
<p> </p>
<p><strong><em>[6 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A scientist has 100 female fish and 100 male fish. She measures their lengths to the </span><span style="font-family: times new roman,times; font-size: medium;">nearest cm. These are shown in the following box and whisker diagrams.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/fish.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the range of the lengths of <strong>all</strong> 200 fish.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Four cumulative frequency graphs are shown below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/females.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Which graph is the best representation of the lengths of the <strong>female</strong> fish?</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct end points <em><strong>(A1)(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">max = 27 , min = 4</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">range = 23 <em><strong>A1 N3</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Graph 3 <em><strong>A2 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">While there were a large number of candidates who answered both parts of this question correctly, a surprising number did not know how to find the range of all 200 fish in part (a). Common errors included finding the ranges of the male and female fish separately, or averaging the separate ranges of the male and female fish. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Some candidates did not interpret the cumulative frequency graphs correctly, or just seemed to guess which graph was correct. The most common incorrect "guess" was graph 4, likely because this graph had a more familiar cumulative shape. </span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>A bag contains 5 green balls and 3 white balls. Two balls are selected at random without replacement.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Complete the following tree diagram.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2018-02-11_om_09.35.12.png" alt="N17/5/MATME/SP1/ENG/TZ0/01.a"></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that exactly one of the selected balls is green.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>correct probabilities</p>
<p><img src="images/Schermafbeelding_2018-02-11_om_09.51.13.png" alt="N17/5/MATME/SP1/ENG/TZ0/01.a/M"> <strong><em>A1A1A1 N3</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>A1 </em></strong>for each correct <strong>bold</strong> answer.</p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>multiplying along branches <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\frac{5}{8} \times \frac{3}{7},{\text{ }}\frac{3}{8} \times \frac{5}{7},{\text{ }}\frac{{15}}{{56}}\)</p>
<p>adding probabilities of correct mutually exclusive paths <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\frac{5}{8} \times \frac{3}{7} + \frac{3}{8} \times \frac{5}{7},{\text{ }}\frac{{15}}{{56}} + \frac{{15}}{{56}}\)</p>
<p>\(\frac{{30}}{{56}}{\text{ }}\left( { = \frac{{15}}{{28}}} \right)\) <strong><em>A1 N2</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The random variable <em>X</em> has the following probability distribution, with \({\rm{P}}(X > 1) = 0.5\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/M12P1TZ2Q4.jpg" alt> </span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the value of <em>r</em> .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Given that \({\rm{E}}(X) = 1.4\) , find the value of <em>p</em> and of <em>q</em> .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to substitute \({\rm{P}}(X > 1) = 0.5\) <em><strong>(M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(r + 0.2 = 0.5\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(r = 0.3\) <em><strong>A1 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks] </span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution into \({\rm{E}}(X)\) (seen anywhere) <em><strong>(A1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(0 \times p + 1 \times q + 2 \times r + 3 \times 0.2\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct equation <em><strong>A1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(q + 2 \times 0.3 + 3 \times 0.2 = 1.4\) , \(q + 1.2 = 1.4\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(q = 0.2\) <em><strong>A1 N1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of choosing \(\sum {{p_i} = 1} \) <em><strong>M1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(p + 0.2 + 0.3 + 0.2 = 1\) , \(p + q = 0.5\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct working <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(p + 0.7 = 1\) , \(1 - 0.2 - 0.3 - 0.2\) , \(p + 0.2 = 0.5\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(p = 0.3\) <em><strong>A1 N2</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Exception to the <strong><em>FT</em></strong> rule. Award <em><strong>FT</strong></em> marks on an incorrect value of <em>q</em>, even if <em>q</em> is an inappropriate value. Do not award the final <em><strong>A</strong></em> mark for an inappropriate value of <em>p</em>. </span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[6 marks] </span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">The majority of candidates were successful in earning full marks on this question. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In part (b), a small number of candidates did not use the correct formula for \({\rm{E}}(X)\) , even though this formula is given in the formula booklet. There were also a few candidates who incorrectly assumed that \(p = 0\) , forgetting that the sum of the probabilities must equal 1. There were a few candidates who left this question blank, which raises concerns about whether they had been exposed to probability distributions during the course. </span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">A school collects cans for recycling to raise money. Sam’s class has <span class="s1">20 </span>students.</p>
<p class="p1">The number of cans collected by each student in Sam’s class is shown in the following stem and leaf diagram.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-01-31_om_18.02.34.png" alt="M16/5/MATME/SP1/ENG/TZ1/08"></p>
</div>
<div class="specification">
<p class="p1">The following box-and-whisker plot also displays the number of cans collected by students in Sam’s class.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-01-31_om_18.04.35.png" alt="M16/5/MATME/SP1/ENG/TZ1/08.b+c+d+e"></p>
</div>
<div class="specification">
<p class="p1">There are <span class="s1">80 </span>students in the school.</p>
</div>
<div class="specification">
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-01-31_om_18.13.01.png" alt="M16/5/MATME/SP1/ENG/TZ1/08.e"></p>
<p class="p1">The mean number of cans collected is <span class="s1">39.4</span>. The standard deviation is <span class="s1">18.5</span>.</p>
<p class="p1">Each student then collects <span class="s1">2 </span>more cans.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the median number of cans collected.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) Write down the value of \(a\).</p>
<p class="p1">(ii) The interquartile range is <span class="s1">14</span>. Find the value of \(b\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Sam’s class collected <span class="s1">745 </span>cans. They want an average of <span class="s1">40 </span>cans per student.</p>
<p class="p1">How many more cans need to be collected to achieve this target?</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">The students raise $0.10 for each recycled can.</p>
<p class="p1">(i) Find the largest amount raised by a student in Sam’s class.</p>
<p class="p1">(ii) The following cumulative frequency curve shows the amounts in dollars raised by all the students in the school. Find the percentage of students in the school who raised more money than anyone in Sam’s class.</p>
<p class="p1"><img src="data:image/png;base64,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" alt></p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) Write down the new mean.</p>
<p class="p1">(ii) Write down the new standard deviation.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">valid approach <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p2"><span class="s1"><em>eg</em>\(\,\,\,\,\,\)</span>between 10th and 11th, \(\frac{{8 + 8}}{2}\)</p>
<p class="p2">median \( = 38\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></span></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> \(a = 20\)</span> <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N1</em></strong></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>valid approach <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\({Q_3} - {Q_1},{\text{ }}{Q_1} + 14,{\text{ }}b - 30 = 14\)</p>
<p class="p1"><span class="Apple-converted-space">\(b = 44\) </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">valid approach <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>x\(\,\,\,\,\,\)\(40 \times 20,{\text{ }}\frac{{x + 745}}{{20}},{\text{ }}40 - \frac{{745}}{{20}}\)</p>
<p class="p1">correct working <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(800 - 745,{\text{ }}20 \times 2.75\)</p>
<p class="p1"><span class="s1">55 </span>(more cans) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>(i) <span class="Apple-converted-space"> </span>most cans in Sam’s class \( = 50\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p2"><span class="s2">5 ($</span>) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>correct value of 64 or 16 <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p2">valid approach <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\(\frac{{64}}{{80}},{\text{ }}80\% ,{\text{ }}80 - 64,{\text{ }}\frac{{16}}{{80}}\)</p>
<p class="p2"><span class="s2">20% <span class="Apple-converted-space"> </span></span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p2"><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><span class="s1">(i) <span class="Apple-converted-space"> </span>41.4 </span>(exact) <span class="Apple-converted-space"> </span><strong><em>A1 N1</em></strong></p>
<p class="p2">(ii) <span class="Apple-converted-space"> </span>18.5 <span class="Apple-converted-space"> </span><span class="s2"><strong><em>A1 <span class="Apple-converted-space"> </span>N1</em></strong></span></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Generally, candidates were very successful with this question, appearing to move easily between the three different representations of data. The main conceptual errors appeared in part d) where a percentage of 100 was found (instead of 80) and in part e) where the new standard deviation was often given as 20.5. Arithmetic errors seemed to be the other factor, with a surprising number of candidates finding in part c) that \(800 - 745 = 15\).</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Generally, candidates were very successful with this question, appearing to move easily between the three different representations of data. The main conceptual errors appeared in part d) where a percentage of 100 was found (instead of 80) and in part e) where the new standard deviation was often given as 20.5. Arithmetic errors seemed to be the other factor, with a surprising number of candidates finding in part c) that \(800 - 745 = 15\).</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Generally, candidates were very successful with this question, appearing to move easily between the three different representations of data. The main conceptual errors appeared in part d) where a percentage of 100 was found (instead of 80) and in part e) where the new standard deviation was often given as 20.5. Arithmetic errors seemed to be the other factor, with a surprising number of candidates finding in part c) that \(800 - 745 = 15\).</p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Generally, candidates were very successful with this question, appearing to move easily between the three different representations of data. The main conceptual errors appeared in part d) where a percentage of 100 was found (instead of 80) and in part e) where the new standard deviation was often given as 20.5. Arithmetic errors seemed to be the other factor, with a surprising number of candidates finding in part c) that \(800 - 745 = 15\).</p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Generally, candidates were very successful with this question, appearing to move easily between the three different representations of data. The main conceptual errors appeared in part d) where a percentage of 100 was found (instead of 80) and in part e) where the new standard deviation was often given as 20.5. Arithmetic errors seemed to be the other factor, with a surprising number of candidates finding in part c) that \(800 - 745 = 15\).</p>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>Pablo drives to work. The probability that he leaves home before 07:00 is \(\frac{3}{4}\).</p>
<p>If he leaves home before 07:00 the probability he will be late for work is \(\frac{1}{8}\).</p>
<p>If he leaves home at 07:00 or later the probability he will be late for work is \(\frac{5}{8}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><strong>Copy</strong> and complete the following tree diagram.</p>
<p><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAVgAAAEKCAYAAABXKk28AAAgAElEQVR4Ae2dD3BT95XvvzI7JCTEpARSJMwrQ1hMEpzSmHjz4mwtcCrHk6bhwQIbiu3WhibTECgplrELCZMlDrYZqAkhvBA7gAkUqD3wsq+Awa4zzyTA2g2JmYK0boYdjOQ+/hSEHgF2pd+bc6VrS7Yly/p7r3TujMfSvb8/5/f5XR39dO75naMRQgjwwQSYABNgAmEnkBT2FrlBJsAEmAATkAiwguUbgQkwASYQIQKsYCMElptlAkyACbCC5XuACTABJhAhAqxgIwSWm2UCTABwmmuRo5mPWvPtIeCwwXxsM8p2nodzCLWUWJQVrBJnhWViAolMwNaGmoL1aHeoHwIrWPXPIY+ACTABhRJgBavQiWGxmEB8ErgLa3sDNhSkQaPRuP5mlqC22Qw7DdjWjJKp2ai0WtFY9CiG6crQbHMZCpzW09hZkuOup8PMklo0m22KxsQKVtHTw8IxgXgi4IS9/X0sfPEQhr3WCIcQEOIOLGvuQ132G9jWfh1InoWK800warUw1JyDw1KOWclJcF5qwJL0IjSPexMWB9U7jw9ST2CRvgwNl+4qFhIrWMVODQvGBOKNwDWc3v8JTHkFKMrQwqV8hkOrX4A8w3kc+6rbx0OtK2jZXI7atBX4zfJMaKWKyZhaWIHdeSexdHMrlLqO/bt4m0IeDxNgAkolMAazKtpggQ3m5kM4fp2eYl3FqffeQmULYFjgQ27b1zha1w5t3kSM81oSjkRK6iRYVx9H22/00krXRwsxO+0lbsyk4I6ZABNQPIETJ07g22+/DUFOJ+znd6JANwqp2Vtx6jrZVh9GzgcNqDEAHSaLyw7rowdrZTZGyXZb6f8IpBYd8FFaGad5BauMeWApmIBiCZjNZlRVVeHy5cuora3FiBEjgpPVacb+5aU4lluPru1zMF5e3tGDrQ4rMN1fs2STbcbhwqlu04K/ssq5Jg9RORKxJEyACSiCwLVr17BlyxYsWrQIubm5OHjwIEaPHh28bHYLTB1AWuZjbjuqqynnzeu44q/V5CeQk6dDx4k/w+q18+A62jf8GJqcWpi9zvtrLLrXWMFGlzf3xgRUQaChoQHPP/+8JOtnn32GOXPmhC63W1E21u1Di9X15N9pPYHqsrdQa/VofqQOqWn3uU7cssPuHAP9sjLkHn4LZdUn3ErWBnNDJVYWA1XlczBFoZpMoWJ5wOaXTIAJRI3AmTNnMHv2bJw8eRKkZJcuXRq8SaCf1GOg/9VW7Mj4HNm6eyR/1pRVX2BCwVbUG9N7SydNQm5JHu6SH+z9hdjfeRtJ42ejuqUaWd1vQzeM/GdHQX9oNF5v24430h/srauwVxoOuK2wGWFxmEAMCHR1dWHz5s0ge2txcTEyMzNjIEX8dckr2PibUx4REwiYAHkF7Nq1CxMmTMDTTz+NvXv3snINmN7gBVnBDs6ISzCBuCTQ2NiIrKws0Or16tWrkp01aA+BuCQU+qDYTSt0htwCE1AVAdntioTevXs3pkyZoir51SQsK1g1zRbLygRCIEBuV9u2bZPcrdatWweDwRBCa1w1EAJsIgiEEpdhAiomQHZW8gh46KGHkJKSAnK7YuUanQnlFWx0OHMvTCAmBGh7K+3CIjPAxYsXJQUbE0EStFNWsAk68Tzs+Cbg6Xa1du1aTJ/udx9qfMOI4ejYRBBD+Nw1Ewg3ATIH0PZW2nlFble0vZWVa7gpB94eK9jAWXHJOCPQE1HfK0KTO8p+FM6FGyfZWcntio4jR46EZ3truIVMsPbYRJBgE87D7SUghOh9o+JX5HZlNBoxduxYdrtS2DyyglXYhLA4TCBQAuR2tX79erS0tGDTpk28AytQcFEsxyaCKMLmrphAOAh4ul1NmzZNcrvi2AHhIBv+NngFG36m3CITiBgBcrtasWIF9Hq9tL01pPisEZOSG5YJsIKVSfB/JqBgAvL2VsoqwNtbFTxRfURjBdsHCL9lAkoiQHbWPXv2SBGvVq1axZ4BSpqcAGRhG2wAkLgIE4gFgYhkFYjFQBK4T17BJvDk89CVSYCyCtDuK9reyuYAZc5RoFKxgg2UFJdjAhEm4Lm9lbMKRBh2lJpnE0GUQHM3TMAXAc4q4IuM+s+zglX/HPIIVEyAswqoePICEJ1NBAFA4iJMINwE2O0q3ESV2R4rWGXOC0sVpwQ4q0CcTqyPYbGJwAcYPs0EwknAc3srZxUIJ1llt8UrWGXPD0sXBwQ4q0AcTGKQQ2AFGyQ4rsYEBiPg6XbFWQUGoxWf19lEEJ/zyqOKIYG+WQX27t3LWQViOB+x7JoVbCzpc99xR2CgrAIjRoyIu3HygAIjwCaCwDhxKSbglwBnFfCLJ2EvsoJN2KnngYeDAGcVCAfF+G2DTQTxO7c8sggS8HS74qwCEQSt8qZ5BavyCWTxo0+AswpEn7lae2QFq9aZY7mjToDsrB999BHoP4cRjDp+VXbIJgJVThsLHU0CZGfdsmULFi1ahKeffhoHDx6UYrVGUwbuS50EWMGqc95Y6igR4KwCUQIdp92wiSBOJ5aHFRoBzioQGj+u7SLACpbvBCbgQUB2uyI7K2cV8ADDL4MiwCaCoLBxpXgjIGcVeOihhyQ7K21vzczMjLdh8niiTIAVbJSBc3fKI8BZBZQ3J/EiEZsI4mUmeRxDJkBmgKqqKly+fJndroZMjysEQoAVbCCUuExcEeCsAnE1nYoeDJsIFD09LFw4CcjbW59//nlwVoFwkuW2fBHgFawvMnw+rgh4ZhUg31ZSsHwwgUgTYAUbacLcfkwJcFaBmOJP+M7ZRJDwt0B8AuCsAvE5r2obFStYtc0YyzsoAdntigoeOXIEc+bMAWcVGBQbF4gAATYRRAAqNxkbAmQOWLp0KcaOHctuV7GZAu61DwFewfYBwm/VS+C+++5DRkYGvvrqK1y4cEG9A2HJ44aARggh4mY0PBAmAEjxWuUNBJWVlRxakO+KmBFgBRsz9NxxpAl4Zh5YtWoVRo8eHekuuX0m4EWATQReOPhNPBGgYC2fffaZFLyFgrjs2rUL5F3ABxOIFgFWsNEizf3EhAB5D5AXwdWrV3H27FlkZWWBVrZ8MIFoEGAFGw3KYejDaa5FjmY+as23w9CaDeaGMszUaKDR6JBTex7OMLSq5CbIPED2WMqlRfbZJUuWSLZaJcvMsqmfACtY9c/hkEfgNP8er8/9FJPqL8AhLDhaOBWJciNMmTJFyqk1b948KcdWeXk5KPgLH0wgEgQS5XMVCXYqb/MejHnw/oRRrH0ny2AwSPZZiklA9lmKT8D22b6U+H2oBFjBhkowqvX/E9fO/gEbCtKgoZ/3ugJsOGaG3UuGu7C216Fkps5VRpODktpmmO1kBLgNc+18DEstQiPaUZk9FhpdGZptdM0Gc3Otj3rUwRU0l8yAZmYJtm8ogE7qX67rr08v4RT1huyz+fn5uHjxIk6ePImXX36Z7bOKmqE4EIb8YPlQPgGHqUYYAAFtvth4yiIcwiFumuqFUf+I0FedEjelIdwRXfWvCW1PGSHEzQ5Rkz9NaAvrRZfDNU5XW+nC2HTZPfAb4lxNoVc9h6VVbMyfJqDfKNpuUsXLosmYLoBpIr+mQ9wUd4TFdEH6H0ifyicsxJdffileeuklUVxcLEwmkxpEZhkVTgAKl4/FcxNwKcVporD+gnDrSSGEQ9xoKhVabalouuEQ4kaTMGq1wlBzzqOMcJ/vVaj9FKxUr2/bcj25PbeClfuSZybAPuXiavhfX18vnnrqKfHee++Jq1evqkFkllGhBNhEoKpfIY8ic9p3PeymSRiZMhlp1kYcbbsCW9tx1Fl1mD5xjEcZACN1SE2zoO7o17D1G6/TXa9v23I9oMNk6WOGkBuR6w61T7m+Mv+TWxf5z9JBwbnZPqvMeVKDVKxg1TBLQ5LRbVuVXLDIDUsDzbBHUdRo9dHKXXRf6ISvq1TJeuYCuv36cQ21Tx+iKOg02WcpcAwpV9k+e+bMGQVJyKKogQBH01LDLA1JxnmoMe1C4ZR7fdby1pXDMW7iZGjR6bO8dvpEjEsCunyWGLxPn1UVfoG8DMh/ljYnrF27VoprsGzZMs6IoPB5U4p4ClvB3oX19Pso0Lkc4GeWNLiffisFV6zl+AamLk+fASfsXZ3o0BqQM2MMkmc8hzztOZw4+1fvjQP209gwc7KPDQVJfupZYOoA0lJ1GDng0P3V9dfngI0p+iRtu927d6+07ZZMCFu2bGG3LkXPmEKEU45t2CFu/umg2CE9IRdCfoqtNTaJG8oRMmaS9HgR6EtFvYmIOMTNcztEvtbz4dRAXgTnRL3R4OENIES/h1yivxeB7H3Qz4ug70MuEVifMQMXgY7pwRc9AKMHYfRAjA8m4IuAclawzv9A240nkZehlR7QJGn/O4oKXgTqjqNN8tNUyDdSTMWYjarXZ+CbdzKh0QzDA7OakbazHtVzvud+qDUc4+eUo2V3FrpL0jGM7K8PzMOhsa+gbc9rSB/pa7qTMbVwU596v4YpqxqmT5f7qUcwgu0zpiBD6py23ZJ9lrbdHj58GLNnz+ZttyERjd/KCg5XSE7xRXjl2nJ8ujLDx0/U+J0YHpl6CHBYRPXMVbQl9bWkibYc3v3ZzWiufRvvdP4Me95g5eoNh98pjQCHRVTajChHHoUpWCdszWXQPZCK7KJ3seuLP+KLzv6em8rBx5IwARcBz7CIlBuMwiJS8kU+EpuAQk0EtLd9H367sgSVpjmo/7eNmDN+eGLPFI9eVQTMZrMUFvHy5cuSmxdF8eIj8QgoVMG6JoJioOambsX0piOomDUm8WaHR6x6ArSKXb16tfQg7NVXX+W0Naqf0aENQGEmAm/hkyY/gwWGe7xPqvwd7wZS+QQOUXwOizhEYHFWXNEKFnYLTH/5Af4hNVn12Okno9FolHYDcdxR1U/nkAYgh0WktDW07ZbT1gwJn6oLK0bBOi81oEiXg5Kdp2GlvZzOS2hevw3dZa/CoGL7K0XLp10/ixYtknYBHTx4EPSB4yPxCHDamsSbc8Uo2KRRU5D5Iwsqf/YP0A3TQPfzvbg+txofF05TrQ8sBQqhaEx0UHQm2mLJBxOQ09bk5uZKX7z0Bcxpa+LzvlD0Qy61Iic7qxwYZPHixVKAELWOheWOLAEyF9XU1EgpxVetWgVSuvwLJ7LMo9m6Ylaw0Rx0pPqiVQjZWX/xi1+guLg4Au45nqlZ0lCw6YTLnNIzILcfsWeowp7XAWSkdVrRvrPElW1WV4BNp63eQWOon0DK9MjDLwYjQMq0b1hETis+GDUVXfcVpIDPB07g1q1bYufOnQKAFPyD3of/cIibbRuFXr9WNFnuCOHoEk2lL3iki6Ee5bQukGQheXr+DDXC1JsKYQDx/ibaqmYLfWmjsDgo2E6jKNXPFlVtf/MoG0gZj+L8csgEPNPWXLx4ccj1uYKyCHDKmBDn4+jRo1JUpXfeeSey6UUc50SN4RmPPFpySpd5osb0rWsUN/4oqta4FGTvsCitzBrxQt80Mr0FpFdShC2vSFnudDQeijmQMn2a5bdBEKAvaM+0NZH5wg5CMK4yZAJsIgjy1wa5XS1ZsgRbt26VoiqVlZVF1Inc2fk59jWOR2qKR2TW5CeQk3cJ+1ovSD/lnf9vAv6HMRtaz1l1mvH7ir9gzrMTvdPIeI37Njpbj6AxbTJSeiJuuWO9dhxBa+dtKSPt4GW8GuU3QRIgswE9ED1y5IjUArl10QNTPtRHwPOjqD7pYyAx2VnLy8ulp7/z5s0DuV1FfhukWwFqJ2PiuL5bhu+gcd/n6HQCSdpH8EiPgnTBIcXckDIXOZPlDAd3calhKXSaNBQ1/IfLxuq8gNZ9rZAzF3hjbXUp8EDKeFfkdyES8AyLSP6zFBaRN6qECDXK1SOqYKV8UD0PWdz5oaL0Ptwc6Wmv7HaVnJwsuV3RLp3oHHZ0mb4BvFaYgfRMivk0Hv/pDzHe30zThg6/mQsAadPHYGUCEYnLDJkAfYFT2hp6cEreKfQglQLK8KF8AhHNySUEPWNR/0FPdauqqqSVKilZytOkioNWnQ0PI2fPaA9xKUD2FljiY2o8xhX/Lyks4pNPPikF+Z4wYQJ27twJ+hXFbl3KnXt/6xrlSh0lyWiVQKsFUq60cqBVRGyU60ikpE4COjrRZfdOWegPhWQeeFyPGcmDTLOU1ttfeu5AUnj7k4SvhYuAbJ+lbbccFjFcVCPXziCfvMh1rOSWyRxAu2voQcPTTz8tJbubPn16DEW+F5OffR79DBLOK7hw5joMC57B5H4z6TYP5DyBQSM5JE3Eswue7Tc+Z/cFnLE+iwX0gCyQMv1a4BORIkD2WXqwSmlrDhw4wGlrIgU6xHb7fSxDbC+M1d0PY3JqYQ580RZy/xRejp7a2mw26SkuKVkl/ASTIoulfYajbdd6xyjZTp90KcDes65Xsnlghqd5oG8h+b1Lgad55T9zZ6w1PI9npQdkgZSR21PH/1g+I6C+w3GQfXb79u2SfZbiXdAvLt52Gw6yYWpjyI5dUarg6KoXhVoIePhhRrJrk8kkXnrpJbF48WJBr5V3BLLRoFdq8ll9YcCMvO4ssPDMRkv1AtlEEEiZXhn4VXQJRGfDS3THpPbelLnR4OYpUZW/XBjzp0VcwVIK5uLiYmmzQGtrq8Ln846wnNoi8umLBwZh3HFK2nXVX+hvhanG2GcXllzKl4KlTOAWcWpjvtDSDjC9Uexos4h+m78CKSN3xf9jQkBd93RMEEWtUwUGe7mO9g1r8ceZv8ST+xci+8wvYTpciClhNmaQnZVSLs+dO5efxobp1xA3oywCcgzisWPHSiaEyPtrK2v8SpAmzGor1CE5YW/fhc1YiFfTA7EdBtcfuV2RnZWct+lpbH5+viLsrMGNhmsxgYEJkEKljTDkykX2Wdogw/bZgVlF6mxE/WCHLLS9Dds2A8ven4GR8HiYM+SGBq5A3+gfffQR6P+HH36I2HoGDCwjn2UC4SZAG2L+8R//UQqL+NBDD6G+vp7DIoYbso/2FLSCvY727c2YVP4q0vts9/Qhe8CnZbcrz6wCrFwDxscF44AAecJQWMSLFy9Kv9xefvllcFjEyE+sQhQsmQZ+h/rvvYzZYU4PQzuvyBxAB2cViPwNxT0omwBtlKENM7RxhjbQkFsX/aLjI0IEovY4zW9HfuKYSjFNtcIwSLi9vs17xtVUpttVX4n5PROIPgHPsIjkfaDo46ZJNFatEzvk8JwDCCuF1IRHCM8ByvQ/dUOYGqtF6Y5z/b1m+hce0hmFrGDHYFZFG7mMefxdRpMxHTDUwOSw4GjhVD/h9nq/fSKfVaC3L37FBNROgDbS0C87Oih/nHLDIjphO70DBcVfwRFu6LY21BSsR3vYG0ZAOivcw4lIe2Rn3bVrF8iIT9tb6aah4Bh8MAEm4J+AbJ+lbbccFtE/q6FeVcgKdqhie5eX3a4o+AW5XSlle6u3lPyOCSibQPjDIso54uZje3MLdpbkwLU9OQ0FG47C7BW4yAZzcy1KZurcZXJQUtvsLkPtrMbU7HdhxQEUpY6ArqQZtoBwUh67BmwoSHO3q4FmZglqm82wU31bM0qmZqPSakVj0aMYpitDs821N99pPe0hsw4zS2rRbA6s1x7RhmRQUFhhsq3S1lba4sp2VoVNDoujagLhSVvjTjsk7QwsFfWmGxITV763Rzzyyd0Q52oKhVabLzaecu0edFhaxUbayanfKNpu0n5CuS3/9lVvG6x7e7lHu0LcEZamtUKPF3p3Ot5oEkat93Me11b9aSJ/Y6t7t6Qs42uivutOwHOryhVsbLIK9Hwn8QsmEPcE5LCIlLaGAh+RJw4FQgruSIdxzRuYM8UV1y1J+wM8l/EgWo6dhYUWi7Y2fLz6NHK3vI3lGVrJbpmkzcSK96phNFWhbL+5f3bjgAS5htP7P4EprwBF7naB4dDqFyDPcB7Hvur20e4VtGwuR23aCvxmeaY7BVMyphZWYHfeSSzd3Brg6lllNliys8Yuq0BAM8qFmEBcEegbFjE8mRQ84xv/F2xtx1FnfRSZ077r/VAokDjFfmm7Hp5bKmagu/mQpDsaGrajJHsWihpv+a5p+xpH69oHSKHkktvqFXXOdzN0RTUrWMpFRM7RZIQnJUtO00oII+gfL19lAvFBQA6LGP6A83fRfaETVj+YrGcuoDuokKVO2M/vRIFuFFKzt+LUdWrkYeR80IAawyAB5gFYK7MxyivF1QikFh3wI2n/S8raKttfPilq++bNmyVnaHKO5h1YA0DiU0xAtQSGY9zEydCi0+cIBk7G6bN47wWnGfuXl+JYbj26ts/pzUtHD7Y6rIDfGPpaGGqacThA99DeTr1fKXYFK29vVU5WAW9w/I4JMIFwEHCnh9eew4mzf/W2iQaSjNOfCHL9zMe8Utk7b17HFX/1kp9ATp4OHSf+DKvXypki/f0YmiEkAVCkglVqVgF/c8LXmAATCJJA8gz8fF0GDi99E9WnrS4laz+L2teXozK1GOXzpyAJSRiZMhlpUhdO3LLf8lbGA3XtVpSNdfvQYr3rqmk9geqyt1DraZOQbL33uVq4ZYfdOQb6ZWXIPfwWyqpPuJWsDeaGSqwsBqrK5wQcPlVRCpb2RFPud8oxRE7PlHOIjOx8MAEmEM8E6An9JrTszkJ3STqGkd3zgV/DlFUN06fLe4I/JU3OQUnpDRSl3o/75/4OnV6ry4H4jIH+V1uxI+NzZOvukfxgU1Z9gQkFW1FPu0TlI2kSckvycJf8YO8vxP7O20gaPxvVLdXI6n4bumEaaDSjoD80Gq+3bccb6Q/KNQf9r4iA2+R2tW3bNil25aZNm3gH1qDTxgWYABNQA4GYrmBltyva3kpPJ3l7qxpuGZaRCTCBQAnEzIuAtreuWLECer1e2t7KpoBAp4zLMQEmoBYCUVewnFVALbcGy8kEmECoBKJmIpDdrjirQKhTxvWZABNQC4GoKFjPrAK0t5l8W/lgAkyACcQ7gYiaCGh7K+2+om125HZF//lgAkyACSQKgYgoWHK7Wr9+PVpaWsBuV4lyK/E4mQAT6EsgrCYCsrNyVoG+iPk9E2ACiUogbCtYdrtK1FuIx80EmIAvAiErWHK7ovS/ly9fZjurL8p8ngkwgYQkELSC9dzeum7dOhgMhoQEyINmAkyACfgiMGQbrLy9lVL8JicnS9tbWbn6wsvnmQATSGQCQ1rBerpdkW9r+KObJ/JU8NiZABOINwIBKVjKwyNnFSguLuZoV/F2F/B4mAATiAgBvyYCeXurZ1aBzMzMiAjCjTIBJsAE4o2ATwXLWQXibap5PEyACUSbQD8TAbldGY1GjB07lt2uoj0b3B8TYAJxRaDfCvbWLVe+8O985zu47z53npq4GjIPhgkwASYQHQL9FCylxd67dy+efvppKerVli1bQLZYPpgAE2ACTGBoBPopWKo+YsQISblSaEGbzYasrCyQTZYPJsAEmAATCJxAQEkPPbfDVlZWctjBwPlySSbABBKYQEAKVubjGdBl1apVnFJbBsP/mQATYAIDEBjQRDBAOekU+cBS5leyz1ImWApNyPZZX7T4PBNgAolOYEgKlmDJ9tmrV6/i7Nmzkn2WVrZ8MAEmwASYgDeBIZkIvKu63nn6zdI2Wk4LMxAlPscEmEAiEhjyCrYvJFKoBw8exLx580AZY8vLy0GhDPlgAkyACTjNtcjRzEet+fYQYNhgPrYZZTvPwzmEWkosGrKClQdFIQvJPksRtsg+S9G22D4r0+H/TIAJBEzA1oaagvVodwRcQ7EFw6ZgaYRkn83Pz8fFixdx8uRJts8qdtpZsOAJOGFrLoNOo4FG/suphVntS63ggXBNPwTCqmDlfmgVS/6yH374oZROhmIbkK2WDyagegLOizj+yaew9gxEC8OCZzA5Ip+knk7i6MVdWNsbsKEgrfcLamYJapvNsNMobc0omZqNSqsVjUWPYpiuDM0217eX03oaO0ty3PV0mFlSi2azTdFsInpb0LZbss+SWxfZZ2nbLdtnFX0/sHB+CThh/7IB/3NMNW4IASH9WXC0cCoi+kHyK5OaLjphb38fC188hGGvNcIh8bsDy5r7UJf9Bra1XweSZ6HifBOMWi0MNefgsJRjVnISnJcasCS9CM3j3oTFQezP44PUE1ikL0PDpbuKhRCV+4LiyZJ9lg5KNcP2WcXeDyyYXwLXcHr/J2isrMA7tQ2KXz35HUpMLrr4mfIKUJShdX8pDYdWvwB5hvM49lW3j4daV9CyuRy1aSvwm+WZ0EpaKxlTCyuwO+8klm5uhVLXsVFRsDSXZJ9dunSppFzJPvvyyy+DUtDwwQTUQsBp/l+oqGwH0IjKornITp2Hkobzrp+2ahlECHJSZpPQHlyPwayKNlgqZqC7+ZCkCxoatqMkexaKGl1R/AYUz/Y1jta1Qzt9IsZ5aayRSEmdBGvdcbS5zQgD1o/hSS9xoyGHbJ8ln9m1a9dKsWdp4vhgAkonkDSlEEeFgMPShoM1RuhJ0c5d6fppq3ThQ5CPzHpk3qNforTBKPjDCfv5nSjQjUJq9lacuk621YeR80EDagxAh8ni98vKWpmNUfKDRen/CKQWHQhenCjUjLqClcdE2245LKJMg/+riUCSNh0vFVagydKIUv2fsHH/nxT7EzUUrrRaJXMemfXokN0wg27Tacb+5aU4lluPLsdRVBT+E+bMeQmzdLdg6uh9bDhw+26bbI/tW7aBCwi3nXbgerE9GzMFS8OWt91SWEQ6KCwiTSgfTEANBJK02Vi15meAgn+iBsuRzHdkxiNzHn0mybxHn9eQDrsFpg4gLfMxtx3V1Zrz5nVc8ddw8hPIydOh48SfYfVyh7uO9g0/hkbBbnIxVbAy09GjR0sTuHv3bhw+fBizZ89mty4ZDv9XMIEkjEyZjLS0yU1I6pAAABo4SURBVEgZqYiPUsisyFxHbpVkviMzHrlbklkvLIdbUTbW7UOL1fXk32k9geqyt1DruYAdqUNqmjubyi077M4x0C8rQ+7ht1BWfcKtZG0wN1RiZTFQVT4HUxSKX1Fi0bbb7du3SxNLbl000ezWFZZbmxuJCAEn7F3/F+klP1HsBzzQYZM5QLazklslme/Cn0F6DPS/2oodGZ8jW3eP5M+asuoLTCjYinpjeq+oSZOQW5KHu+QHe38h9nfeRtL42ahuqUZW99vQDaNNHqOgPzQar7dtxxvpD/bWVdirkIO9RGo8NOG0mp07dy527twpxToI+SdKpITldhOAADnIH8dp/AAvppOL0V1YT9fit//6CH619kdeP3nVBoNMAOvXr5d2YS5cuJDjPIdxAhW1gvUcl2yfpaeW9LOF09Z40uHXMSFw49/w2xk6DNNooCvYjBP2WVjztnqVK+2uJHMcLWTIPEd2VjLX8RE+AopdwfYdohwWkc5z2pq+dPg9EwicAJndaMXa0tKCTZs2RcAUELgs8V5SNQpWnghKvrh69Wrpm/fVV1/lb1wZDP9nAoMQILPbgQMHUFBQgPr6euTm5obuGTBIn4l+WbEmAl8Tw2ERfZHh80zANwFamJCZjcxtZHajTQP8TMM3r3BdUd0K1nPg/FPHkwa/ZgL9CbBprT+TaJ5RtYKVQck30dixYyUXL05bI5Ph/4lKgBYf27Ztk6LZrVu3DvTLj4/oE1CdiWAgRHLaGrIpcVjEgQjxuYEI9ATM9trf7hFIO8LnB5Ip1HPy9lbKKkIbBGh7KyvXUKkGXz8uFKw8fA6LKJPg/4EQcMVz9djTPtA+9wieC0TGoZSh7M5kZ6XtrWRnpewibGcdCsHwl40LE8FAWMiYv3nzZmnLLW35C/+ulIF65XNMIPoEyERWVVWFy5cvswtj9PH77TFuFaw8agpaQfuqyYywbNmy8O2rljvg/0wgRgTIzrpnzx7s2rULq1atkjwDYiQKd+uDQFyZCAYaI6Wt4bCIA5Hhc2olINtZPcMIknmMD+URiHsFS8jlbbccFlF5N6C6JHLCbj7qkbAvDQUbjsJs94qhF9EhkZ017GEEIypxYjce9yaCgaaXbFYfffSRZJ8l8wGtcvlgAoMRkBLvPfUW/mvdXrxfOA0jnZfQvLoQixwlOF8xC8mDNRDCdc9nCnzPhgAy2lVFAh+tra3ipZdeEsXFxeLixYuKJuGwnBI7jAYBQGjzt4hTljse8jrEjaZSoQWk61Sm92+eqDF961F2gJcOi2jbYRR6qqfNFxtPWYSjX7E7wtK2Sxj1WgFME/kbW4Wlf6F+teLnxLfCVDNPQFsqmm70DtxhqhGGPufCOeZbt26J9957Tzz11FOivr5e0Hs+1EMgIUwEvr60PNPWTJgwQXpYQPYtxR3209i48G2YcmrhEHfQXnAFJQvfR3vPT9NraDvaCM+YxT1jMDyPZyff2/O2/4vraN/4S6w0PYc9DgFH+yJcLvklNlIK5Z7DnW555TfI2XMBwnEEBZffxcKNp/3mUOqpHhcvhmPcxMnQWr/Gn/5dzmFK8WA78Ze85zAjOfwfJQojSG5XdJB5i7e3qvBGUs93QWQlvXr1qnjnnXeklcLRo0cj29mQWnetnLTGJnGjp95l0WR8RhhqzrlWmjf+KKrWNPZZUdKqdo14QS7TU9f7Rf8VmHs1bKgRJnmh5jgnagzPCGPT5d7KN5qEURvA6ri3hvpf3TwlqmgFry0UNeduCIelUawxftDn10Tow/zyyy97flmZTKbQG+QWYkYAMetZoR3TDb148WLpBlfEzS0pt0d6lanEzVsJOiydovOmrA3dYKV6CwcxD7h/9noqU6reR3lKShh9lWkfJa/Q+Qy3WA5Lq9iYP81lgunLLcTOyExF5ioyW5H5ig/1Ewj/7xoVruI9RVZa2hpn5+fY1/ggpk8cg36T1XgErZROQ/sIHumTE4rqNaTMRY6HeYAe0hTpNNAVNeASPfh2XkDrvtYB8s0TkVbsa70AJ26js/UIGrWTMXHccE9UAO6gcd/n6IzeQ/Q+/Uf/bdIDD+F7aQWoMhqAxtV4ZfWxPon4hi4TmaXIl5XMVJFL1zJ0ubhG6AT6fWZDbzI+WiD7LO3jnjZtGmhfN9nDom+fddn4OjAJqSkjhwCWlOJpPP7TH2K8vxmWs3ym6uC7dTu6TN8AcZTYbwggvYvaz6J2+cfAy8uwsuJTWJpeAd4tCMkWzWEEvRHH2zt/H794G+uQx0P+s7Sfm/Z10/5ueuBAfoiKP2hl2vAwcmZ4p/9IGj8HNRYBS80c/4pX8QOMhYC3Yd7/Noq6UjFNSyv54dDOWoNP24qB4g3Yb749JKHIVZDStVAAbErXUlZWxsHjh0RQHYUjqmBjGa0onPgpTxGlqaEPAu35XrJkSZTSirvTQuMbmLrsAQ9JMg88rh/8ybaUHhnoMFn8eAOMRErqJKCjE109XgsBixJHBd0rea8RJWHk338fGVqvk37f0PZWypZMUd8oRgZlUebwmn6RqfpiRBVsLKMVRWJW5LCI8+bNkz4g5eXlEU8rnjT5GSww3NNnOHfRfaET1gFdsNzmgZwnBnd8T5qIZxc826dtwNl9AWesz2LBsxORhHsx+dnn0S+aqPMKLpy5DsOCZzA5ondRP/FidGI0Mub/FPrGTXhnx1n3F5IN53//Cepz/9nL1j2QgPL2VjI3kdmJzE8cgGggUvF1LiE+GuGeMjltTXJycuTts5ISHI+6o19D9r4EaDV1aWDl5sM8MDADl/JMqzuONpv8pMpt9/VQ3pKST/sMR9uu9TYj2W+fdCvh3tPx+yoJI9Nfw542I8bVGfCAFCs2E+9em48/VM/2a3LhMILxe1cMOjL1O0LEdgRRca2R/C9fEKVNXcIh7ghL01qh128UbX1ds4QQ5FL1gpfPbC8fR1e9KNRCaAvrRVePV9ffRFvVbKEvdfnRkm9nqX62qGr7W29F4RA32zYKvX6taKIdZI4u0VT6gtBXnRI3PUrxS28CinP58xaP30WBAPvBhglypJ3De/0vtUJv3CXavLbKyoMgv1ZjH+UoXyO9OJCCJa1sEac25ru22uqNYkebj62yp7aIfC1twzUI445TfTY29PaT6K+Uu2kl0Wcm+uNPyGAvgy7rQyhA7lyUc568DxYuXMhPhkNgqbaqZGc9fPhwz/wXFRVxRgG1TWKY5WUbbJiBDpS2JsxdcHMKJMBhBBU4KQoQiVewEZwEDosYQbgKaZrnWCEToVAxWMFGYWJodUP+s5y2Jgqwo9QFp2uJEmiVd8MmgihMoGdYRDIhRH/LbRQGmUBdkJ2d07Uk0ISHMFRewYYAj6smFgHPBJqLFy/mHViJNf1BjfbvgqrFlZhAAhHwTNfCKeATaOLDMFQ2EYQBIjcRnwTIlLNlyxYpkwCHEYzPOY70qFjBRpqwwtp3mmuRo5mP2iFFf7LBfGwzynaeh7yhVmHDCrs4chhBm83G6VrCTjdxGmQTQeLMdfAjtbWhpmA9zqzrF/Il+DYVWpPcrija1dixY6XoaRzpSqETpRKxWMGqZKJYzMgSILcr2oHX0tKCTZs2caSryOJOmNbZRKCKqbbB3FyLkpk6uGLs5qCkthnmnvisV9BcMgOamSXYvqEAOor0pCtDc0+ELH+DvAtrewM2FKS529ZI7dQ2m10h+WzNKJmajUqrFY1Fj2KYR7tO62nsLMlx19NhZkktms1yzC8nbM1l0GlyULJ9PQp0Gmg0M1DSfMWfMFG/RnZWStdCYQTJzsphBKM+BXHdIStYxU+vDedrV0C/6DOMq2iHQwg4LG9i3GfLkfpitUfqbgAtf0Dr6GKYxR1YWl5BxqCppN3puF88hGGvNUptC6q75j7UZb+BbZS6O3kWKs43wajVwlBzDg5LOWYlJ4Hyey1JL0LzuDdhcQgIcR4fpJ7AIn0ZGi7d9aDaiLpWLUrNDjgs+/CLDO8sCx4Fo/5StrOSlwBlreC02FGfgvjvMPrxZbjHIRGQMrxOE4X1F1wpuuXK0nmtO9ssZXhNF9CWiqYbPXEI5ZJe/70zxLrqeacEp+halKbbI5OtV1/UnLu/fllVPdtzZ75Fune6by9pYvOGwwjGhnsi9so2WEV/hzphazuOOuujWDftu95ZZd3pXuqkdC9jghzFGMyqaIMFZII4hOPXHQCu4tR7b6GyBTAs8NGs7WscrWuHNm8ixnn9BnKll7GuPo623/wQM3xUj9VpsrNu27YNBw8exLp160CB0/lgApEk4PXxiGRH3HYwBNypYfxUtZ65gO6gfaecsJ/fiQLdKKRmb8Wp69TQw8j5oAE1hsFydQHWymyMkiL7k32V/kYgteiAH2ljc4nsrLS9leysKSkpkp2VlWts5iLReuUVrKJnfDjGTZwMLTp9Sqmd7lpFdvks4eeC04z9y0txLLceXds9Ms3Sg60OKzDdT12QTbYZhwuneq+se6o4PVLc9JyM+gsKtLNixQro9XrJzkoJLPlgAtEiwCvYaJEOqp8kJM94Dnnaczhx9q/eTv5STiwgLVWHkUG1DUBuI/MxaD3uBOfN6/D7rD/5CeTk6dBx4s+weq2er6N9w4+hyamF2et8sAIGX0/2Z6UoZh9++KGUFZiVa/A8uWZwBDw+VsE1wLUiTCB5Bn6+LgOHl76J6tNWl5K1n0Xt68tRmVqM8vlTfKwgA5DLrSgb6/ahxep68u+0nkB12VuotXrUl+y997lO3LLD7hwD/bIy5B5+C2XVJ9xK1gZzQyVWFgNV5XMwJUZ3FtlZaXsrpcUmtyuyt06f7ncp7jFQfskEwksgRh+D8A4ivltLxtTCTWjZnYXuknQMI1vnA7+GKasapk+XI31kKFM4BvpfbcWOjM+RrbtHsqOmrPoCEwq2ot6Y3os1aRJyS/Jwl/xg7y/E/s7bSBo/G9Ut1cjqfhu6YWR/HQX9odF4vW073kh/sLduFF9RtCsKI/jVV19JdlZyu+KDCcSSAIcrjCV97jvsBDgnWtiRcoMhEAhl+RNCt1yVCUSGwEA50TjAeWRYc6uDE+AV7OCMuIRKCXAcV5VOXByJzQo2jiaThzIwAc9MBMuWLZN8YQcuyWeZQHgJsIkgvDy5NQUSIC+CvXv3Sl4FZEIgLwM2GyhwouJQJFawcTipPKT+BEaMGCEFczly5Ih0MSsrS9rd1b8kn2EC4SPAJoLwseSWVESANiJ89NFHoP9r165lX1kVzZ2aRGUFq6bZYlnDTsBzK+2qVavAu73CjjihG2QTQUJPPw8+MzNT2pRAu74oGAwF32b7LN8X4SLACjZcJLkd1RKQ7bMUdJtcu8g+S8G4+WACoRJgE0GoBLl+3BEguywFibl8+bIUJIYTH8bdFEdtQKxgo4aaO1IbAVrFrl69Wgp1yPZZtc2eMuRlE4Ey5oGlUCABCspNSRCnTZsm2WcpzgHbZxU4UQoWiRWsgieHRYs9AbLP5ufnS8G6T548KdlnyfOADyYQCAE2EQRCicswATcBOZD32LFjUVxcDLbP8q3hjwArWH90+BoT8EFAts/Onj0br776KvvP+uCU6KfZRJDodwCPPygCsn02OTlZCvLN9tmgMMZ9JV7Bxv0U8wAjTYDDIkaasHrbZwWr3rljyRVGwDMs4uLFi9k+q7D5iYU4bCKIBXXuMy4JUFhESrJI224p6SKHRYzLaR7SoFjBDgkXF050ApR1d1NBmpQgUjOzDA1mWz8kFHOWwyL2w5KQJ1jBJuS086CDImD/Cgcbh+GfP+6AcFhw6ieXsFS/Hs02Z7/mKCrX0qVLsXv3bpD/LHkbkAmBj8QiwDbYxJpvHm3QBG7jLy1ncO8Pn8Z4eVlia0bJVCOw+wgqZo3x2zJtTqD4BuQ3y2lr/KKKq4vyrRJXg+LBMIHwE7gXj+g9lCsAZ/cFnEn9KeZnjB60OwqLyGlrBsUUdwVYwcbdlPKAIk/ABnNzLUrfsaJkz2tIHxnYx0gOi0j2WZvNxmERIz9RMe+BTQQxnwIWQFUEJLNANiqtJLUBxvpqrJkzFSODGASHRQwCmsqqsIJV2YSxuMog4LSeRt1v1+BnlRYU1v8rts/5HgJbx/aXn9PW9GcSL2eCvSfiZfw8DiYQFIEkbQYK3q1GjeEqDp/6C+xBteKqxGlrQoCn8KqsYBU+QSyeggkkTcSzC54Ni4CyfZbS1pw9e5bDIoaFauwbYQUb+zlgCVRLwI4u0w3k/sMjQdlgBxo2+c9WVlZK/rPk1rVkyRIptfhAZfmc8gmwglX+HLGESiDg/A80FM3AzJI6tFvvArgLa/P7qOheiGLDhKDtr76GRv6ytO123rx50rbb8vJyXLt2zVdxPq9QAqxgFToxLJbCCCR9B49lfh+mynzM0N0DjW4JPrn+Y+z4uABTA3TTCmZEcljElJQUTlsTDMAY12EvghhPAHfPBAIlwGERAyWlnHKsYJUzFywJEwiIAIdFDAiTIgqxiUAR08BCMIHACQwUFpHts4Hzi2ZJVrDRpM19MYEwEqCwiJRWnI7nn38enLYmjHDD1BSbCMIEkptRHwGNRhNToYUQYevf0z67du1a0CqXj9gTYAUb+zlgCZhA2AhwWMSwoQxLQ2wiCAtGboQJKIMAh0VUxjzIUrCClUnwfyYQJwTkbbectib2E8omgtjPAUvABCJKgMMiRhSv38Z5BesXD19kAp4EnLCbj2KDnPRQk4aCDUdhtvfPyeVZK9avadvt9u3bUVxcLG27NRqNvO02SpPCCjZKoLkb9RNwXjqI5fqV6Mjai5tCQDiOoODaRuj/pQX9c8sqb7wcFjH6c8IKNvrMuUdVEriNzqO/Qy1eRME/PeaKnpU0HvqfLUBa3XG0DZBZVonDlO2zFBaRXLuysrLQ2NioRFHjQiZWsHExjTyIyBMYjnETJ0Nr/Rp/+nd5veqEvasTf8l7DjOS1fVRorCIZWVlUljEAwcOSGnFyVbLR3gJDFtLXsl8MAEmMAgBDe4Z+wDEF5tQ/N43+G8vzMT3//P/oPwTJ0p/9RImPTBskPrKvPzQQw/hJz/5CUjhLl26FJcvX8bjjz8OWunyEToBdX3thj5eboEJBE9gZAbe2HMAG390GkWPjsKwn13EondfRYZ2ePBtKqSmKsIi2s04tuEd7DTf9knNaa5FjmY+av2U6V/ZBvOxzSjbeR7hflzJCrY/bT7DBHwSSHrgIXwvrQBVRgPQuBqvrD4Ga7g/lT57j+wFWrXm5+eD7LMnT55UWNoaJ2ynd6Cg+Cs4wo3B1oaagvVoD3vDCHsg9nAPndtjAsohYD+L2uUfAy8vw8qKT2FpegV4twALN54OKemhcgbokoTT1oRxRgQfTIAJBEDgW2GqmSdgqBEmh1zcIW62bRR6zBM1pm/lk3H3v76+Xjz11FPivffeE7du3RrC+BziRlOp0GKe+LDpj2KH0UDRbQQwTeRXHRGmmz0ghRA3hKmpRhj1WncZgzDWNLnLyO1QXdef1tgkbgwgicNUIwxe83FHWNrqRVX+tJ660BtFTZNJ3KT6N5qEUdvbLrSloumGSy6H5ZSHzFqhN9aIJtNAvQ4giPsUmwjC+GXFTcUzAUpw+E2fASZh5N9/HxnaPqfj7K1nWMRvv/02iNEdwC/+pREPFB0ARRBzWDZi/P9+Da9sa3Ov/G04X7sC+kWfYVxFOxxSmTcx7rPlSH2xGu12IHnWOpxvKoUW81Bj+haWillIHlQSJ+zt72Phi4cw7LVGqV0h7sCy5j7UZb+Bbe3XqWFUnG+CUauFoeYcHJZyzEpOgvNSA5akF6F53JuwOASEOI8PUk9gkb4MDZcoJ1tgByvYwDhxqYQnMBoZ838KfeMmvLPjbK9i+P0nqM/9Z+RMvjeuCZF9lrwMyHww9CMdxjVvYM4Ul0pM0v4Az2U8iJZjZ2Eh+7WtDR+vPo3cLW9jeYZWslsmaTOx4r1qGE1VKNtvDvLh0zWc3v8JTHkFKHK3CwyHVr8AeYbzOPZVt492r6Blczlq01bgN8szoZW0ZDKmFlZgd95JLN3cGvDGElawQ79buEZCEkjCyPTXsKfNiHF1Bjyg0UCjycS71+bjD9WzMZ4/SUO4K0YiJXUS0NGJLvt/wdZ2HHXWR5E57bveD4VG6pCaBnSYLEHauMdgVkUbLBUz0N18SApI3tCwHSXZs1DUeMu3vLavcbSuHdrpEzHOa15dcluHsLHEq7rvHvkKE2AC0uonPQ8Vf7RIP3WF6MDOlTmYEsGssvFP/S66L3TC6meg1jMX0B2Up4YT9vM7UaAbhdTsrTh1nRp5GDkfNKDGMLjitlZmY5T0RUpfpvQ3AqlFB/xI2v/S3/U/xWeYABNgAtEi4N4hh06fHfZfSfos6n3Bacb+5aU4lluPru1zen9l2JpR0mEF/CZ9IJtsMw4XTvVeVXv3MOg7XsEOiogLMAEmEDkCSUie8RzytOdw4uxfvW2idgtMHUBaqs4V+2GoQsj1Mx9z21FdDThvXscVf20lP4GcPB06Tvy5j4/zdbRv+DE0ObUwB7iiZgXrDzRfYwJMIPIEkmfg5+sycHjpm6g+bXUpWfI5fn05KlOLUT5/CpKQhJEpk5EmSePELfstb2U8kJRuRdlYtw8tVteTf6f1BKrL3kKtp01CsvXe52rhlh125xjol5Uh9/BbKKs+4VayNpgbKrGyGKgqn4MpAWrOAIsNJD2fYwJMgAmEgwA9od+Elt1Z6C5JxzCydz7wa5iyqmH6dDnS3TbupMk5KCm9gaLU+3H/3N+hc9BV5Bjof7UVOzI+R7buHsmOmrLqC0wo2Ip6Y3qv4EmTkFuSh7tFj2LY/YXY33kbSeNno7qlGlndb0M3jOyvo6A/NBqvt23HG+kP9tYd5BVnNBgEEF9mAkyACQRLgFewwZLjekyACTCBQQiwgh0EEF9mAkyACQRLgBVssOS4HhNgAkxgEAL/Hy5pK03h1+CLAAAAAElFTkSuQmCC"></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that Pablo leaves home before 07:00 and is late for work.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that Pablo is late for work.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that Pablo is late for work, find the probability that he left home before 07:00.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Two days next week Pablo will drive to work. Find the probability that he will be late at least once.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><img 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"><em><strong>A1A1A1 N3</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for each bold fraction.</p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>multiplying along correct branches <em><strong>(A1)</strong></em><br><em>eg </em>\(\frac{3}{4} \times \frac{1}{8}\)</p>
<p>P(leaves before 07:00 ∩ late) = \(\frac{3}{32}\) <em><strong> A1 N2</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p> </p>
<p>multiplying along other “late” branch <em><strong>(M1)</strong></em><br>eg \(\frac{1}{4} \times \frac{5}{8}\)</p>
<p>adding probabilities of two mutually exclusive late paths <em><strong>(A1)</strong></em><br>eg \(\left( {\frac{3}{4} \times \frac{1}{8}} \right) + \left( {\frac{1}{4} \times \frac{5}{8}} \right),\,\,\frac{3}{{32}} + \frac{5}{{32}}\)</p>
<p>\({\text{P}}\left( L \right) = \frac{8}{{32}}\,\,\left( { = \frac{1}{4}} \right)\) <em><strong>A1 N2</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>recognizing conditional probability (seen anywhere) <em><strong> (M1)</strong></em><br><em>eg</em> \({\text{P}}\left( {A|B} \right),\,\,{\text{P}}\left( {{\text{before 7}}|{\text{late}}} \right)\)</p>
<p>correct substitution of <strong>their</strong> values into formula <em><strong>(A1)</strong></em><br><em>eg</em> \(\frac{{\frac{3}{{32}}}}{{\frac{1}{4}}}\)</p>
<p>\({\text{P}}\left( {{\text{left before 07:00}}|{\text{late}}} \right) = \frac{3}{8}\) <strong><em> A1 N2</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid approach <em><strong>(M1)</strong></em><br><em>eg</em> 1 − P(not late twice), P(late once) + P(late twice)</p>
<p>correct working <em><strong>(A1)</strong></em><br><em>eg</em> \(1 - \left( {\frac{3}{4} \times \frac{3}{4}} \right),\,\,2 \times \frac{1}{4} \times \frac{3}{4} + \frac{1}{4} \times \frac{1}{4}\)</p>
<p>\(\frac{7}{{16}}\) <strong><em> A1 N2</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: 'times new roman', times; font-size: medium;">The weekly wages (in dollars) of 80 employees are displayed in the cumulative </span><span style="font-family: 'times new roman', times; font-size: medium;">frequency curve below.</span></p>
<p><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAArgAAAMECAIAAADb3cHCAAAgAElEQVR4nOyd/3tU9Z3285+cuTxiCQ9YNEhx3SUi0kVKs5hKzAXqVpFKbLWmbMs2yeIibrfB3aBYyS48blIoFtHtJLoNXxI7VJFVJF0oykPTFKsS0oUmirmciHFynh+OZCbJ5J3J+zOT+Zwzr/s611WSm7lz98wh5+V7zvmcIi9oisfj7e3t+W6BEEIIFYSK8l1gymqJRheVlsbj8XwXQQghhMKvgIFCPB5fVFrqOpGWaDTfXRBCCKHwK2Cg0BKNuk7EdSIMFRBCCKFpUJBAYWSc4G8MFRBCCKFcK0igMDJOcJ1IbU0NQwWEEEIo1woMKPjjhMqKCh8Uurq6GCoghBBCuVZgQMEfJxw9etQHBc/zGCoghBBCuVZgQGFLfX1lRYXneSOg4A8Vurq68l0NIYQQCq0CAwp9fX3+H0ZAIfWbCCGEEMqFAgMKI0oFBYQQQgjlVIACQgghhCYUoIAQQgihCQUoIIQQQmhCAQoIIYQQmlCAAkIIIYQmFKCAEEIIoQkFKCCEEEJoQgEKCCGEEJpQgAJCCCGEJhSggBBCCKEJBSgghBBCaEIBCgghhBCaUIACQgghhCZU7kFhoOuVhvvnOBHXicxZs+2Vrksp3uXezj11S2e5zoK1Tx3tTWSUBygghBBC06Ycg0LivZZ1C+as2X1mIOF5l3tjjy8r3hi75BNBYqBz27Klj8d6L3uJntjG8mUNxwYyiAQUEEIIoWlTbkEh0dW0wllYF7t45eszzWW3fvFl6p89z7sUqyte3dw1OGkmoIAQQghNm3I8UbgUqyuelRwVXIrVzf1iopDoalrhpJLBxVjtrSuazkz6+QOggBBCCE2bcn2NwoedDeWus2Bt09sDiZ7Ypo3bjvmXIgx2Na12kx9DeJ53MVa70C1r6pqMFAAFhBBCaNqU+4sZE71vPeVfzDhmfjAGC6YGCsJWWbmKjY2NjY2t0LYcncan4fbIS13Rn2xt2LDMibj+pYuel1NQSPuqjlcOTxQ4kfXuH99794/vpbWE92OiNEWBjz766Df/czJbBXTWb/7n5EcffZStDkKaotv//b/P9vb+KVtpum4T7YTs/j/N7ruQ3SMku4ei4t+C0EGXpthv1r4L2f1/Om2/kXQ7Z6qHQda75esw6Orq2vCDDa4TmV08q6+vb6JXmSjXoHDpTNPfPRx9L+F5id6OR5fOcpdu6xxI5Oijh+weEICCZzcoPPnkU++++2620gAFnQUoCAV0aYACoJCJFY/H9/7856V/+Zf+ObHijpVfW7Z8opcYahruekj5xGHg2Nal1/lXLCa6mlakgkLiTHPZdYYXMwIKuh8kWICCnAYoAApCAV0aoAAoyFbb/oMPfec7/qlwdvGsLVu29Pf3ywUMlVNQSFyKbZwz0a0NObg9ElDQ/SDBAhTkNEABUBAK6NIABUAhrRWPx//j2WdvuvHG+SXzXCdy9+q7XnvttQwLGCq3EwVv4NjWpbOuLLiUGDize+3cR1p6/MsUsr/gEqCg+0GCBSjIaYACoCAU0KUBCoDCGGvkKgTXicydc+0TTzT4I4TMCxgqx6DgeYneY7try9yJlnA+1ri2OOI6ZXW7jpkv4Qwo6H6QYAEKchqgACgIBXRpgAKg4FvxePzAgQMjVyHcvfqu9vZ24VUBBoWsC1DQFdBZgIKcBigACkIBXRqgACj09fXt2Pns7OJi/yqExx57LHWEAChMLkBBV0BnAQpyGqAAKAgFdGmAQiGDwptvvnn3XXf5p7mlX/3qvn37Mk8DFJLy92DHK4fzuFVWrirwAjZ02LChZu/eFwp8J1DAhg4UyHsBGzoYFoi2vFxdvX5+ybz5JfNqa+qiLS8rCuTotBtUUEhrdTBRYKKgTWOioLOYKAgFdGlZfBeYKKjTpu0w6Ovr+4faupFPGXbu2OF/ypDdf4yGAhQABckCFOQ0QAFQEAro0gCFAgGFU6dOPf30dv+MdsvNN7/80kuG3QCFpAAFXQGdBSjIaYACoCAU0KUBCqEHhVgs9vXly0fuZfhDd3dWugEKSQEKugI6C1CQ0wAFQEEooEsDFMIKCvF4/Be/+MVNN97oOpEZV11VW1PTtv9gFrsBCkkBCroCOgtQkNMABUBBKKBLAxRCCQqN27df+39mj1yIkItugEJSgIKugM4CFOQ0QAFQEAro0gCFMIGCf63i3Xfd41+IMOZ2R0AhVwIUdAV0FqAgpwEKgIJQQJcGKIQDFPr6+kbWXb77rnv8RRVz2g1QSIp1FGwoYEMH1lGggCUdKJD3AjZ0SC3w9NPblyxe4jqR6ur1ihUR1AVydNoNKiiktTqYKDBR0KYxUdBZTBSEArq0LL4LTBTUabrD4NSpUyPrKqbezpD3f4yGAhQABckCFOQ0QAFQEAro0gCFIILCqVOnbll0i387w/1r1oy54zHv/xgNBSgACpIFKMhpgAKgIBTQpQEKwQKFU6dOVd55p+tEZl49o7amJu0DoPP+j9FQgAKgIFmAgpwGKAAKQgFdGqAQFFAYQQR/UYSVK+80STO3AIWkAAVdAZ0FKMhpgAKgIBTQpQEK9oPCGETwpwjWHgbmAhQABckCFOQ0QAFQEAro0gAFy0Hhvm/eOwYR5AJyWha7AQpJAQq6AjoLUJDTAAVAQSigSwMU7AQFf10E/6bHtA9osPYwMBegAChIFqAgpwEKgIJQQJcGKNgGCv7qiv6pp7p6fdpnOAkFhB8EKORKLLhkQwEbOrDgEgUs6UCBvBfIXYddu/b4CzCve6BKXjop7zsBUEiKiYKugM5ioiCnMVFgoiAU0KVl8V1goqBO63jl8C9+8YvZxcX+MxpGFmAO4mFgLkABUJAsQEFOAxQABaGALg1QyDsoHDhwoLp6vf+kxzHPaAjiYWAuQAFQkCxAQU4DFAAFoYAuDVDIIyiM3Pe4ZPGSkYdBGxYQXgUo5EqAgq6AzgIU5DRAAVAQCujSAIW8gMLIwx79+x7b9h/MVgHhVYBCrgQo6AroLEBBTgMUAAWhgC4NUJhmUIjH443bt8+65kv+fY/+0ghhOgzMBSgACpIFKMhpgAKgIBTQpQEK0wkKBw4cWHDDfP+KxePHj5t0s/YwMBegAChIFqAgpwEKgIJQQJcGKEwPKHR3d2/e/E/+FYv79u0z72btYWAuQAFQkCxAQU4DFAAFoYAuDVDINSjE4/F//tGPXCcyv2RebU1NPB7PSjdrDwNzBRUUOvK9rkWBF7ChAwsuUcCSDhTIe4HMO0RbXs5wAaXA7QRAISkmCroCOouJgpzGRIGJglBAl5bFd4GJQupLuru7/VsfR1ZHyG43aw8DcwEKgIJkAQpyGqAAKAgFdGmAQtZBIR6P/+u//MvIrY856mbtYWAuQAFQkCxAQU4DFAAFoYAuDVDILihEW17272v4m6+XjXmeE6CQoQAFQEGyAAU5DVAAFIQCujRAIVug0NfXd/ddd80vmTe7eNZLL72U627WHgbmAhQABckCFOQ0QAFQEAro0gCFrIDC3p//3F9DafPmf/LXUMp1N2sPA3MBCoCCZAEKchqgACgIBXRpgIIhKJw7d+7ry5f7dz8eP348u7s6iIeBuQAFQEGyAAU5DVAAFIQCujRAwQQUGrdv9y9a3Lxpk0lamA4DcwUVFIStsnIVGxsbG1tBbStX3ln6V6WuE5l3fck3vrEy733ysuXotBtCUEj7KiYKOouJgpw20U5goqBLY6KgsyYqUDgThcbt22cXz5px1VVbtmwxT9N1s/YwMFdQQSGtxUcPuh8kWICCnAYoAApCAV0aoDCll3R3d9/210tdJ7Jg/lfG3P2oLqDrZu1hYC5AAVCQLEBBTgMUAAWhgC4NUMj8JSNXJGzbtm2qh0HWu1l7GJgLUAAUJAtQkNMABUBBKKBLAxQyecm5c+f8QcItN9/sDxIABUAhKUBBV0BnAQpyGqAAKAgFdGmAwqQvGVkjIfWKBEABUEgKUNAV0FmAgpwGKAAKQgFdGqAgv+Tuu+7y10h4++23M+kAKJgLUAAUJAtQkNMABUBBKKBLAxQm+v6rr77qPyE69cFOk3YAFMwFKAAKkgUoyGmAAqAgFNClAQrjvx+Px//5Rz9ynciSxUtee+21KXUAFMwVVFDoeOVwHrfKylUFXsCGDhs21Ozd+0KB7wQK2NCBAjkt0Lb/YG1NnetEqqvXt+0/WJg7IcMCOTrtBhUU0lodTBSYKGjTmCjoLCYKQgFdWhbfhRBMFPzrFmdcddWuXbvkNCYKgEJSgIKugM4CFOQ0QAFQEAro0gAF//vxePyh73xnzHWLgIJgAQpJAQq6AjoLUJDTAAVAQSigSwMUOl453N3dfdONN7pO5DsPfjsej2eSBigACkkBCroCOgtQkNMABUBBKKBLAxT2Pv+iv97ivn37Mk8DFACFpAAFXQGdBSjIaYACoCAU0KUVMijE4/ENP9gwv2Te/JJ5U31wA6AAKCQFKOgK6CxAQU4DFAAFoYAurWBB4dy5c0sWL3adSG1NXerHDRmmAQqAQlKAgq6AzgIU5DRAAVAQCujSChMUXn311ZG7G3Q7B1AAFJICFHQFdBagIKcBCoCCUECXVoCg4D8Ecu6ca48fP57hSzLvACiYK6ig0JHvdS0KvIANHVhwiQKWdKCASYEnnmhwnci6B6qExZRCvxOyVSBHp92ggkJaq4OJAhMFbRoTBZ3FREEooEvL4rtg+UShbf9B/6KEzZs2ZfgSxbuQ3V0dxMPAXIACoCBZgIKcBigACkIBXVqBgMKrr75afnv5VO+BBBQEC1BIClDQFdBZgIKcBigACkIBXVohgMLen//cdSLlt5f7FyVkWEC2AAVAISlAQVdAZwEKchqgACgIBXRp4QYFf6UE14ks/epX2/YfnFIB2QIUAIWkAAVdAZ0FKMhpgAKgIBTQpYUYFOLx+G1/vdR1Ivfft0ZRQLYABUAhKUBBV0BnAQpyGqAAKAgFdGlhBYXu7u4FN8x3ncjOHTt0BWQLUAAUkgIUdAV0FqAgpwEKgIJQQJcWSlA4deqUv57SoUOH1AVkC1AAFJJiHQUbCtjQgXUUKGBJBwrIBfY+/+L8knnlt5fvff7Fgt0J01MgR6fdoIJCWquDiQITBW0aEwWdxURBKKBLy+K7YMNEYdeuPa4TmV8yr7+/37CAbDFRABSSAhR0BXQWoCCnAQqAglBAlxYmUPBvcKi44460D3kCFACFXAlQ0BXQWYCCnAYoAApCAV1aOEAhHo9X3nmn/yjIbBWQLUABUEgKUNAV0FmAgpwGKAAKQgFdWghAIR6P+2szb6mvz+7JGFAQLEAhKUBBV0BnAQpyGqAAKAgFdGlBB4Xu7m6fEvzbIAEFXRqgYCRAQVdAZwEKchqgACgIBXRpgQaF7u5u/zbI1tZWOU1RQLYABUAhKUBBV0BnAQpyGqAAKAgFdGnBBYURSkh9ggOgoEsDFIwEKOgK6CxAQU4DFAAFoYAuLaCg8Oabb8665kuzrvnSmOc8AQq6NEDBSCy4ZEMBGzqw4BIFLOlAgRUryv0lldr2HyzYnWBDgRyddoMKCmmtDiYKTBS0aUwUdBYTBaGALi2L78L0TBRaW1tnXj0j7ZJKQpqigGwxUQAUkgIUdAV0FqAgpwEKgIJQQJcWLFBobW11ncjca7+clhKENEUB2QIUggsKF2O1C/1Te3Ira+pK+O7l3s49dUtnuc6CtU8d7U1klAgo6AroLEBBTgMUAAWhgC4tQKDgU8L8knkrV9451Z+iKCBbgEJgQeFSrK54NCU4s1Y0nUl4nuclBjq3LVv6eKz3spfoiW0sX9ZwbCCDSEBBV0BnAQpyGqAAKAgFdGlBAQWfEpYsXhyPx6fnMJAtQCGgoJC4FPvJY7GelEnBxVjtQ81dg57neYkzzWW31sUufuFcitUVr/7CEgUo6AroLEBBTgMUAAWhgC4tEKCQSglCASFNUUC2AIWAgsLl3q73UocEia6mFVc+d0h0Na1wUsngYqz21ivDBkmAgq6AzgIU5DRAAVAQCujS7AeFMZQgFBDSFAVkC1AIKCiM0WBX07qq6HuJL/682i3eGLs0AgYXY7ULUy5fmFCAgq6AzgIU5DRAAVAQCujSLAeF8ZQgFBDSFAVkC1AIBSgkzjSXP3aFDMZjwdRAQdjSvgpQ0FmAgpwGKAAKQgFdms2g8MIL/zmeEoQCQpqigGwBCmEAhURX0x21sUtffJVDUKisXMXGxsbGlt1txYrymVfPuOkvbsp7E7aJthydvqcNFAa7mh5KXrrIRw8ZWEwUZIuJglBASGOiwERB0a21tXXmjGvWPVA1ZpYgFxB+ylQLTGpN9TDIejdrDwNzTRcojPrcwfP8ixlTQSFxprnsOi5mTBWgIFuAglBASAMUAIWpdrtyXcKtx4//ZkoFhJ8ypQKZWIBC4EFh9OcO/re4PXISC1CQLUBBKCCkAQqAwpS6jayq9N57703PbyRAQWcFHRTGfO7giwWXJrEABdkCFIQCQhqgAChk3u2tt97yKaG/v3/afiMBCjor4KCQONO87iedA+M/Vbjce6xxbXHEdcrqdh1jCecxAhRkC1AQCghpgAKgkGG37u5u/8nR/nMcAAVdAV1a4YFCVgUo6AroLEBBTgMUAAWhgC7NElAYoYTu7u5J0wAFoYAuDVAwEqCgK6CzAAU5DVAAFIQCujQbQKGnp2fWNV+acdVVx48fzyQNUBAK6NIABSP5oNDxyuE8bpWVqwq8gA0dNmyo2bv3hQLfCRSwoUP4ClRXr59fMm/v8y8GZQ/Y0MGGAjk67QYVFNJaHUwUmCho05go6CwmCkIBXVoW3wXF/9N4PF5dvd51Iq2trZmnMVEQCujSsvuP0VCAAqAgWYCCnAYoAApCAV1afkHhvnvvdZ3Izh07ppQGKAgFdGmAgpEABV0BnQUoyGmAAqAgFNCl5REUNvxgg+tEnn56+1TTAAWhgC4NUDASoKAroLMABTkNUAAUhAK6tHyBwlNPPuk6kW9XPZj330iAgs4CFJICFHQFdBagIKcBCoCCUECXlhdQ8JdfLFu+XPgpgIKugC4NUDASoKAroLMABTkNUAAUhAK6tOkHBX/5xZtuvNF/4FPefyMBCjoLUEgKUNAV0FmAgpwGKAAKQgFd2jSDwpjlF4WfAijoCujSAAUjAQq6AjoLUJDTAAVAQSigS5tOUIjH4zcvXDjjqqvefvvtSX8KoKAroEsDFIzEgks2FLChAwsuUcCSDsEtMNWFlazdAzZ0sKFAjk67QQWFtFYHEwUmCto0Jgo6i4mCUECXlsV3Qf5/+vTT210nsm3btgx/ChMFXQFdWnb/MRoKUAAUJAtQkNMABUBBKKBLmx5Q+Glzs38zZOY/BVDQFdClAQpGAhR0BXQWoCCnAQqAglBAlzYNoODf5lBdvX5KPwVQ0BXQpQEKRgIUdAV0FqAgpwEKgIJQQJeWa1Dwb3OYO+fatv0Hp/RTAAVdAV0aoGAkQEFXQGcBCnIaoAAoCAV0aTkFhdTbHLL4LgAKQgFdGqBgJEBBV0BnAQpyGqAAKAgFdGk5BQX/mU/+kyEBBUAhQwEKgIJkAQpyGqAAKAgFdGm5AwX/aQ41P/yhrhugoCugSwMUjAQo6AroLEBBTgMUAAWhgC4tR6Bw6NChkac56LoBCroCujRAwUgsuGRDARs6sOASBSzpYH+Btv0Hy28vL7+9vG3/wVDuARs62FAgR6fdoIJCWquDiQITBW0aEwWdxURBKKBLy+K74P8/jcfjSxYvdp1I6jrNim5MFHQFdGnZ/cdoKEABUJAsQEFOAxQABaGALi3roFD9yCOuE9m3b59hN0BBV0CXBigYCVDQFdBZgIKcBigACkIBXVp2QeGFF/5zqisw5v03EqCgswCFpAAFXQGdBSjIaYACoCAU0KVl8V14+9TbpX+18KYbb4zH4+bdAAVdAV0aoGAkQEFXQGcBCnIaoAAoCAV0adl6F+Lx+NeWfe36L8/t7u7OSjdAQVdAlwYoGAlQ0BXQWYCCnAYoAApCAV1att4F/9KEn/3suWx1AxR0BXRpgIKRAAVdAZ0FKMhpgAKgIBTQpWXlXWhtbXWdSPUj1Vn8fwoo6Aro0gAFI7GOgg0FbOjAOgoUsKSDbQXa9h9csnjJ3XfdUzh7wIYONhTI0Wk3qKCQ1upgosBEQZvGREFnMVEQCujSzN8F/7FP3d3d2f1/ykRBV0CXlt1/jIYCFAAFyQIU5DRAAVAQCujSDN+FTY/+o+tEftrc7GX7/ymgoCugSwMUjAQo6AroLEBBTgMUAAWhgC7N5F146623XCdy1+rV/peAgjot0IdB1gUoAAqSBSjIaYACoCAU0KWp34V4PD7rmi/NnXPtyKoJgII6LbiHQS4EKAAKkgUoyGmAAqAgFNClqd+F++6913Uir7322sj3AQV1WnAPg1wIUAAUJAtQkNMABUBBKKBL070LP21udp1IzQ9/mPp9QEGdFtDDYKKXGApQABQkC1CQ0wAFQEEooEtTWCtX3jnz6hk3L1w45vuAgjotiIcBoJCUDwrClvZVgILOAhTkNEABUBAK6NIU1k1/cZPrRMYv1QwoqNOCeBgACklNCgqVlavY2NjYCmT72rLlrhP52rLleW/ClvctR6fdoIJCWouPHnQ/SLCYKMhpE+0EJgq6NCYKU7W6u7tdJ7Jg/lfS/n0mCuq0YB0GkxYwFKAAKEgWoCCnAQqAglBAlzYl6+aFC2dePWPlyjvT/n1AQZ0WrMNg0gKGAhQABckCFOQ0QAFQEAro0jK3nnrySX8RxokKAArqtAAdBpkUMBSgAChIFqAgpwEKgIJQQJeWoeV/6PCN28uFAoCCOi0oh0GGBQwFKAAKkgUoyGmAAqAgFNClZWj5Hzr09PQIBQAFdVpQDoMMCxgKUAAUJAtQkNMABUBBKKBLy8Qa+dBBLgAoqNMCcRhkXsBQgAKgIFmAgpwGKAAKQgFd2qRW6ocOcgFAQZ1m/2EwpQKGCioodLxyOI9bZeWqAi9gQ4cNG2r27n2hwHcCBWzoMM0F1j1QNb9kXtv+gwW7B+zsYEOBHJ12gwoKaa0OJgpMFLRpTBR0FhMFoYAuTbb8Zzrs+Pd/z6QAEwV1muWHwVQLGApQABQkC1CQ0wAFQEEooEsTrLb9B9M+0wFQEDoACuYCFAAFyQIU5DRAAVAQCujSBKu2pi7tMx0ABaEDoGAuQAFQkCxAQU4DFAAFoYAubSLr0KFDrhPZvGlT5gUABXWatYeBYAEKSQEKugI6C1CQ0wAFQEEooEtLa8Xj8VnXfOnuu+6Jx+OZFwAU1Gl2HgayBSgkBSjoCugsQEFOAxQABaGALi2tVfPDH7pOZO/zL06pAKCgTrPzMJAtQCEpQEFXQGcBCnIaoAAoCAV0aeOtt99+23Ui991771TfBUBBnWbhYTCpBSgkBSjoCugsQEFOAxQABaGALm285a/W3N/fDygACoIFKCTFgks2FLChAwsuUcCSDjktsGvXHteJ7Nq1p2D3QFA62FAgR6fdoIJCWquDiQITBW0aEwWdxURBKKBLS7Xi8XjqwglTfReYKKjTrDoMMrQAhaQABV0BnQUoyGmAAqAgFNClpVrfrnrQdSJvvfWW/CpAQegAKJgLUAAUJAtQkNMABUBBKKBLG7Heeust14l8u+rBSV8FKAgdAAVzAQqAgmQBCnIaoAAoCAV0aSOWfw1j6sIJgAKgIFiAQlKAgq6AzgIU5DRAAVAQCujSfGvfvn2uE/lpc3MmrwIUhA6AgrkABUBBsgAFOQ1QABSEArq0jlcO+9cwzrvu+gxfBSgIHQAFcwEKgIJkAQpyGqAAKAgFdGkdrxz212EcuYZx0lcBCkIHQMFcgAKgIFmAgpwGKAAKQgFdWtv+g/46jJm/ClAQOgAK5goqKHTke12LAi9gQwcWXKKAJR2yW6C2pm5+yby2/QcLdg8EtIMNBXJ02g0qKKS1OpgoMFHQpjFR0FlMFIQCijT/lsgt9fVTehUTBaFDdg/46TkMdBagkBSgoCugswAFOQ1QABSEAoq0edddX357edpnSQuvAhSEDoCCuQAFQEGyAAU5DVAAFIQCU03zb4nctWvPVNMABaEDoGAuQAFQkCxAQU4DFAAFocCU0vxbIhfcMD+L7wKgoE4DFFIFKAAKkgUoyGmAAqAgFJhS2pb6ev+WSEABUNBZgEJSgIKugM4CFOQ0QAFQEApkntbT0zPjqqu+cXu5/IMABUBBsACFpAAFXQGdBSjIaYACoCAUyDzNf0pkd3e3/IMABUBBsACFpFhHwYYCNnRgHQUKWNLBsEDb/oPzS+Zt3vxPBbsHwtHBhgI5Ou0GFRTSWh1MFJgoaNOYKOgsJgpCgQzTypYvd51IT0/PpD9oqu8CEwV12vQfBuYWoJAUoKAroLMABTkNUAAUhAKZpI1fYQlQABR0VghA4XJv5/5fNNw/x4nMqY1dSn5zT93SWa6zYO1TR3sTGQUBCroCOgtQkNMABUBBKJBJWtny5TOvnpG6whKgACjorICDQqL3rafun1N8/9b//HXXwAgOJAY6ty1b+nis97KX6IltLF/WcGwggzBAQVdAZwEKchqgACgIBSZNe+2118Yv2AwoAAo6K9Cg8GFnQ/mcNY1v9V4e9e3EmeayW+tiF7/48lKsrnh1c9fgpHGAgq6AzgIU5DRAAVAQCkyaNu+662cXF49ZsBlQABR0VnBBITHQuW1Z8SMtPZfHGl1NK5xUMrgYq711RdOZST9/ABR0BXQWoCCnAQqAglBATvMXbN63b1/mPwhQABQEK7CgkDjTXHbdstpn/asT3OL7t3Z0DXie5w12Na12izfGLo2AwcVY7UK3rKlrMlIAFHQFdBagIKcBCoCCUEBOm3fd9fOuu35KPwhQABQEK6igkOhqWuGU1e061pvwPO/SmaZ1c5zyrZ0fpsMCQGGsAAXZAhSEAl4fRc4AACAASURBVEIaoGADKOzatSftOEH+QYACoCBYAQWFxKXYxjmpY4PEmeayWW5ZU1fCFBSErbJyFRsbG5vN27zrS+ZdX5L3Gmwh23J0Ls8pKIz/fGHkO58YfvQgbGlfxURBZzFRkNMm2glMFHRpBTJRmOjqhEl/EBMFJgqCFVBQSFyKbZwz6orFwa6m1T4NJLqaVowdNlzHxYypAhRkC1AQCghpgELeQWHunGvXPVA11QKCBSgIHQAFc+UUFPybHhdURd+7cvpPubWB2yMnswAF2QIUhAJCGqCQX1DYtm2b60SiLS9PtYBgAQpCB0DBXDkGBe9yT/SROcXrms9c8rzLvcca15Zt6/xizSUWXJrEAhRkC1AQCghpgEIeQSEej8+8esZNN944Pe8CoKBOAxRSlWtQ8DzvUlfHtrXFEdeZtax2T+eoZZcu9x5rXFsccZN3RkwuQEFXQGcBCnIaoAAoCAXGv2pLfb3rRA4cOAAoKArIFqAQaFDIsgAFXQGdBSjIaYACoCAUGP+q2cWzbrrxRjkNUAAUdBagkBSgoCugswAFOQ1QABSEAmNetXPHDn+cIKcBCoCCzgIUkgIUdAV0FqAgpwEKgIJQYMyr5l13vT9OkNMABUBBZwEKSfmg0PHK4TxulZWrCryADR02bKjZu/eFAt8JFLChQyYF/KUY9z7/YsHugdB3sKFAjk67QQWFtFYHEwUmCto0Jgo6i4mCUCD1VWOe7DA97wITBXVajg6DnHYDFJICFHQFdBagIKcBCoCCUGDkVeOXYgQUFAVkC1AAFJICFHQFdBagIKcBCoCCUGDkVeMfFAkoKArIFqAAKCQFKOgK6CxAQU4DFAAFoYD/qrRPdgAUFAVkC1AAFJICFHQFdBagIKcBCoCCUMB/1c0LF84uLo7H4xmmAQqAgs4CFJICFHQFdBagIKcBCoCCUMDzvGjLy64TeerJJzNPAxQABZ0FKCQFKOgK6CxAQU4DFAAFoYDnedXV62dePWPMOEFOAxQABZ0FKCTFOgo2FLChA+soUMCSDhMV8McJTz+9vWD3QEF1sKFAjk67QQWFtFYHEwUmCto0Jgo6i4mCUODu1XctWbykr69vSmlZfBeYKKjTsngY6NKy+4/RUIACoCBZgIKcBigAChMV6OnpcZ3IE080TDUNUAAUdBagkBSgoCugswAFOQ1QABQmKvDtqgddJ9K2/+BU0wAFQEFnAQpJAQq6AjoLUJDTAAVAIW2BeDzuOpH77r03v+8CoKBOAxRSBSgACpIFKMhpgAKgkLbAlvp614mcOnUKUAAUdGmAgpEABV0BnQUoyGmAAqCQtsDMq2eULV+uSwMUAAWdBSgkBSjoCugsQEFOAxQAhfEFdu3a5TqRAwcO6NIABUBBZwEKSQEKugI6C1CQ0wAFQGF8gdRHQAEKgIIuDVAwEgsu2VDAhg4suEQBSzqkFtj7/IuuE9m1a0/B7oGC7WBDgRyddoMKCmmtDiYKTBS0aUwUdBYThTEFypYvT12zOb/vAhMFdZrhYWCelt1/jIYCFAAFyQIU5DRAAVBILfCH7m7XiWx69B9N0gAFQEFnAQpJAQq6AjoLUJDTAAVAIbWAv8hS6prNgAKgoEsDFIwEKOgK6CxAQU4DFACFkQIjiywZpgEKgILOAhSSAhR0BXQWoCCnAQqAwkgBf5Glt956yzANUAAUdBagkBSgoCugswAFOQ1QABRGCsydc+2SxYvN0wAFQEFnAQpJAQq6AjoLUJDTAAVAwS+wb98+14m0traapwEKgILOAhSSAhR0BXQWoCCnAQqAgl+gbPny2cXFWUkDFAAFnQUoJMWCSzYUsKEDCy5RwJIO3/jGSteJPP309oLdA3kvYEMHGwrk6LQbVFBIa3UwUWCioE1joqCzmCh4nnfLolvG3BVpkpbFd4GJgjpN0c3mf4yGAhQABckCFOQ0QAFQiMfjM6+eMeauSHWazgIUhA6AgrkABUBBsgAFOQ1QABS2bds2/q5IdZrOAhSEDoCCuQAFQEGyAAU5DVAAFOZdd/2C+V/JVprOAhSEDoCCuQAFQEGyAAU5DVAocFB46623XCdS9vW/yUqa2gIUhA6AgrkABUBBsgAFOQ1QKHBQqLjjjplXz7D2XQAU1GmAQqoABUBBsgAFOQ1QKGRQ6OnpcZ1I9SOPWPsuAArqNEAhVUEFBWGrrFzFxsbGlutt6V/f5jqRlSvvzHsTNjZ/y9FpN4SgkPZVTBR0FhMFOW2incBEQZcWrInCyMMdrH0XmCio05gopCqooJDW4qMH3Q8SLEBBTgMUChYUDh06NPJwB2vfBUBBnQYopApQABQkC1CQ0wCFggWFijvumHXNl+LxuFAg8zRDC1AQOgAK5gIUAAXJAhTkNEChMEHBv4xxww82yAUyTDO3AAWhA6BgLkABUJAsQEFOAxQKExS21Ne7TqS7u1sukGGauQUoCB0ABXMBCoCCZAEKchqgUJigMHIZo1wgwzRzC1AQOgAK5gIUAAXJAhTkNEChAEHBv4xx3759kxbIJC0rFqAgdAAUzAUoAAqSBSjIaYBCAYKCvxqjfxmjXCCTtKxYgILQAVAwV1BBoeOVw3ncKitXFXgBGzps2FCzd+8LBb4TKDDNHdr2H5xfMu+JJxqs2gkUsKGDDQVydNoNKiiktTqYKDBR0KYxUdBZhTZR8B8qferUqUwKTJqWLYuJgtAhu7taUUCXlt1/jIYCFAAFyQIU5DRAodBAYd511990440ZFpg0LVsWoCB0ABTMBSgACpIFKMhpgEJBgYL/UOkd//7vGRaQ06b6EsECFIQOgIK5AAVAQbIABTkNUCgoULj/vjWuE+nr68uwgJw21ZcIFqAgdAAUzAUoAAqSBSjIaYBCQYHCzKtn3HfvvZkXkNOm+hLBAhSEDoCCuQAFQEGyAAU5DVAoHFDYsfNZ14m8+uqrmRcQ0gAFRQHZAhQAhaQABV0BnQUoyGmAQuGAQnX1+tnFxVMqIKQBCooCsgUoAApJAQq6AjoLUJDTAIUCAQX/KVCbHv3HKRUQOgAKigKyBSgACkmx4JINBWzowIJLFJi2Dk8/vX1+yby2/Qft3AkUsKGDDQVydNoNKiiktTqYKDBR0KYxUdBZBTJRmDvn2urq9VMtIHSYnneBiYI6TdHN2sPAXIACoCBZgIKcBigUAij4yyfs2rVnqgWEDoCCooBsAQqAQlKAgq6AzgIU5DRAoRBAwV8+IYhnCEBBnQYopApQABQkC1CQ0wCFQgCFmVfPuGv16iCeIQAFdRqgkCpAAVCQLEBBTgMUQg8Ku3btcp3IgQMHgniGABTUaYBCqgAFQEGyAAU5DVAIPShU3HHHrGu+JKQJBYRXAQqKArIFKAAKSQEKugI6C1CQ0wCFcINCf3+/60SqH3lESBMKCK8CFBQFZAtQABSSYh0FGwrY0IF1FCiQ6w5PP73ddSITLZ9gz06ggA0dbCiQo9NuUEEhrdXBRIGJgjaNiYLOCvdE4eaFCxfcMF9OEwoIr5qed4GJgjpN0c3aw8BcgAKgIFmAgpwGKIQYFMYs2xzEMwSgoE4DFFIFKAAKkgUoyGmAQohBoeaHP3SdSF9fn5wmFBBeBSgoCsgWoAAoJAUo6AroLEBBTgMUQgwKc+dcu2Tx4knThALCqwAFRQHZAhQAhaQABV0BnQUoyGmAQlhB4e2333adyI5///dJ04QCwqsABUUB2QIUAIWkAAVdAZ0FKMhpgEJYQWHM5w5CmlBAeBWgoCggW4ACoJAUoKAroLMABTkNUAgrKMy8esY3bi/PJE0oILwKUFAUkC1AAVBIClDQFdBZgIKcBiiEEhQOHTrkOpHW1tZM0oQCwqsABUUB2QIUAIWkWHDJhgI2dGDBJQrkqENtTd2SxUsCtBMoYEMHGwrk6LQbVFBIa3UwUWCioE1joqCzQjlR8B8XmWGaUEB41fS8C0wU1GmKbtYeBuYCFAAFyQIU5DRAIXygsPf5F8d/7iCkCQWEVwEKigKyBSgACkkBCroCOgtQkNMAhfCBQm1NnetE4vF4hmlCAeFVgIKigGwBCgEHhUuxuuKIf4J3ndXNXYNXjMu9nXvqls5ynQVrnzram8goDFDQFdBZgIKcBiiEDBTi8fj8knnrvvWtzNOEAsKrAAVFAdkCFAINCpd7oo/Mca6AQllT1xdAkBjo3LZs6eOx3steoie2sXxZw7GBDOIABV0BnQUoyGmAQshAYdeuXa4TefXVVzNPEwoIrwIUFAVkC1AIMigMHNta9ljs0rhxQeJMc9mtdbGLX3x5KVZXnDpsmFCAgq6AzgIU5DRAIWSgUHHHHeWjl0+YNE0oILwKUFAUkC1AIbigkLgU2zjHmbWs9tmWWFfqwCDR1bRi1McQF2O1t65oOjPp5w+Agq6AzgIU5DRAIUygEI/HXSfyxBMNU0oTCgivAhQUBWQLUAgsKCTONJfNunJ1wqxltdGuAZ8EBruaVrvFG1MmDRdjtQtTPpiYUICCroDOAhTkNEAhTKDgf+4QbXl5SmlCAeFVgIKigGwBCoEFBV+J3s6Xnq1bOst1Zl25EGE8FkwNFIStsnIVGxsb21S3m/7iprnXfjnvNdjY1FuOzuHTAgq+Ej2xjWVXpgg5BIW0r2KioLOYKMhpE+0EJgq6tDxOFPzPHaofeSRM/ynJREGdFqbDwFzTCAqef8XiwrrYRT56yMQCFGQLUBAKCGmAQtq0kfsdwnSGABTUaWE6DMw1vaCQONNc9k3/AsZEV9OKVFBInGkuu46LGVMFKMgWoCAUENIAhbRpFXfcMeuaLym6CQV0aYACoKCzwgIKl2KPboxd8v/M7ZGTWYCCbAEKQgEhDVAYnzbyuYOim1BAlwYoAAo6K6igkOjtfPmlTn/JxUTv0W21DbHeyyMmCy7JFqAgW4CCUEBIAxTGp6WusxSmMwSgoE4L02FgrlRQGB784+/+ODicxfREb8ejS2e5TsQtvn/rf/76yr2RI7rce6xxbXHEdcrqdh1jCecxAhRkC1AQCghpgML4tJHPHRTdhAK6NEABUNBZ0wMKQ73Rv6t47MU3zvYP5einZUOAgq6AzgIU5DRAIQSgkPq5g6KbUECXBigACjpr2kDhX7Ye/FXLMxsf/s7mnW2d7w/YCAyAgq6AzgIU5DRAIQSgsG/fvtTnO4TpDAEoqNPCdBiYKxUUvOFPBz8d9jzPGx688P9ee/GZH1Q9/M+7209+MDCUzc8jDOWDQscrh/O4VVauKvACNnTYsKFm794XCnwnUMC8Q21NXfnt5YHeCRSwoYMNBXJ02i2a4PvDQwPvdUYbHiid5TpuybLvbn0+9tveT2zgBSYKugI6i4mCnMZEIQQThZlXz0h9rrRiv1n7LjBRUKeF6TAwVyoofN5/ZP/r/UODF0/HmurumOu6TuSa0nVbo2+cvfjnD062N9c9+N2fvHo+39MFQEFXQGcBCnIaoBB0UDh06JDrRA4cOKDuJhTQpQEKgILOmrZrFB4pvb2s9KqI67gl5XXN7b/tHUy5FWG4e0/FDcvrX+/PKyoACroCOgtQkNMAhaCDwv33rXGdSDweV3cTCujSAAVAQWdNGyg85DqzFn2rIfrGHz4aPznoP/T9qyPuVd9ru/B5jtpkIkBBV0BnAQpyGqAQdFCYefWMu1avNukmFNClAQqAgs6aNlD4h++/+LuBiQYGg6f3fOvW5Y915PfTB0BBV0BnAQpyGqAQaFDwP3dobW016SYU0KUBCoCCzpqmBZeG/vyn3vMn2/f/pndo2POGPzvb/m87Xjxy9mMbrmEcEaCgK6CzAAU5DVAINCh858Fvj/ncQdFNKKBLAxQABZ01XbdHXjj8+LLZrnPb1k5/MeXhoQu/3lJW+fhrf7KHFQAFXQGdBSjIaYBCoEFh7pxrv3F7uWE3oYAuDVAAFHTW9IDCp+/u+daiNY/+5Lk3epMfLnz67p573XmPH/k4swWWcy/WUbChgA0dWEeBAiYdoi0vu05k1649IdgJFLChgw0FcnTaTQWFgc6Gf3jhg89G/4VP391zr+useObkJzlqMFUxUdAV0FlMFOQ0JgrBnSjU/PCHrhPp6+sz7CYU0KVl8V1goqBOC9NhYK5UUBg6H/3Rljf6Uj5lGB7qP/bMyi+7TkYPgJ4eAQq6AjoLUJDTAIXggsLcOdcuWbzYvJtQQJcGKAAKOmu6rlHof33LytV1O18+cryz841ftTT9aG3pLNdxb3j4v85bc5ECoKAroLMABTkNUAgoKLz62uuuE3nqySfNuwkFdGmAAqCgs6YJFDxveOj8r7asvME/GbtOxHVmLXrouXcG8rlwwhgBCroCOgtQkNMAhYCCwtNPP+M6kXPnzpl3Ewro0gAFQEFnTRsoeJ7neUP973YebotGW34Ze/N07+Bw8mFRNghQ0BXQWYCCnAYoBBQU/vaeby64YX5WugkFdGmAAqCgs6YXFMYq3tX0ZEuvLY+cBhR0BXQWoCCnAQpBBIX33nvPdSKbHv3HrHQTCujSAAVAQWdNFygMnX+zefPapSUpHz3420OAQqoABV0HQCHr3Wz+3WQtKPy0+aeuEzl16lRWugkFdGmAAqCgs6YHFD47H/3uTCfizr3t7jX3r01uq5bNfRhQSBWgoOsAKGS9m82/m6wFhXvuuvsb5Xdkq5tQQJcGKAAKOmva1lG4zf1642/jY9ZWGjrf2pTfB0GligWXbChgQwcWXKKArsP8knlPPNEQpp1AARs62FAgR6fdVFAYjp/cXvY3TV3j1mDkYsYxYqKg68BEIevdbP6PGEWa4t+C0CFtmv8gqP/6ZVu2ugkFdGlZfBeYKKjTwnQYmKto1FfDH73TvOmfYr2jqYCLGccKUNB1ABSy3s3m3012gsL99625/svXHT/+m2x1Ewro0gAFQEFnTQ8oDHY1rR53GSMXM6YRoKDrAChkvZvNv5vsBIWZV89Yu+b+LJ4+hQK6NEABUNBZ0/TRw2e/a668qmTZ6jVrUy9mvK+89CpAYZQABV0HQCHr3Wz+3WQhKLz99tuuE9m7dy+goOgGKOgK6NKsBQXPG/6grfHQuNWabbyYMa0FKOh+kGABCnIaoBAsUPAfBPX+++8DCopugIKugC7NYlDwhocunGh9ZuN3G470e95w//GfbtnR9k6fLcMEz/MABW0BnQUoyGmAQrBA4aYbb1yyeHF2T59CAV0aoAAo6KxpAoXh/te3LJvtOhF3TbTX/87AyZ1/u3rzr87ZwwqAgq6AzgIU5DRAIUCg0NPT4zqRf/7RjwAFXTdAQVdAl2YtKFx+f1/VV+758b72pu9XfQEKnvfZB/secOc9fuTjcTdN5kmAgq6AzgIU5DRAIUCgsG3bNteJdHd3Awq6boCCroAuzVpQGOhsqN5x+hNvqHPrd0ZA4dN399zrOrdt7RzIUYOpigWXbChgQwcWXKLAlDpUV69f90BVKHcCBWzoYEOBHJ12x0wU6neOBoXhj/97yy0zXGd1c9dgjhpMVUwUdAV0FhMFOY2JQoAmCq4TqX7kkUnTFPvN2neBiYI6LUyHgblSQcEbvnDo0YefOdQZ3XRf08mz77x56NnvL53tOu4ND//XuFsh8iZAQVdAZwEKchqgEBRQeOmll1wn8uqrr06aFqYzBKCgTgvTYWCuotFffj5wet8Pls5OWWpp1qKHnntnwJZ7Iz1AQVtAZwEKchqgEBRQuP++Na4Ticfjk6aF6QwBKKjTwnQYmKto/LeGBy/8vvNwWzTa0nro9dO9g0OfDg5ZM08AFLQFdBagIKcBCkEBhblzrv3G7eWZpIXpDAEoqNPCdBiYKw0ojNbwZ6eb1u/5vS33PAAK2gI6C1CQ0wCFQICCvyDjT5ubM0kL0xkCUFCnhekwMFcqKHx+4Zffuybtsx5uebIzbstQAVDQFdBZgIKcBigEAhS2bNniOpG+vr5M0sJ0hgAU1GlhOgzMlQoKQ73Rh0uWrhr1oIfVt5XMve3uNZtazn6aowZTFaCgK6CzAAU5DVAIBCjcvHDhghvmZ5gWpjMEoKBOC9NhYK7RE4W2ptbzo9dgHP7w5Paax+1bmVHYKitXsbGxsY1srhNZ+te35b0GG1uutxyddosm/RvDZ/dUztsU67flxodJQSHtq5go6CwmCnLaRDuBiYIuLRcThV27drlO5M0338wwLUz/KclEQZ0WpsPAXJOBwvDghcP1t9q3MmNai48edD9IsAAFOQ1QsB8U/BsjM08L0xkCUFCnhekwMNfojx4mupjRvolCWgtQ0P0gwQIU5DRAwX5QmHn1jPvuvTfztDCdIQAFdVqYDgNzpYJCuosZ1zxc1/Dc4bMf23LPA6CgLaCzAAU5DVCwHBT8GyP3/vznmaeF6QwBKKjTwnQYmGv0RGH8xYzjNPzp4Kd5pQZAQVdAZwEKchqgYDkobN60KfXGyEzSwnSGABTUaWE6DMxVNPlfGaV4V9OTLb35vAcCUNAV0FmAgpwGKFgOCjcvXHjTjTdOKS1MZwhAQZ0WpsPAXKmgMNjVtHrSewpc5yFAAVDQdQAUst7N5t9NNoCC60Q2PfqPU0oL0xkCUFCnhekwMFcqKAx/2rn1L52SZavXJK9RuK+89Prb7k5esrBq2dyHAQVAQdcBUMh6N5t/N+UdFHbsfHbMjZGZpIXpDAEoqNPCdBiYKxUU4qd3PFh3qGc0BQye3fN3Vft+/9kXXw6db21qu5DPOyB8UOh45XAet8rKVQVewIYOGzbU7N37QoHvBAoIHWpr6spvLy+EnUABGzrYUCBHp91UUBjobHhs/LRgqLOh5Orvtpz/zLNDTBR0BXQWEwU5jYmCzROFJYuXjLkxMpM0xX6z9l1goqBOC9NhYK5UUPi8/9Dm6ugfR5HCcN+b9WUuCy6NFqCg6wAoZL2bzb+b8gsKaW+MzCQtTGcIQEGdFqbDwFxFo74aPL3ngYoH/nXPwSPHOjuPvxmL7qitKHEi13x9+0meHpkiQEHXAVDIejebfzflFxQ2b9o0v2TemBsjM0kL0xkCUFCnhekwMFfRmK+HB07teeirqeszXrPs0ZbfXbIFEwAFbQGdBSjIaYCCtaCw9KtfXfdAlSItTGcIQEGdFqbDwFxjQcHzPG948OLvO2O/bG2JtsU6uy8OJnL0s3UCFHQFdBagIKcBCnaCQjwed53I009vV6SF6QwBKKjTwnQYmCsdKNgtQEFXQGcBCnIaoGAnKLz00kuuE4m2vKxIC9MZAlBQp4XpMDDXGFAYHrpwovWZjd9tONLvecP9x3+6ZUfbO335XDZhnAAFXQGdBSjIaYCCnaDgPzFSlxamMwSgoE4L02FgrlGgMNz/+pZls10n4q6J9vrfGTi5829Xb/7VOXtYAVDQFdBZgIKcBijYCQpz51z7jdvLAQVAQZ0WpsPAXKmgcPn9fVVfuefH+9qbvl/1BSh43mcf7HvAnff4kY9tuVKBBZdsKGBDBxZcokDaDm37D7pOZMfOZwtnJ1DAhg42FMjRaTcVFAY6G6p3nP7EG+rc+p0RUPj03T33so7CGDFR0HVgopD1bjb/R4wiTfFvYXyHbdu2uU7k3LlzujTFfrP2XWCioE4L02FgrjEThfqdo0Fh+OP/3nLLDNdZ3dw1mKMGUxWgoCugswAFOQ1QsBAUKu6449r/M1udFqYzBKCgTgvTYWCuVFDwhi8cevThZw51Rjfd13Ty7DtvHnr2+0tnu457w8P/dd6ahRQABV0BnQUoyGmAgoWgMOOqq6ofeUSdFqYzBKCgTgvTYWCuotFffj5wet8Pls5Oeaj0rEUPPffOQD6fAjVGgIKugM4CFOQ0QME2UHjttddcJ9La2qpOC9MZAlBQp4XpMDDXaFAY6n/3t787/+Gfft95uC0abWk99Prp3kFrZgm+AAVdAZ0FKMhpgIJtoLB50ybXifgrNwMKgII6LUyHgblSQeHT9/dVzXRmrWg6Y8sdDukEKOgK6CxAQU4DFGwDhZtuvHHJ4sUmaWE6QwAK6rQwHQbmGgUKZ/esuebqh154b8x1i8ODf/zdH60ZLAAKugI6C1CQ0wAFq0DBX7l506P/aJIWpjMEoKBOC9NhYK6iUV8NnYtt+fv6/zrz0VAKFgz975EfP97Sa8uSS6yjYEMBGzqwjgIFxnTYtWuP60Ta9h8stJ1AARs62FAgR6fdIs8bGvjzR4PDnucNdjWtTrmMMXV7yDZQSGt1MFFgoqBNY6Kgs6yaKPgrNxumKfabte8CEwV1WpgOA3MV9b/xr8uv+upjR/o8b/iz3zVXXlWybPWatWvuT9lWLZv7MKCQKkBB1wFQyHo3m383TT8o+Cs3G6aF6QwBKKjTwnQYmKtoz53X3XDPT964MOR53nD89M9/FH137NUIQ+dbm9ou2HKHJKCgK6CzAAU5DVCwBxR6enpcJ/LUk08apoXpDAEoqNPCdBiYq2hrxc7Tn/loMNQbfSzdUs2f9x/Z/3o/oJAUoKDrAChkvZvNv5umGRR27tjhr9xsmBamMwSgoE4L02FgrqIjTz5z6L1LQ543ISgM9XX+5Ak+ekgVoKDrAChkvZvNv5umGRQq7rhjdnGxeVqYzhCAgjotTIeBuYqGzndsXjZ7gmsYuZgRUAAUAAXJsgcUZl49Y923vmWeFqYzBKCgTgvTYWCuIs8bHrpwsq25oe4799+99C9Ky785+krG+9euvq0EUBgtQEHXAVDIejebfzdNJyi8/fbbIys3G6aF6QwBKKjTwnQYmKso5c8TXaPwyekd/wIopApQ0HUAFLLezebfTdMJClu2bBlZudkwLUxnCEBBnRamw8BcqaAwPHj2f357cTwQ2LgyY0e+17Uo8AI2dGDBJQqMdFj3QNW6B6oKdidQwIYONhTI0Wm3aPK/YpmYKOgK6CwmCnIaEwVLJgquE/EfLW2epthv1r4LTBTUaWE6DMwFKAAKkgUoyGmAgg2g0TLtrQAAIABJREFUsGJFuetEDhw4kJW0MJ0hAAV1WpgOA3NNIygk3mtZt3D0oykv93buqVs6y3UWrH3qaG9mz6wEFHQFdBagIKcBCjaAwpJbv+o6kXg8npW0MJ0hAAV1WpgOA3NNGyhc7ok+MmfUM6wTA53bli19PNZ72Uv0xDaWL2s4Nv5CyvECFHQFdBagIKcBCjaAwrzrS0YeLW2eFqYzBKCgTgvTYWCu6QGFxEDnT9Zu2LC2OAUUEmeay26ti1384q9citUVr27uGpww44oABV0BnQUoyGmAQt5BYcyjpQ3TFN08i98FQEGdFqbDwFxjQGF46MKJ1mc2frfhSL/nDfcf/+mWHW3v9JneGTlwbOuan3T2dNSlgEKiq2mFk0oGF2O1t47+YCK9AAVdAZ0FKMhpgELeQeGll15yncirr76alTRFN8/idwFQUKeF6TAw1yhQGO5/fYu/SuOaaK//nYGTO/929eZfnTNghQ87G/5ua+eH3qVYCigMdjWtdos3xi6NgMHFWO1Ct6ypazJSABR0BXQWoCCnAQp5B4X13/ve3Gu/PP4CBV2aoptn8bsAKKjTwnQYmCsVFC6/v6/qK/f8eF970/ervgAFz/vsg30PuPMeP/JxZpcajlVioPMna/2LD0aBwngsmBooCFvaVwEKOgtQkNMAhbyDwi0331z6V6XZSlN08yx+FwAFdVqYDgNzpYLCQGdD9Y7Tn3hDnVu/MwIKn767517XuS3dio0ZaODYto0v9fjn/ukChcrKVWxsbAWyuU7ka8uW570GG5sNm+Y0nYHGTBTqd44GheGP/3vLLTNcJ6PLDMfpw86nnmjpufzFV3z0MMWXMFGQLSYKQgEhLUwTha6uLteJrFx5Z1bSdN08i98FJgrqtDAdBuZKBQVv+MKhRx9+5lBndNN9TSfPvvPmoWe/v3S267g3PPxf5xUrOF+K1RWn/Y/+1c1dg4muphWpoJA401x2HRczpgpQkC1AQSggpIUJFFqi0QerqqZ6hgAUdN0ABV0BXZq9oOB5nw+c3veDpalPnZ616KHn3hn4PAs/atREgdsjJ7cABdkCFIQCQlqYQKG2puZnu3cDCoCC0AFQMFfR+G8ND174fefhtmi0pfXQ66d7s/Y0qDGgwIJLk1mAgmwBCkIBIS1MoOA6kRMnTgAKgILQAVAwVyoofH6hbcfPzyquRchMY0HB87zLvcca1xZHXKesbtcxlnAeI0BBtgAFoYCQFhpQ8C9QiMfjgAKgIHQAFMyVCgpDvdGH3Lkrf/BM9I2z/aaLLOVMgIKugM4CFOQ0QCGPoOBfoCB0ABQABXVamA4Dc42aKPQf2f96/0Dvbzv2/PN3v/lww77Xzlwc1C2fkEMBCroCOgtQkNMAhTyCgn+BgtABUAAU1GlhOgzMVTTB94eHPvzDG9FnNqyp2rSz7TfnP8nWhQrm8kGh45XDedwqK1cVeAEbOmzYULN37wsFvhMKuYDrRHbsfLbAdwIF7OlgQ4EcnXZHgcLwp4OffkEEn3109o3o1u8un+u6TsSd++ALZz/NUYOpiomCroDOYqIgpzFRyNdEYeQCBaGD4l+WoptQQJeWxXeBiYI6LUyHgblSQWGoN7p1R+cfTr/2fMO3br3GibiOW1L+g+3RN85++FmOfrxCgIKugM4CFOQ0QCFfoDBygYLQAVAAFNRpYToMzDUGFB66snzCDXd8/5nW4+8NDNnzmcMXAhR0BXQWoCCnAQr5AoWRCxSEDoACoKBOC9NhYK5xoDC3om5nW+f7l7jrAVDwAIXJ0gCFfIGCv4KC3AFQABTUaWE6DMw1BhT+ZcfpT3L0k7IlQEFXQGcBCnIaoJAXUEi9QEHoACgACuq0MB0G5koFhdSLGVPl3zaZjVWcsyFAQVdAZwEKchqgkBdQSL1AQegAKAAK6rQwHQbmKuqK1n//sdauwWHP+/SD2H9sbWgYt21+eGl1S68tn0UACroCOgtQkNMAhbyAQuoFCkIHQAFQUKeF6TAwV9GOJTNcp2LH6bjnJT4+8vhX0jzsMeI6DwEKqQIUdB0Ahax3s/l3U+5AIfUCBaEDoAAoqNPCdBiYq2jg7BuHjpwd8D9xuHzimXu3HXn/fO8onT3SUG8bKHTke12LAi9gQwcWXCrMAjt2Pus6kbb9Bwt5J1DAwg42FMjRabdo9JefD/zpwsCoyxQuf3Tx0uCf//fP1twnyURBV0BnMVGQ05goTP9E4We7d6deoCB0UPzLUnQTCujSsvguMFFQp4XpMDBXKih8fuGXT2w/Oeauh0/P//JH1c2/HbCFEwAFZQGdBSjIaYDC9IPCg1VVqRcoCB0ABUBBnRamw8BcqaAw1Bt9bGvnwOi/kBg8/WzFVWv2sIRzigAFXQdAIevdbP7dlAtQiMfjYy5QEDoACoCCOi1Mh4G5ijzP8z77Q9s/VJSkv4wx4joRF1AYLUBB1wFQyHo3m3835QIUTpw4kbqCgtwBUAAU1GlhOgzMNTJR+PTC8WcfuH52afk31665f9T2nY3b287w0UOqAAVdB0Ah691s/t2UC1D42e7dtTU1GXYAFAAFdVqYDgNzFaX8efD9fY89Ofajh08vnn3/I2uuZPQABW0BnQUoyGmAwjSDQmVFRUs0mmEHQAFQUKeF6TAwV9Gor4Y+HRzLBJ9//MaT924+fMEaVAAUdAV0FqAgpwEK0wkKfX19rhPp6urKsAOgACio08J0GJhrDCicizWsKb1q/GUK/opMVghQ0BXQWYCCnAYoTCcoHD16NO2/fUABUBA6AArmSgWFz/sP/f1Mp2TZ6nvvXvoXVy5WWLXs+lvu3vSfpwfsetZDR77XtSjwAjZ0YMGlQitQXb1+3QNVBb4TKGBtBxsK5Oi0mwoKA50N67a80TfsDX/yxrb10feGPc/zBt/fV/Pgvt9/lqOfP3UxUdAV0FlMFOQ0JgrTOVGorKhob2/PvIPiX5aim1BAl5bFd4GJgjotTIeBuVJB4ZPf/uSxFz74zPM87/KJ7VX/8c7gsOd5Xm907dXfbTlvCyoACroCOgtQkNMAhWkDhQMH210ncu7cucw7AAqAgjotTIeBuVJBIfHxkR99dXVNw9b/aPvdxfO/rK2s+/lrxw/vq1txjXPbuIWY8iZAQVdAZwEKchqgMG2g8G//tmNRaemUOgAKgII6LUyHgbmKRn01dC5Wv6rEmX1705nPh862PLTQPytf8/XtJ+O23PYAKOgK6CxAQU4DFKYNFKof+V7j9sYpdQAUAAV1WpgOA3MVjfvO0MCfP/I/c/CG/ve37S+1Hjr+vjVXMnqAgraAzgIU5DRAYdpA4cb5Xzl69OiUOgAKgII6LUyHgbnGg8J4fd5/ZP/r/bawAqCgK6CzAAU5DVCYnjPEbzo7XSfS19c3pQ6AAqCgTgvTYWCuoljj1q0NDeK2+eGl1S29QzlqMFUBCroCOgtQkNMAhek5Qzy3Z0/lnZUTpQEKgILQAVAwV9GRTV+d8FlQye0h20ChI9+3qxZ4ARs6sI5C4RQov728tqauwHcCBSzvYEOBHJ12iy6f/Lc1Db9+v1fQ2SMN9baBQlqrg4kCEwVtGhMFnTUN/ynpP1r6+edfmCiNiQITBaFDdnd1EA8DcxV5w5989NFl6a8MDQ786U9/tua5UICCroDOAhTkNEBhGs4Q/qOlhRMeoAAoCB0ABXMVTfYXhj873bR+z+8TOfr5UxegoCugswAFOQ1QmIYzROP2xr9bvx5QECxAQegAKJgrFRQ+v/DL712T9hqFW57sZB2FFAEKug6AQta72fy7KVu/hSsrKlpaWgAFwQIUhA6AgrlSQWGoN/pwydJVa9fcn9xW31Yy97a712xqOftpjhpMVYCCroDOAhTkNEAh12eIc+fOuU7k/50+DSgIFqAgdAAUzDV6otDW1Hp+9EWLwx+e3F7z+K/O2XIpI6CgLaCzAAU5DVDI9Rmivb29sqJCPuEBCoCC0AFQMFfRpH9j+OyeynmbYiy4lCJAQdcBUMh6N5t/N2Xlt3BtTU3j9kZAQbYABaEDoGCuyUBhePDC4fpb7XsolLClfRWgoLMABTkNUMj1GcJ1IkePHgUUZAtQEDoACuZKBYWJL2a0b6IgbJWVq9jY2EKwfW3ZcteJrFx5Z96bsLEFYsvRaTcVFNJdzLjm4bqG5w6f/diWex746EFbQGcxUZDTJtoJTBR0aWO+/7Pdux+sqvIm+y/jiTowUWCioE4L02FgrtEThfEXM9onQEFXQGcBCnIaoJDTM0RlRUVLNOoBCpNZgILQAVAwV9Hkf8UyAQq6AjoLUJDTAIXcnSH6+vpcJ3Lu3DkPUJjMAhSEDoCCuTIBBR4zPVaAgq4DoJD1bjb/bjL8Ldze3r6otNT/M6AgW4CC0AFQMNcYUPjso67XXtixjcdMe4CC53mAwmRpgELuzhBb6usbtzf6fwYUZAtQEDoACuZKBYXh+MntZVcF4zHTaS1AQfeDBAtQkNMAhdydIfwbI/0/AwqyBSgIHQAFc6WCwkBnw23uvO///MTZ8zxmGlDwPA9QmCwNUMjRGcJ/YmQ8Hve/BBRkC1AQOgAK5koFhaELv/y7mffs+2DsrZDDQ3/+Xx4znSpAQdcBUMh6N5t/N5n8Fm7c3ujfGOkLUJAtQEHoACiYq2jUV0PnYptrd74zZhHGofOtTW0X7LqYseOVw3ncKitXFXgBGzps2FCzd+8LBb4TwlpgyeIlTzzRUOA7gQLB6mBDgRyddkeDwvClruijZaXl949ac2nVsrkP89FDqpgo6DowUch6N5v/I0aR5n/ff2Kkf2OkLyYKssVEQeiQ3V0dxMPAXKM/ejhUe0P6dZG5mHGUAAVdB0Ah691s/t2k/i3sPzEy9fuAgmwBCkIHQMFcqaAw0Nlwm3vL4+3nPxl9PUK8q+lJQCFVgIKuA6CQ9W42/25S/xZ+sKpq5MZIX4CCbAEKQgdAwVypoDAc73zy1oo97467bHH40oeXbLmWEVBQFtBZgIKcBihk/QwRj8ddJ3LixInU7wMKsgUoCB0ABXMVjfpq+KN3mjc/+UYfEwUPUPA8D1CYLA1QyPoZ4ujRo+P/gQMKsgUoCB0ABXOlgsJgV9PqCZ7dzDUKowQo6DoAClnvZvPvJt1v4S319Vvq68d8H1CQLUBB6AAomCsVFIY/+11z5VUly1avWctdD4CC53mAwmRpgELWzxCLSktHFmQcEaAgW4CC0AFQMFfRqK+GP2hrPHR+zOUIQwNnX2QdBevul81vARs6sI5C+Ars2Pms60SiLS8X8k6gQEA72FAgR6fdosn+wvBnp5vW7/l9Ikc/f+pioqAroLOYKMhpTBSyeyjW1tSlLsg4IiYKssVEQeiQ3V0dxMPAXKmg8PmFX37vmrTXKNzyZGfcltseAAVdAZ0FKMhpgEJ2D8Uli5e0RKPjvw8oyBagIHQAFMyVCgpDvdGHS5auGnWBwurbSubedveaTS1nP81Rg6kKUNAV0FmAgpwGKGTxUBy/IOOIAAXZAhSEDoCCuUZPFNqaWs+Pvmhx+MOT22se/9U5Wy5lBBS0BXQWoCCnAQpZPBRbotEli5ek/fuAgmwBCkIHQMFcRZP+jeGzeyrnbYr123UxY1oLUND9IMECFOQ0QCGLh+KDVVW1NXVp/z6gIFuAgtABUDDXZKAwPHjhcP2tzm1bOwcm+ZvTJUBBV0BnAQpyGqCQrUOxr6/PdSI7dj6b9u8DCrIFKAgdAAVzpYLCxBczMlEYLUBB1wFQyHo3m383TSmtvb19UWmp7tQOKAAKQgdAwVypoJDuYsY1D9c1PHf47Me23PMAKGgL6CxAQU4DFLJ1KG6pr2/c3ggo6CxAQegAKJhr9ERh/MWM9okFl2woYEMHFlwKTYG2/Qf9zx0KeSdQIOgdbCiQo9NuCigMDVy8ODAKE4b+t+v3fbaBAxMFXQGdxURBTmOikJVD0X8QVDweV/xbEDro0hT7zdp3gYmCOi1Mh4G5iuInd323fOE1TuSa0m/9+LnjF1M/Yxi+1NX65OPPnRqw54MHQEFbQGcBCnIaoJCVQ3HkQVCAgs4CFIQOgIK5irzh7j0Vtz6w87/PD6Zdpjn+hz1//8Nxz3/IowAFXQGdBSjIaYBCVg5F14n4D4ICFHQWoCB0ABTMVTT8/osP/sOv+gUQ+OTIY8ue+e1lW1ABUNAV0FmAgpwGKJgfiidOnPA/dxBeAijIFqAgdAAUzFXUf+jHjx/pF//OBy1r7m/uGsxRg6kKUNAV0FmAgpwGKJgfio3bG/3PHYSXAAqyBSgIHQAFcxX1Rh+bZDGl4T++cM/tLLiUKkBB1wFQyHo3m383ZZi2qLS0vb1dfgmgIFuAgtABUDBX0SdHfnz3nm7hc4XhvkM/KK7YcTqeowZTFaCgK6CzAAU5DVAwPBT9zx36+vrklwAKsgUoCB0ABXMVDfcd+sGSTYfOT/BwyOEPTz6z+poFW978xK5rFDryfbtqgRewoQPrKISgQHX1+vLbywt8J1AgHB1sKJCj026R5w28s+OemaUPP/PL4+/2D6bgwNDA+8db6r95g/PlyqbTn+Xo509dTBR0BXQWEwU5jYmC4aGY+rmD8BImCrLFREHokN1dHcTDwFxFnud5Q+dij91+jRNxHffKEs7fvOOvZrlOxHXcG771XNegLeMED1DQFtBZgIKcBiiYHIpjPncQXgIoyBagIHQAFMxV9MX/Dv1v53Mb75jrjnoW1NyKuuY3eofMKCHR+9ZT989xIq5TVhc9M/qSyMu9nXvqls5ynQVrnzram3Ydh3ECFHQFdBagIKcBCiaHYuP2xgerqjJ5CaAgW4CC0AFQMFdR6hfDgxd+f/LIodZoS7Qt9ubpDwbMl2/+8ET0P9/qvex5l3uPNa4tXlgXu3jFSgx0blu29PFY72Uv0RPbWL6s4Vgmd1YACroCOgtQkNMABZNDccznDsJLAAXZAhSEDoCCuYom/ysGSvzh6K97Ll/56mKsduGc2tilL7wzzWW3JrnhUqyueHUmqzUACroCOgtQkNMABfWhOP5zB+ElgIJsAQpCB0DBXLkFhVFKnGkuq9za+eEXX3U1rXBSyeBirPbWFU1nJv38AVDQFdBZgIKcBiioD8XxnzsILwEUZAtQEDoACuaaHlBIDHTFmmsfejTWc4UDBruaVrvFG2OXRsDgYqx2oVvW1DUZKQAKugI6C1CQ0wAF9aE4/nMH4SWAgmwBCkIHQMFc0wAKF2O1C10n4jqzltVGuwYSyW+OwgJAYawABdkCFIQCQpoNoLBj57PjP3cQXgIoyBagIHQAFMw1PRMFz0v0du6qXeZE5qyL9iQ8c1AQtsrKVWxsbDZvN/3FTXOv/XLea7CxhWzL0Ql8ukDB80Z/3GD60YOwpX0VEwWdxURBTptoJzBRkNPml8wb/7mD8BImCrI1UQEmCuq0MB0G5ppOUPASXU0rrsBB6p89z7/U8TouZkwVoCBbgIJQQEjLOyikvd9BfgmgIFuAgtABUDDXdIJC4lJsY8kXHz1we+TkFqAgW4CCUEBIyzsoNG5vLL+9fEovARRkC1AQOgAK5sopKFzuiT4yZ2nt7s7ehOclejseLXuo+cylKy4LLk1iAQqyBSgIBYS0vIPCotLSJ55omNJLAAXZAhSEDoCCuXIKComBM7vXFvuXDixY2xDt7L08+i/4yzVGXKesbtcxlnAeI0BBtgAFoYCQll9Q8D93aNt/cEo/BVCQLUBB6AAomCunoJATAQq6AjoLUJDTAIWpFmjc3rilvj6LpyihA6AAKKjTwnQYmAtQABQkC1CQ0wCFKRWIx+OuEzl69CigACgoCsgWoAAoJAUo6AroLEBBTgMUplTg6NGjrhOJx+OAAqCgKCBbgAKgkJQPCh2vHM7jVlm5qsAL2NBhw4aavXtfKPCdEKwC6x6oWvdAVYHvBAqEtYMNBXJ02g0qKKS1OpgoMFHQpjFR0FmZFxj53EHRjYmCbDFREDpkd1cH8TAwF6AAKEgWoCCnAQqZF2hvb19UWqrrBijIFqAgdAAUzAUoAAqSBSjIaYBC5gUerKpq3N6o6wYoyBagIHQAFMwFKAAKkgUoyGmAQoYF+vr6XCdy4sQJXTdAQbYABaEDoGAuQAFQkCxAQU4DFDIs0BKNVlZUqLsBCrIFKAgdAAVzAQqAgmQBCnIaoJBhgcqKip/t3q3uBijIFqAgdAAUzAUoAAqSBSjIaYBCJgXOnTvnOpFz586puwEKsgUoCB0ABXMFFRQ68n27aoEXsKED6ygEpUB19fry28sLfCdQIPQdbCiQo9NuUEEhrdXBRIGJgjaNiYLOyqTAotLS9vZ2k25MFGSLiYLQIbu7OoiHgbkABUBBsgAFOQ1QmLTAyLLNJt0ABdkCFIQOgIK5AAVAQbIABTkNUJi0wJb6+i319YbdAAXZAhSEDoCCuQAFQEGyAAU5DVCQC6Qu22zSDVCQLUBB6AAomAtQABQkC1CQ0wAFuUDqss0m3QAF2QIUhA6AgrkABUBBsgAFOQ1QkAukLtts0g1QkC1AQegAKJgLUAAUJAtQkNMABaHA3udfHLN8groboCBbgILQAVAwF6AAKEgWoCCnAQpCgerq9anLNpt0AxRkC1AQOgAK5goqKHTke12LAi9gQwcWXLK8wPySeU880VDgO4EChdPBhgI5Ou0GFRTSWh1MFJgoaNOYKOisiQqkXT5B3Y2JgmwxURA6ZHdXB/EwMBegAChIFqAgpwEKExWoralZ90BVtroBCrIFKAgdAAVzAQqAgmQBCnIaoJC2QF9fn+tEdux8NlvdAAXZAhSEDoCCuQAFQEGyAAU5DVBIW6AlGq2sqMjib2FAQbYABaEDoGAuQAFQkCxAQU4DFNIWqKyoaIlGAQVdGqAAKOgsQCEpQEFXQGcBCnIaoDC+wIkTJ1wn0tfXByjo0gAFQEFnAQpJAQq6AjoLUJDTAIXxBbbU19fW1OjSAAWdBSgIHQAFcwEKgIJkAQpyGqAwpoB/GaP/FChAQZcGKAAKOgtQSIoFl2woYEMHFlyysMDmzf80v2Rege8EChRmBxsK5Oi0G1RQSGt1MFFgoqBNY6Kgs8YUqKyo+Nnu3eo0xb+F8R0M0xT7zbZ3YURMFNRpYToMzAUoAAqSBSjIaYBCaoGRyxjVaYCCzgIUhA6AgrkABUBBsgAFOQ1QSC0wchmjOg1Q0FmAgtABUDAXoAAoSBagIKcBCiMFUi9jVKcBCjoLUBA6AArmAhQABckCFOQ0QGGkgL8ao2EaoKCzAAWhA6BgLkABUJAsQEFOAxRGCvirMRqmAQo6C1AQOgAK5gIUAAXJAhTkNEDBL+A/VHrkMkZ1GqCgswAFoQOgYK6ggoKwVVauYmNjm85t3vUlC+Z/Je812NgKfMvRaTeEoJD2VUwUdBYTBTltop1QUBMF/zLGrq4u8zQmCjprogJMFNRpYToMzBVUUEhr8dGD7gcJFqAgpwEKlZWrGrc3jrmMUZ0GKOgsQEHoACiYC1AAFCQLUJDTAIWVK+9cVFra3t6elTRAQWcBCkIHQMFcgAKgIFmAgpwGKCy59auLSkvj8XhW0gAFnQUoCB0ABXMBCoCCZAEKchqgMLt4VuP2xmylAQo6C1AQOgAK5gIUAAXJAhTktAIHha6urvF3RZr8IEBBZwEKQgdAwVyAAqAgWYCCnFbgoLClvn7e9SXZShO+DyjIFqAgdAAUzAUoAAqSBSjIaYUMCv5dkV9btjwrafL3AQXZAhSEDoCCuYIKCh2vHM7jVlm5qsAL2NBhw4aavXtfKPCdkMcC1dXrlyxekvc9UODvAgXs6WBDgRyddoMKCmmtDiYKTBS0aUwUpmTF43H/rsjp+U9JJgqyxURB6JDdXR3Ew8BcgAKgIFmAgpxWsKDQ3t7u3xUJKAgFdGmAAqCgswCFpAAFXQGdBSjIaQULCpUVFT/bvVtXQNENUJAtQEHoACiYC1AAFCQLUJDTChMUUp8VCSgIBXRpgAKgoLMAhaQABV0BnQUoyGmFCQq1NTUjiywBCkIBXRqgACjoLEAhKUBBV0BnAQpyWgGCgr/I0sizIgEFoYAuDVAAFHQWoJAUoKAroLMABTmtAEGhcXtjbU2NSQFFN0BBtgAFoQOgYC5AAVCQLEBBTis0UPAXWTp69KhJAUU3QEG2AAWhA6BgrqCCQke+17Uo8AI2dGDBpekv4C+yZNUesKEDBfJewIYONhTI0Wk3qKCQ1upgosBEQZvGRGFSy19kKXWcoCug6MZEQbaYKAgdsrurg3gYmAtQABQkC1CQ0woKFFqiUX+RJcMCim6AgmwBCkIHQMFcgAKgIFmAgpxWOKDQtv/gotLSlmjUvICiG6AgW4CC0AFQMBegAChIFqAgpxUOKDz99Pbx4wRdAUU3QEG2AAWhA6BgLkABUJAsQEFOKxxQWLJ4ycgiS4YFFN0ABdkCFIQOgIK5AAVAQbIABTmtQEAhdc1m8wKKboCCbAEKQgdAwVyAAqAgWYCCnFYgoFBZUVFdvT5bBRTdAAXZAhSEDoCCuYIKCh35vl21wAvY0IF1FKanwI6dz7pOJNrysp17wIYOFMh7ARs62FAgR6fdoIJCWquDiQITBW0aE4WJrNqami319RO9iomCUECXprCYKAgdsrurg3gYmAtQABQkC1CQ00IPCiOPgAIUgniGABTUaWE6DMwFKAAKkgUoyGmhB4Xamhr/EVCAQhDPEICCOi1Mh4G5AAVAQbIABTkt3KCQ+kRpQCGIZwhAQZ0WpsPAXIACoCBZgIKcFm5QGBknCK8CFIQCujRAAVDQWYBCUoCCroDOAhTktBCDQuo4QXgVoCAU0KUBCoCCzgIUkgIUdAV0FqAgp4UYFFLHCcKrAAWhgC4NUAAUdBagkBSgoCugswAFOS2soDBmnCC8ClAQCujSAAVAQWcBCkmx4JINBWzowIJLuStw910UKzhaAAAgAElEQVT33H3XPYHYAzZ0oEDeC9jQwYYCOTrtBhUU0lodTBSYKGjTmCiMWOPHCcKrmCgIBXRpCouJgtAhu7s6iIeBuQAFQEGyAAU5LZSgMObqBPlVgIJQQJcGKAAKOiu4oJAY6Dq0dc0C14m4TlndrmO9iVT3cm/nnrqls1xnwdqnjo62JhSgoCugswAFOS18oJB2nCC8ClAQCujSAAVAQWcFFRQSPfu3PXWoayDheZd7jzWuLZ61rOHYwBVzoHPbsqWPx3ove4me2MbyFEsSoKAroLMABTktfKCQdpwgvApQEAro0gAFQEFnBRQUBv/w6zd6knOCwa6m1W7xxtilhOd5XuJMc9mtdbGLX5iXYnXFq5u7BicNBRR0BXQWoCCnhQwUdu3ak3acILwKUBAK6NIABUBBZwUUFMYocSm2cc4VUEh0Na1wUsngYqz21hVNZyb9/AFQ0BXQWYCCnBYyULj7rnvSjhOEVwEKQgFdGqAAKOis8IBCybpoT8IbO13wPM+7GKtd6JY1dU1GCoCCroDOAhTktDCBwkRXJ8ivAhSEAro0QAFQ0FnhAIWLsdr7t3Z+eOXPY7BgaqAgbGlfBSjoLEBBTgsTKNTW1Nx91z1TTQMUhAK6NEABUNBZIQCFxEDnT9YmL1fMIShUVq5iY2Ob0va1ZctdJ1L29b/JexM2Njb1lqPz93SBwsCxbbV7zgyMUAAfPUxuMVGQLSYKQgEhLe33KysqGrc3Zvc/Ypgo6NKYKDBR0FkBB4XEey8/tS+FEjzPv5gxFRQSZ5rLruNixlQBCrIFKAgFhLTx3z969KjrRPr6+gAFRTehgC4NUAAUdFaQQSHRE3tqR6z38hdfDry9u+noJY/bIye3AAXZAhSEAkLa+O/74wRdN0BBKKBLAxQABZ0VWFAYONNSWzb6GoJZV8YGLLg0iQUoyBagIBQQ0sZ8v7293R8n6LoBCkIBXRqgACjorGCCQuK9lnULxl1smDo28JdrTLu684QCFHQFdBagIKcFHRTi8fii0tKWaFTdDVAQCujSAAVAQWcFExRyI0BBV0BnAQpyWtBBoSUaXVRaGo/H1d0ABaGALg1QABR0FqCQFKCgK6CzAAU5LdCg0NfXlzpO0HUDFIQCujRAAVDQWYBCUj4odLxyOI9bZeWqAi9gQ4cNG2r27n2hwHeCSYHq6vVLFi8J+h6woQMF8l7Ahg42FMjRaTeooJDW6mCiwERBm1ZoE4W+vj7XiRw9etSwGxMFoYAuLYvvAhMFdVqYDgNzAQqAgmQBCnJacEGhtqbmwaoq826AglBAlwYoAAo6C1BIClDQFdBZgIKcFlBQmOj5T4CCoptQQJcGKAAKOgtQSApQ0BXQWYCCnBZQUKisqEj7OGlAQdFNKKBLAxQABZ0FKCQFKOgK6CxAQU4LIig8/fT2kRWWzLsBCkIBXRqgACjoLEAhKUBBV0BnAQpyWuBAIR6Pzy+Z97Pdu7PVDVAQCujSAAVAQWcBCkkBCroCOgtQkNMCBwot0ej8knkjKyyZdwMUhAK6NEABUNBZgEJSgIKugM4CFOS0YIGCf0vkE080TLWAYAEKQgFdGqAAKOgsQCEpFlyyoYANHVhwaaoF1j1QZb7Ckm17wIYOFMh7ARs62FAgR6fdoIJCWquDiQITBW1a6CcKI7dEZvcIYaIgFNClZfFdYKKgTgvTYWAuQAFQkCxAQU4LEChUVlRsqa/XFRAsQEEooEsDFAAFnQUoJAUo6AroLEBBTgsKKLREoyO3RAIKgIKuG6CgK6BLAxSMBCjoCugsQEFOCwQoxOPx1KdEAgqAgq4boKAroEsDFIwEKOgK6CxAQU4LBCg0bm+srKgYuSUSUAAUdN0ABV0BXRqgYCRAQVdAZwEKcpr9oOBfw3jixAmTAoIFKAgFdGmAAqCgswCFpAAFXQGdBSjIafaDwoNVVWMe6wAoAAq6boCCroAuDVAwEqCgK6CzAAU5zXJQaG9vH/9YB0ABUNB1AxR0BXRpgIKRWHDJhgI2dGDBJblAtOXl+SXzNm/+p3DvARs6UCDvBWzoYEOBHJ12gwoKaa0OJgpMFLRp4ZsorHugKvUaRpMCgsVEQSigS8viu8BEQZ0WpsPAXIACoCBZgIKcZi0ojKzDmJUCggUoCAV0aYACoKCzAIWkAAVdAZ0FKMhpdoJCPB6vrKhY90BVtgoIFqAgFNClAQqAgs4CFJICFHQFdBagIKfZCQot0eii0tK2/QezVUCwAAWhgC4NUAAUdBagkBSgoCugswAFOc1CUDh37pzrRNrb2/P+uwlQ0KUBCoCCzgIUkgIUdAV0FqAgp1kICiMLJ+T9dxOgoEsDFAAFnQUoJAUo6AroLEBBTrMNFPyFE86dOyekAQqAgq4boKAroEsDFIzEOgo2FLChA+sojCkwPQsn2LYHbOhAgbwXsKGDDQVydNoNKiiktTqYKDBR0KaFYKJQW1PzYFXyTocsHoqCxURBKKBLy+K7wERBnRamw8BcgAKgIFmAgpxmDyj4HzqkLpyQ999NgIIuDVAAFHQWoJAUoKAroLMABTnNElCIx+OLSkt/tnt3JmmAAqCg6wYo6Aro0gAFIwEKugI6C1CQ0ywBhS319eNXa8777yZAQZcGKAAKOgtQSApQ0BXQWYCCnGYDKJw4cSLtas15/90EKOjSAAVAQWcBCkkBCroCOgtQkNPyDgorV945/kMHOQ1QABR03QAFXQFdGqBgJEBBV0BnAQpyWt5BYcH8r6R9RKSQBij8//bO/7WR+8zjf4oEaqEyZ2jYsHWvic2S3Z7j9nSOk7giTkKKbWotBNbdQIpG1XK7G6h3D7nr7dmwJSfXWpdN2uyIXbJ7W2mZGoKueNFFhZCKgbQstdAWI1whOkYcw9wPI82MpNHH8jMja0bzfjM/JHpWj9+az4w+Lz2fLwNQoHkDKNAM0LIBFCxJBQXGYfougAItBFBgZxssKHSudOglG0ABoEDzBlCgGaBlAyhY0pGgEA6/gQPH0B+vvvr6N7729YkXJwbuBAcOHA45+tTtuhUUTEMYeqD9IUYIFQV2tm4n4QQqClw0Gp6dff31Hxw3GyoKqCjQvKGiQDNAy4aKgiUBFGgGaCGAAjvboEBBe6aDk7+bAAq0bAAFgAItBFDQBVCgGaCFAArsbAMBhb29vTMTE2meZxhgZAMoABRo3gAKNAO0bAAFSwIo0AzQQgAFdraBgML5SER7poOTv5sACrRsAAWAAi0EUNAFUKAZoIUACuxsJw8Kt1OpMxMTlUqFbYCRDaAAUKB5AyjQDNCyARQsCaBAM0ALARTY2U4YFERRDPj8mUzmSAOMbAAFgALNG0CBZoCWDaBgSQAFmgFaCKDAznaSoKA++enaykovBhjZAAoABZo3gALNAC0bQMGSAAo0A7QQQIGd7SRB4drKypmJibZNGJ383QRQoGUDKAAUaCGAgi4VFLKPfz/AIxx+w+MGnODh/fejd+78xiMn4fr1RMDn39radlorDNyAEzzAwMANOMGDEwz0qdt1KyiYhrKoKKCiQM3m5IrCfz/KjL/woroeskcDjGz2XiGoKDAM0LLZ2AqoKJCzDdNlYF0ABYACKwRQYGc7AVCQJCn8g/Di/MKxDDD+EEABoEDzBlCgGaBlAyhYEkCBZoAWAiiws50AKGysb3z79Le6fVInfzcBFGjZAAoABVoIoKALoEAzQAsBFNjZ+g0KuVwu4PN/9NFvbGwFgAJAgeYNoEAzQMsGULAkgALNAC0EUGBn6ysoqFs1306l7G0FgAJAgeYNoEAzQMsGULAkgALNAC0EUGBn6x8oSJKkbdUMUAAosEMABYYHgIJ1ARQACqwQQIGdrX+gsLG+oW3VDFAAKLBDAAWGB4CCdbkVFLKDXq7qcQNO8DDE+yjcvLke8Plv/fJD57fCwA04wQMMDNyAEzw4wUCful23goJpKIuKAioK1GzOqSjs7e0FfP7bqVQv2Zz8I4aQjXAvMDzQshHOm2NbARUFcrZhugysC6AAUGCFAArsbLaDgiRJ4dlZLhrtMZuTv5sACrRsAAWAAi0EUNAFUKAZoIUACuxstoMCF412PtABoABQYIcACgwPAAXrAigAFFghgAI7m72gwEVjAZ9/b2+v92xO/m4CKNCyARQACrQQQEEXQIFmgBYCKLCz2QgK6t5KuVzuWNmc/N0EUKBlAygAFGghgIIugALNAC0EUGBnswsU1L2VlpcvHjebk7+bAAq0bAAFgAItBFDQBVCgGaCFAArsbLaAgjqB8XwkcjKtAFAAKNC8ARRoBmjZAAqWBFCgGaCFAArsbLaAgjaBEaAAUKCFAAoMDwAF63IrKGQHva+Fxw04wcNwbLi0vHzx9Knn73z0W5e2wsANOMEDDAzcgBM8OMFAn7pdt4KCaSiLigIqCtRsA6kotE1gPJlWsPcKQUWBYYCWzcZWQEWBnG2YLgPrAigAFFghgAI7mxVQEEUx4POned6KNyd/NwEUaNkACgAFWgigoAugQDNACwEU2NnIoKAuc9hY37DozcnfTQAFWjaAAkCBFgIo6AIo0AzQQgAFdjYaKGiPkG7bgRGgAFCghQAKDA8ABesCKAAUWCGAAjsbDRTORyKd+zTTvDn5uwmgQMsGUAAo0EIABV0ABZoBWgigwM5GAIXl5YtnJiY692mmeXPydxNAgZYNoABQoIUACroACjQDtBBAgZ3tuKBwO5UK+PyFQsEub07+bgIo0LIBFAAKtBBAQRf2UXCCASd4cN0+Cjdvrgd8/ps314epFQZuwAkeYGDgBpzgwQkG+tTtuhUUTENZVBRQUaBm63dFoVAoqIshB94K9l4hqCgwDNCy2dgKqCiQsw3TZWBdAAWAAisEUGBn6xEUjIshB94KAAWAAs0bQIFmgJYNoGBJAAWaAVoIoMDO1gsoqJTARaP98Obk7yaAAi0bQAGgQAu5GhTkmriT/iSx+M24UJVbQ/Vyfjs2ORLwjS3eyJVl8/e3CaBAM0ALARTY2Y4EBe3JkNpiyIG3AkABoEDzBlCgGaBl8xYoyGLytXcWfhj0B4JtoCDX8mtTk1eFcl2RS0J8ZiqxW+shIUCBZoAWAiiws7FBwXRjpYG3AkABoEDzBlCgGaBl8xYoKIqiKIdicq4dFOTiZuhsTNhv/G9ViAXnNsXDI3MBFGgGaCGAAjsbAxSePXumbqxUqVT6583J300ABVo2gAJAgRYaQlCQxeS0z0gG+wJ3djpZPHL8AaBAM0ALARTY2Rig8N5F842VBt4KAAWAAs0bQIFmgJYNoGD6yr7AjQdCSfEoUgAo0AzQQgAFdrZuJyHG/XT8hRePtf0izZuTv5sACrRsAAWAAi00fKDQiQXHAwXGEQ6/gQPHAI/v/PN3Aj7/9PTMwJ3gwIHDU0efuvAhBAXTd6GiQAuhosDO1nkSNtY3Aj7/r36VsvGToqKAigI71M0AKgrkbMN0GVgXhh4ACqwQQIGdre0kqJSQy+Xs/aQABYACOwRQYHgAKFjXoEBBkcXktPEVubgZeg6TGY0CKLBDTgMFjRLI2QAKAAVaCKDA8ABQsK6BgQKWRx4ZAiiwQ44ChTTPa5RAzgZQACjQQgAFhgeAgnUNDhSw4dJRIYACO+QcUMjlckZKIGcDKAAUaCGAAsMDQMG6+g8KVSEW1GYajrQOLtTLuxuLQX/AF4pt7WIL5zYBFNghh4BCJyWQswEUAAq0EECB4QGgYF0nU1GwUwAFmgFaCKDAznbu7Hc7KYGcDaAAUKCFAAoMDwAF6wIoABRYIYAC4y3G2YvWs9G8Ofm7CaBAywZQACjQQgAFXSooZB//foBHOPyGxw04wcP770fv3PnNoP768vLFgM8/+S8ve7wVBm7ACR5gYOAGnODBCQb61O26FRRMQ1lUFFBRoGY7ljetltDtJKCiQMtGuBcYHmjZCOfNsa2AigI52zBdBtYFUAAosEIAhc63GEccAAoABYYBWjaAAkCBFgIo6AIo0AzQQgCFtre0zUsAKAAUGAZo2QAKAAVaCKCgC6BAM0ALARS0t0iSdG1lpe3J0QAFgALDAC0bQAGgQAsBFHQBFGgGaCGAgvoWSZLORyJtlKAAFAAKTAO0bAAFgAItBFDQBVCgGaCFAAqKojx4+MiUEhSAAkCBaYCWDaAAUKCFAAq6AAo0A7QQQGFvb+/cS+fORyKVSqUzClAAKDAM0LIBFAAKtBBAQRf2UXCCASd4OIF9FO589NvTp56feWXmwcNHzjwJMOAEDzAwcANO8OAEA33qdt0KCqahLCoKqChQs3V6KxQKZyYmrq2sPHj4qFs2VBRQUWAYoGWzsRVQUSBnG6bLwLoACgAFVsizoKA+6mljfYOdDaAAUGAYoGUDKAAUaCGAgi6AAs0ALeRNULidSgV8/jTPH5kNoABQYBigZQMoABRoIYCCLoACzQAt5DVQkCSp81FPAAVGCKDAMEDLBlAAKNBCAAVdAAWaAVrIU6Dw7NkzdRmkKIo9ZgMoABQYBmjZAAoABVoIoKALoEAzQAt5BxT++1Fm/IUXz0cinZslABQYIYACwwAtG0ABoEALARR0ARRoBmghj4BCoVD49ulvLc4vSJJ0rGwABYACwwAtG0ABoEALARR0ARRoBmghL4BCmucDPn+M+ynBG0ABoMAwQMsGUAAo0EIABV3YcMkJBpzgwfqGSw8ePlpevhjw+W/eXHfpSYABJ3iAgYEbcIIHJxjoU7frVlAwDWVRUUBFoedslUrFOHWR5g0VBVQUGAZo2WxsBVQUyNmG6TKwLoACQIEVGlZQEEXxzMSE8QkOAAVaCKDAMEDLBlAAKNBCAAVdAAWaAVpoKEFBnZSwsb5hnLoIUKCFAAoMA7RsAAWAAi0EUNAFUKAZoIWGDBQkSVr6USTg82cyGVu8ARQACgwDtGwABYACLQRQ0AVQoBmghYYJFPb29sKzs6dPPd+5UwLZG0ABoMAwQMsGUAAo0EIABV0ABZoBWmhoQCGTyQR8fi4a7fYoSIACLQRQYBigZQMoABRoIYCCLoACzQAtNASgIEnStZUV7SFP9noDKAAUGAZo2QAKAAVaCKCgC6BAM0ALuR0U1OEG4+MbAAoABYACzRtAgWaAlg2gYEnYcMkJBpzg4cgNl65c+SDg87/15tsPHj4a1pMAA07wAAMDN+AED04w0Kdu162gYBrKoqKAioKiKIrCp+9z0ajp6gZ7vaGigIoCwwAtm42tgIoCOdswXQbWBVAAKLBCbgSFQqFw+tTz4dlZ09UNAAWAAkCB5g2gQDNAywZQsCSAAs0ALeQuUJAkaWN9I+Dzc9GY6XMgbfcGUAAoMAzQsgEUAAq0EEBBF0CBZoAWchEoiKKozlssFAr2flKAAiMEUGAYoGUDKAAUaCGAgi6AAs0ALeQKUJAk6XYqFfD5r62sqIUEgAIjG0ABoEDzBlCgGaBlAyhYEkCBZoAWcj4oGAsJVrIBFGghgALDAC0bQAGgQAsBFHQBFGgGaCEng8L16//x89Wfq/stag+BJGcDKNBCAAWGAVo2gAJAgRYCKOhSQYFxhMNv4Bj6I/Sv//ZPwZFvfO3r35v6/sDN4MCBA4cTjj51u0MICqbvQkWBFnJgRUFb2jDzysyf/vQni9kseut2ElBRoGVDRYEW6mYAFQVytmG6DKzLraBgGsLQA+0PMUJOA4VcLndmYkKdkUB4zLTt3gAKAAWGAVo2gAJAgRYCKOgCKNAM0ELOAYVKpaJutng7lVKXNgAUGAYY2QAKAAWaN4ACzQAtG0DBkgAKNAO0kBNA4cHDR2meD/j85yMR7dlOCkCBaYCRDaAAUKB5AyjQDNCyARQsCaBAM0ALDRwUCoXCuZfOnZmY6HxqA0CBYYCRDaAAUKB5AyjQDNCyARQsCaBAM0ALDRAUtLGG5eWLpvsxAxQYBhjZAAoABZo3gALNAC0bQMGSAAo0A7TQQEBB22lRHWvolg2gwDDAyAZQACjQvAEUaAZo2QAKlgRQoBmghU4eFDKZjLquIZfLsbMBFBgGGNkACgAFmjeAAs0ALRtAwZJUUMg+/v0Aj3D4DY8b6IeHra3tcy+dC/j8V6588ODhoyP//fvvR+/c+c2QnQQYcKMHGBi4ASd4cIKBPnW7bgUF01AWFQV3VhREUXzrzbfVpzq1bcbMyIaKAsMAI5u9VwgqCgwDtGw2tgIqCuRsw3QZWBdAAaDACvUbFCqVirrN4ltvvm1c+thLNoACwwAjG0ABoEDzBlCgGaBlAyhYEkCBZoAW6h8oaIgQnp0tFAoEbwAFhgFGNoACQIHmDaBAM0DLBlCwJIACzQAt1A9QkCRJ3UApPDt75IxFRgigwDDAyAZQACjQvAEUaAZo2QAKlgRQoBmgheztov7nD7u/3t5uW9RA9gZQYBhgZAMoABRo3gAKNAO0bAAFSwIo0AzQQnZ1UWoV4dunvzX+wotpnu/cQAmgQPPm5O8mgAItG0ABoEALARR0ARRoBmgh611UpVK5nUqpVYTV1RvPnj2zyxtAgWGAkQ2gAFCgeQMo0AzQsgEULAmgQDNAC1npoozTFXO5nCRJ9nafAAWGAUY2gAJAgeYNoEAzQMsGULAkbLjkBANsD1tb28vLFwM+/7mXzt28ud4nA9hwCQYc4gEGBm7ACR6cYKBP3a5bQcE0lEVFYdAVhUKhoD7JiYtG26YrsrMRvKGiwDDAyGbvFYKKAsMALZuNrYCKAjnbMF0G1gVQACiwQj12UZIk5XK58Oysurvi1tb2cbMBFGjenPzdBFCgZQMoABRoIYCCLoACzQAtdGQXZZyruLG+oW7AbG9nDFAAKAAU2CGAAsMDQMG6AAoABVaI0UV9b+r711ZW1LmKmUzGuOIRoABQoGUDKNBCAAWGB4CCdQEUAAqsUGcXJUlSJpNRRxm4aLRQKFjJZsUbQIFhgJENoABQoHkDKNAM0LIBFCwJoEAzQAsZu6hCoaAudzwzMXE7lXr11detZLPuDaDAMMDIBlAAKNC8ARRoBmjZAAqWBFCgGaCF/vfzP/7lL39J87xWQtDWMhA8ABRs9+bk7yaAAi0bQAGgQAsBFHRhH4WTMfDg4aObN9ffevNtdTuEK1c+4NP3HXUSsI8CDDjEAwwM3IATPDjBQJ+6XbeCgmkoi4qCHSfBOMQQ43762Wef2eUBFQXbvTn5RwwhG+FeYHigZSOcN8e2AioK5GzDdBlYF0ABoNAIqXxwZmJC3QvhyE2XAQoKQAGgwDRAywZQACjQQgAFXQAFmoFuIZUPTp96PuDzn49EMpmMuheCKoACOxtAAaDAMEDLBlAAKNBCAAVdAAWagbaQsX5wPhK5fj1h5ANNAAV2NoACQIFhgJYNoABQoIUACroACjQDiqI8ePgol8tdW1lR+YCLRrX6wcl0UQAF2705+bsJoEDLBlAAKNBCAAVdAIXjGtjb20vz/PlIRJ2fqM0/6MUDQIGdDaAAUGAYoGUDKAAUaCGAgi6AQi8G1Kc0aYML4dnZ26nUrV9+eFwPAAV2NoACQIFhgJYNoABQoIUACrqcAAoED/aCgqkBSZIKhQIXjanFA3XxgnFyoo1dFOMMELLRvNl7JRC8MQycDCjQWsHe28TeVrDrXiBnI3hzciucGCiczGXACJ1Yv8B4i2MvA+tyKyhkB7qvxcA9aAYePHx065cfctHYzCsz6oszr8xcufLB1tb2cJ8BJ3iAASd4gAEneICBgRvQPPSj23UrKJiGst6oKFQqFdWAVjk4H4ncTqUKhYIkSfaeBFQUaAZQUaBlO+69QDOAigLNGyoKNAOEbLQQQEGXB0FBkiRRFNM8ry1YUI80z4uiaP0PMUIABZoBgAItG0CBFgIoABSO9GBFAAUngoJKBr/e3l6+8GP1aUxq2WBjfSOTyezt7Z3YSQAo0AwAFGjZAAq0EEABoHCkBysaMCjI5d0UFwr4/KPzG0/K9V7eMpSgsLe3l8vl0jzPRaMaGSzOLyxf+LFp2QCgwPYAUGBkAygAFGjeAAo0A4RstJBFUJAkyXTbPWXAoFDbXZ2cuSSUZKVeFq5OTa7la/KRb3I7KKjVAg0LtHkGAZ+fi0Zvp1KZTEYlgxO7LRkhgALNAECBlg2gQAsBFAAKR3o4UuqzADfWNzpxYYCgcCgm50Y5odr4332BOzudLB5JCi4Chb29PVEUM5lMmueXfhTholGNCc5MTKhYkOb5ra1tU44DKLBDAAWHfzcBFAjZaCGAAkDhSA9HqlKpqKzQiQuDAwW5uBl6zkAGclWIj4aS4lGk4DRQEEVRFMVCoZDm+TTPv/Xm21w0apxyGJ6d5aLR5eWLaZ4vFAq9Tz8EKLBDAAWHfzcBFAjZaCGAAkDhSA89yhQXBgYKspic9o3HhH3thaoQH/XNbYqH7DdqHTAOHDhw4MCBo6/HxvrGoEChEwt6BQUFrIADBw4cOHD0/+CiUVEUXQkKEARBEATZK0mS0jyvPV5YGyh339ADBEEQBEH2qlAodCKCqsFOZjxrAIVDMTkX6GEyIwRBEARB9krd/Ldzur3ixuWREARBEASdmAYICsQNlyAIgiAIOjENFBQURS7n1ubHAr6RKW4739sWzhAEQRAEnZgGDAoQBEEQBDlZAAUIgiAIgroKoABBEARBUFcBFCAIgiAI6iqAgjNVFYV7dxMLp/Tlo03J5fwWN+XzB4ILa7vl1lUi9XJ+OzY5EvCNLd7IlV29gqQmPk4sjPr8phNd5fJuigsFfP7R+Y0nbXNgWefHbZLLT26oJyEU44u1lhirrVnnx7WSS3wk2Lohm0fuBUVRlH2BG2/uqjtiXEbulXuhoXo5//BuYmHU5zcsrffCvaBuSNi2v7LhdmC0tR2XAUDBgToUk0s/nA+33gyqDvKJ8FQ8W5YVuZy9NBlezR80Q3ItvzY1eVUo1xW5JMRnphK7tY7U7pD89P6NW4/FqqKtizEunVXuNFcAAAwzSURBVGWtqmWcH9fpoMB/8qRcV5R6eXdjMdiyjSmrrYdy1bH8NL001vLN6JF7QVGUBiSZdQ9euRcURWlyc3Bh9ZMdUf+YHrkXjKTYPPT9CRltbc9lAFBwquTiZmikDRRkMTkdjAtV9epofTB3206XVSEWdOt+2PJXuZ2Szv6tu32z9ulinR+3qfUk7AvcuP6pWW09lPuYHeQTF37CvWPc4t0j94KiKIpykE8sGDBRk1fuBUVRFOUgn5gxq5p4416o7qz+e9ZQDZCrwuXXemhruy4DgIJTZQIKHbtcG+4KWUxOt/zecvNd0aaqENN+T8vFzdBzhs9lvPRZ58fdkoubIf2nAKutWefHpZJr+V8sJv5QankWjJfuhaoQC/oDk9wmb/wl7al7Qa7l16aCF9Kl9rEDj9wLcvmrr9qb/p0e7gXbLgOAglPVCQqdr1SFWFAdsDwUk3MBnRyVRqnKnXdFu6pC7JuN7wjWs8RY58e9kmuisMm9e0koNT8Fq62H8Flrtd3V+V/ka//X8kE8dC8cisk5vdo8GU+LzbqSd+4FubgZem6K+1CdnRAILqxmxZqieO5eaEoWk68t8SW9ftalre27DAAKTlUvbay/0vlV6OovR6PkqnB5ujHuyHw6Oev8uFTawOTIFMc3f00y2nr4nt5+kE+8t5o/aP8gnrsX6uX8p5tcKODzN+freOhekMXktC8U21Jn4lWLyaVR38xq/sBj94KmQzH5butoS5e2tu8yACg4VQAFVY0flIYxNm98OepqTloebfyG8M6Xo1zL37rEP1UHW70NCqrqZeHqVONXsnfuBbkqxEeNZQO5uBkaCYSSouyde8Egubg5c1k/GwAFTwtDD4qiKAf5G1c3i4Zz4J1ya4uM7euZcmttdy1+rzniInt16KFN+rRWz9wLnQ2qvfIPr9wLBsli8rUe+wUMPQy/ukxmNL5inMjTOru1cyKPG1Uv3buVKrauD22b5NwyW4d1ftwuY/uy2pp1ftwl04Xj2qowr90Lmg7F5Ds9tPUw3QudHbz+Sb1xLxjVOu7QeKVbW9t2GQAUnCoPL49UFEVR6mXh1po+g69a3Nreaf6e9sySME1yVYifapm+5IElYS1q7y28dC8YtS/Erzc/tWfuhaoQC45FGoNQSsfSBi/dC23jDuprWB7pXZmBgmc2mamKfHyq2+4inthkpl7iL4xOcql8WVYUuZy9FHrXMATjkU1mjOr8WemNe0Eu5+89bOxMKpef3Ihf1unZI/eC0rgdgkubxWpj/7GQ9km9dS+0jzs0hA2XvKi2omv7trWNbX2bvYhB6hZ+/oA+Q9iNqpf4Cx0157Zta3Nr82Omuzszz4+LJNeKqcXGZnxjiwm+/WMy25p1ftwqswHm4b8XFEUuCfFQ8zL4WBDb+wgP3AuqqmJ2bTFouqe7d+6FQzHJmff0jLa24zIAKEAQBEEQ1FUABQiCIAiCugqgAEEQBEFQVwEUIAiCIAjqKoACBEEQBEFdBVCAIAiCIKirAAoQBEEQBHUVQAGCIAiCoK4CKEAQBEEQ1FUABQiCIAiCugqgAEEulFzO37uzOh9qfY5cI1YTd+4mFk6Z7Anvcsnl/NbPVts/cr2c/3STm+nheT9VUfjYcNJsPVHVndU4L7r8UQIQZCqAAgTZLbm4GRrRHlQxygl/F5PTJs+t2Be48fZHXvX2B5pPAxk3AYWqEAs2/u5QgYJcEuLvxvhi2+Od5Ma5HTkKFDpOms0nSn1Y0VXB9Q8UgKB2ARQgqB9SH21l6L1qX2zOj7U/4ks5FJPvqM92O6bUbs8MFJRujx51tQ7yiXcMDxpukSwmp48GBe1fGk6azSdKrhVTkaVUEXUFaLgEUICg/qijE5LF5LRvrLW32xe4d1vRodfsngIFWUxOdy+6OAYUFJX8enECQS4SQAGC+qRDMTkXCMaFarPXUIckjB1eVYjNHGvQQZOnQGFf4M4yel8ngYIii8lpY6NDkPsFUICgfqm9W2oMimujD3JVuPya3r1VxezaojpqPr/2WNQ6L9PXjaCgjb43h9v1/k8PaaP4zUH9trH5QzE5p02qqOrTLNR3NfM0usB6Ob8dm1T/Qahl3kBNfJxYGO18vSU0tpj4XXPen1wT+djkSCC4sMp/trP7pUmfXRViwbn2uotczm9xUz5/ILj0X59cbwWFqigkG/aCC6tZUbNxBChoOX3+wGQ83TjVzSmQ2T8+ubEwGlzaLFYVpSry8Smff3Q+kd7NPfm8YjBW3AydNQc4CHKnAAoQ1De19ENyVbgeS1yebpnMqI07HORvXF7bVecqVIvJpdFGl9zt9ZaKglzO/Wci+USbRtfa/8nl7KXJ5wz9aL3Ec5HkF22zAhWlXhauTum/hg/F5Jyh/nGQ3/pU/W9ZTE4HL6RL9YYlHX32Be5sY8pFY05Gc/JgbXeNu9VwWPtic35MZ5qZ99KluqIoivz0/lauAxTkqhAfbfuNLpeE+EzjD8mlnRtL39XPar3EXzg1r04UULvzscXmh2WCwqGYnBtd4kua+VBSlDXSGlu8kStXd1cnw6v5A1lMvqb+S0WRS79L7RiwQC5uhp7D6AM0TAIoQFD/ZBx92Bfi14WDL/XRB+O4Q3MGvuEYjwn7XV/XQeFvteJ2rG1hXntFvbWvlYubM5fNa+Mtv4bVP9GEgOrO2laj8zNW12UxOa3/GyEWHG95++RavqZhTesHCcaFqiyLyengwiqf7z6ds2MER5Fr+TUD0LQOPbR4UBTlIJ+YCbS4ZYKC4b+bkNQ5ynMoJudG5xNpc9f7Ajc+ROM+EARQgKB+Su+ZqsKluFBtVPjnNsV/GMYd5KoQHzWZrNft9WbIN7Z47eriNztGxDuH3mu7q82igiwmX+vajRk6S/np/a1P7nJnpxK7NUWuCh+mWov/cjl//5PEonEwpRMUGqHukwzkp+mlsYDPPzqfuCuIHUUOxQwU9gVu3DjVwwAKrXCj22iWXo6eo1Av5x/eVYdIuoKCIpf4SFAdQ/lYENvOZbs9CHK7AAoQ1E81uqLsX4Xrl/S+amQ6ufNYH3fo7N4U5utKCygER9oXWJr1fyX+wmgwLlT/xl5noVULZPHTVL7SLEX8Tbjxsd7zyeUnNxZOzSfu3suXioaKgrHsL5eEeKjZXzL7Trmcv/ehOqVgtFnPN4gACsZO/RigIJdza/MvLybu3M//uciqKDRc30+qExo6V7KgogANlQAKENRXqf1ceHH+Z3rlPzQSOPvy1Ezr8oegPzDJpbRqdu0LIV/p+nrL0ENqMagPw2t/or2vUosK167/hL3OojH6IDZmJKhFgk/uaOMObXMXWoYeFEWpfZFKXP7J5EjANzLFbecb0yZUty2v1D7/LN9SCKmXdzcWg53rODrnKLShQyO5YehhZCqxWzN7OwsUWuYWsIcejO7KT24sjLZzDGuNBgS5TgAFCOqv1FUGhm67Yy8mRWkMpRuH8Buj+91eN/Ze9bJw1Thlr8uqP/XvjnXbtqipQzE5FzgbvnTvqbZ95Ojke3f1IsS+wI03bDSWCYzHsvmdnT/LykE+wZmXK2q7q5Mjhg/S7Murf0g3ZwLKJT7SubpBMVn1oFb+R+c3npTrSu3LdOLdKX2Bhjq/srnmovbF5vzL2kdmgYKBMOTybooLBYLxx1/mdr76h0mVYufTnQYZ1Ev8hZbhIax6gIZOAAUI6rM6ew72er+25ZEmr+tLGQO+kenkl3/XpgoG48Jfs/r8x7Zqf1WIdU5o6PTbshOAyTwJuZy9NDnSWOVYysaCI1McL9Zks6mX+iLJRu/btjyy+oVw72FareEbCyct6vyNLtfE363Oq5MbUl/mP5ye5DbvaW/WF5QGJrnNxtSHjpWiRrehpCirvKWuqCyWhPioLxTjP883TvXIdydf1ueUfC7cz/KbXKi1cNJ59iBoGARQgCCviDmN0Rapv+bbWMF0jsXxxN6Z0UnCzozQEAqgAEHekPw0vbRkvc9mqfZleuuz1ppAvXTv4x0bfl4f5BPvLJrs/eAsySU+ElLHhiBoeARQgKDhlj7LwWxNgY3aF7izrXV4uSbu3GdskXAsySUhPudgVpBrIh+b/xmeHgkNnwAKEDTcasxhNOya3C8ZVgz6A76RKS7ZsceARVVF/vqqM+cJVndW4y2TFSBoaPT/v2hM6A1cN0YAAAAASUVORK5CYII=" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">(i) Write down the median weekly wage.</span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Find the interquartile range of the weekly wages.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">The box-and-whisker plot below displays the weekly wages of the employees.<br></span></p>
<p><br><img src="data:image/png;base64,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" alt></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;"><span style="font-family: 'times new roman', times; font-size: medium;">Write down the value of</span></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) \(a\) ;</span></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \(b\) ;</span></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) \(c\) .</span></span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b(i), (ii), (iii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">Employees are paid \($\ 20\) per hour.</span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">Find the median number of <strong>hours</strong> worked per week.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">Employees are paid \(\$ 20\) per hour.</span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">Find the number of employees who work more than \(25\) hours per week.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">(i) median weekly wage \(= 400\) (dollars) <em><strong>A1 N1</strong></em></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">(ii) lower quartile \(= 330\), upper quartile \(= 470\) <em><strong>(A1)(A1) </strong></em></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{IQR}} = 140\) (dollars) (accept any notation suggesting interval \(330\) to \(470\)) <em><strong>A1 N3</strong></em></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">Note: Exception to the <em><strong>FT</strong></em> rule. Award <em><strong>A1(FT)</strong></em> for an incorrect IQR </span><span style="font-family: 'times new roman', times; font-size: medium;"><strong>only</strong> if both quartiles are explicitly noted.</span></p>
<p><em><strong><span style="font-family: 'times new roman', times; font-size: medium;">[4 marks]</span></strong></em></p>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">(i) \(330\) (dollars) <em><strong>A1 N1</strong></em></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \(400\) (dollars) <em><strong>A1 N1</strong></em></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">(iii) \(700\) (dollars) <em><strong>A1 N1</strong></em></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;"><em><strong>[3 marks]</strong></em></span></p>
<div class="question_part_label">b(i), (ii), (iii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">valid approach <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">e.g. </span><span style="font-family: 'times new roman', times; font-size: medium;">\({\rm{hours = }}\frac{{{\rm{wages}}}}{{{\rm{rate}}}}\)</span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">correct substitution <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">e.g. </span><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{400}}{{20}}\)</span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">median hours per week \(= 20\) <em><strong> A1 N2</strong></em></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;"><em><strong>[3 marks]</strong></em></span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">attempt to find wages for 25 hours per week <em><strong> (M1)</strong></em></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">e.g. \({\text{wages}} = {\text{hours}} \times {\text{rate}}\)</span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">correct substitution <em><strong> (A1)</strong></em></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">e.g. \(25 \times 20\)</span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">finding wages \(= 500\) <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">65 people (earn 500\( \leqslant \)) <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">15 people (work more than 25 hours) <strong><em>A1 N3</em></strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">Many candidates answered this question completely correctly, earning full marks in all parts of the </span><span style="font-family: 'times new roman', times; font-size: medium;">question. In parts (a) and (b), there were some who gave the frequency values on the <em>y</em>-axis, rather </span><span style="font-family: 'times new roman', times; font-size: medium;">than the wages on the <em>x</em>-axis, as their quartiles and interquartile range.</span></p>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">Many candidates answered this question completely correctly, earning full marks in all parts of the </span><span style="font-family: 'times new roman', times; font-size: medium;">question. In parts (a) and (b), there were some who gave the frequency values on the <em>y</em>-axis, rather </span><span style="font-family: 'times new roman', times; font-size: medium;">than the wages on the <em>x</em>-axis, as their quartiles and interquartile range.</span></p>
<div class="question_part_label">b(i), (ii), (iii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">For part (c), the majority of candidates seemed to understand what was required, though there were a </span><span style="font-family: 'times new roman', times; font-size: medium;">few who used an extreme value such as \(700\), rather than the median value.</span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">In part (d), some candidates simply answered \(65\), which is the number of workers earning \(\$ 500\) or </span><span style="font-family: 'times new roman', times; font-size: medium;">less, rather than finding the number of workers who earned more than \(\$ 500\). It was interesting to note </span><span style="font-family: 'times new roman', times; font-size: medium;">that quite a few candidates gave their final answer as \(14\), rather than \(15\).</span></p>
<div class="question_part_label">d.</div>
</div>
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