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</div><h2>SL Paper 2</h2><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The heights of certain plants are normally distributed. The plants are classified into three categories. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The shortest \(12.92\% \) are in category A. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The tallest \(10.38\% \) are in category C. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> All the other plants are in category B with heights between \(r{\text{ cm}}\) and \(t{\text{ cm}}\) . </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;">Complete the following diagram to represent this information.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/big_bang.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 the mean height is \(6.84{\text{ cm}}\) and the standard deviation \(0.25{\text{ cm}}\) , find the value of <em>r</em> and of <em>t</em>. </span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</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;">Let \(A\) and \(B\) be independent events, where \({\text{P}}(A) = 0.3\) and \({\text{P}}(B) = 0.6\).</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: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find \({\text{P}}(A \cap B)\).</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 \({\text{P}}(A \cup B)\).</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 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;">On the following Venn diagram, shade the region that represents \(A \cap B'\).</span></p>
<p style="font: normal normal normal 21px/normal 'Times New Roman'; text-align: center; margin: 0px;"><img src="images/maths_4c.png" alt></p>
<div class="marks">[1]</div>
<div class="question_part_label">c(i).</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 \({\text{P}}(A \cap B')\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c(ii).</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The following frequency distribution of marks has mean 4.5.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/bbc3.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 <em>x</em>.</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><span style="font-family: times new roman,times; font-size: medium;">Write down the standard deviation.</span></p>
<div class="marks">[2]</div>
<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;">Let the random variable <em>X</em> be normally distributed with mean 25, as shown in the </span><span style="font-family: times new roman,times; font-size: medium;">following diagram.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/son.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The shaded region between 25 and 27 represents \(30\% \) of the distribution.</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 \({\rm{P}}(X > 27)\) .</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 standard deviation of <em>X</em> .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</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 table shows the average weights ( <em>y </em>kg) for given heights (<em>x </em>cm) in a population of men.</span></p>
<p style="font: normal normal normal 21px/normal 'Times New Roman'; text-align: center; margin: 0px;"> </p>
<table class="block_black_border" style="height: 106px; width: 347px;" border="0">
<tbody>
<tr>
<td><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Heights (<em>x</em> cm)</strong></span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">165</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">170</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">175</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">180</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">185</span></td>
</tr>
<tr>
<td><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Weights (<em>y</em> kg)</strong></span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">67.8</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">70.0</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">72.7</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">75.5</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">77.2</span></td>
</tr>
</tbody>
</table>
</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;">The relationship between the variables is modelled by the regression equation \(y = ax + b\).</span></p>
<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;">Write down the value of \(a\) and of \(b\).</span></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 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 relationship between the variables is modelled by the regression equation \(y = ax + b\).</span></p>
<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;">Hence, estimate the weight of a man whose height is 172 cm.</span></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 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;">Write down the correlation coefficient.</span></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 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;">State which <strong>two </strong>of the following describe the correlation between the variables.</span></p>
<table border="0">
<tbody>
<tr>
<td><span style="color: #3f3f3f; font-family: 'times new roman', times; font-size: medium; line-height: normal;">strong </span></td>
<td><span style="color: #3f3f3f; font-family: 'times new roman', times; font-size: medium; line-height: normal;">zero </span></td>
<td><span style="color: #3f3f3f; font-family: 'times new roman', times; font-size: medium; line-height: normal;">positive</span></td>
</tr>
<tr>
<td><span style="color: #3f3f3f; font-family: 'times new roman', times; font-size: medium; line-height: normal;">negative </span></td>
<td><span style="color: #3f3f3f; font-family: 'times new roman', times; font-size: medium; line-height: normal;">no correlation </span></td>
<td><span style="color: #3f3f3f; font-family: 'times new roman', times; font-size: medium; line-height: normal;">weak</span></td>
</tr>
</tbody>
</table>
<div class="marks">[2]</div>
<div class="question_part_label">b(ii).</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A company produces a large number of water containers. Each container has </span><span style="font-family: times new roman,times; font-size: medium;">two parts, a bottle and a cap. The bottles and caps are tested to check that they are </span><span style="font-family: times new roman,times; font-size: medium;">not defective.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A cap has a probability of 0.012 of being defective. A random sample of 10 caps is </span><span style="font-family: times new roman,times; font-size: medium;">selected for inspection.</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 probability that exactly one cap in the sample will be defective.</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;">The sample of caps passes inspection if at most one cap is defective. Find the </span><span style="font-family: times new roman,times; font-size: medium;">probability that the sample passes inspection.</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The heights of the bottles are normally distributed with a mean of \(22{\text{ cm}}\) and a </span><span style="font-family: times new roman,times; font-size: medium;">standard deviation of \(0.3{\text{ cm}}\).</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Copy and complete the following diagram, shading the region representing </span><span style="font-family: times new roman,times; font-size: medium;">where the heights are less than \(22.63{\text{ cm}}\).</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/flynn.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Find the probability that the height of a bottle is less than \(22.63{\text{ cm}}\).</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">c(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) A bottle is accepted if its height lies between \(21.37{\text{ cm}}\) and \(22.63{\text{ cm}}\). </span><span style="font-family: times new roman,times; font-size: medium;">Find the probability that a bottle selected at random is accepted.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) A sample of 10 bottles passes inspection if all of the bottles in the sample </span><span style="font-family: times new roman,times; font-size: medium;">are accepted. Find the probability that the sample passes inspection.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">d(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 bottles and caps are manufactured separately. A sample of 10 bottles and </span><span style="font-family: times new roman,times; font-size: medium;">a sample of 10 caps are randomly selected for testing. Find the probability that </span><span style="font-family: times new roman,times; font-size: medium;">both samples pass inspection.</span></p>
<div class="marks">[2]</div>
<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;">A box contains a large number of biscuits. The weights of biscuits are normally distributed with mean \(7{\text{ g}}\) and standard deviation \(0.5{\text{ g}}\) . </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;">One biscuit is chosen at random from the box. Find the probability that this biscuit </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(i) weighs less than \(8{\text{ g}}\) ; </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> (ii) weighs between \(6{\text{ g}}\) and \(8{\text{ g}}\) .</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;">Five percent of the biscuits in the box weigh less than<em> d</em> grams. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> (i) Copy and complete the following normal distribution diagram, to represent this information, by indicating <em>d</em>, and shading the appropriate region. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/8b.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Find the value of <em>d</em>.</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><span style="font-family: times new roman,times; font-size: medium;">The weights of biscuits in another box are normally distributed with mean \(\mu \) and standard deviation \(0.5{\text{ g}}\). It is known that \(20\% \) of the biscuits in this second box weight less than \(5{\text{ g}}\). </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the value of \(\mu \) .</span></p>
<div class="marks">[4]</div>
<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 histogram below shows the time <em>T</em> seconds taken by 93 children to solve a puzzle.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/sunburnt.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The following is the frequency distribution for <em>T</em> .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/cats.png" alt></span></p>
<p> </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>p</em> and of <em>q</em> .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Write down the median class.</span></p>
<div class="marks">[3]</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;">A child is selected at random. Find the probability that the child takes less than </span><span style="font-family: times new roman,times; font-size: medium;">95 seconds to solve the puzzle.</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;">Consider the class interval \(45 \le T < 55\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Write down the interval width.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Write down the mid-interval value.</span></p>
<p> </p>
<div class="marks">[2]</div>
<div class="question_part_label">c(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;">Hence find an estimate for the</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) mean;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) standard deviation.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">d(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;">John assumes that <em>T</em> is normally distributed and uses this to estimate the probability </span><span style="font-family: times new roman,times; font-size: medium;">that a child takes less than 95 seconds to solve the puzzle.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find John’s estimate.</span></p>
<div class="marks">[2]</div>
<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;">Consider the following cumulative frequency table.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img style="display: block; margin-left: auto; margin-right: auto;" src="images/carly.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 \(p\) .</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 style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (i) the mean;</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (ii) the variance.</span></p>
<div class="marks">[4]</div>
<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;">Each day, a factory recorded the number ( \(x\) ) of boxes it produces and the total </span><span style="font-family: times new roman,times; font-size: medium;">production cost ( \(y\) ) dollars. The results for nine days are shown in the following table.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><img style="display: block; margin-left: auto; margin-right: auto;" src="images/plates.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 equation of the regression line of <em>y</em> on <em>x</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;">Use your regression line from part (a) as a model to answer the following.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Interpret the meaning of</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) the gradient;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) the <em>y</em>-intercept.</span></p>
<div class="marks">[2]</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;">Estimate the cost of producing 60 boxes.</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The factory sells the boxes for $19.99 each. Find the least number of boxes that </span><span style="font-family: times new roman,times; font-size: medium;">the factory should produce in one day in order to make a profit.</span></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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Comment on the appropriateness of using your model to</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) estimate the cost of producing 5000 boxes;</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) estimate the number of boxes produced when the total production cost </span><span style="font-family: times new roman,times; font-size: medium;">is $540.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">e(i) and (ii).</div>
</div>
<br><hr><br><div class="specification">
<p align="JUSTIFY"><span style="font-family: times new roman,times; font-size: medium;">A factory makes switches. The probability that a switch is defective is 0.04. </span><span style="font-family: times new roman,times; font-size: medium;">The factory tests a random sample of 100 switches. </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 mean number of defective switches in the sample.</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 probability that there are exactly six defective switches in the sample.</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 there is at least one defective switch in the sample.</span></p>
<div class="marks">[3]</div>
<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 multiple choice test consists of ten questions. Each question has five answers. </span><span style="font-family: times new roman,times; font-size: medium;">Only one of the answers is correct. For each question, Jose randomly chooses one of </span><span style="font-family: times new roman,times; font-size: medium;">the five answers.</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 expected number of questions Jose answers correctly.</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 Jose answers exactly three questions correctly.</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 Jose answers more than three questions correctly.</span></p>
<div class="marks">[3]</div>
<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;">The following is a cumulative frequency diagram for the time <em>t</em>, in minutes, taken by </span><span style="font-family: times new roman,times; font-size: medium;">80 students to complete a task.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/sunday.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 median.</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 interquartile range.</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;">Complete the frequency table below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><img src="images/oned.png" alt></span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The price of a used car depends partly on the distance it has travelled. The following table shows the distance and the price for seven cars on <span class="s1">1 </span>January <span class="s1">2010</span>.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-02-02_om_17.58.43.png" alt="M16/5/MATME/SP2/ENG/TZ2/08"></p>
<p class="p1">The relationship between \(x\) and \(y\) can be modelled by the regression equation \(y = ax + b\).</p>
</div>
<div class="specification">
<p class="p1">On <span class="s1">1 </span>January <span class="s1">2010</span>, Lina buys a car which has travelled \(11\,000{\text{ km}}\).</p>
</div>
<div class="specification">
<p class="p1">The price of a car decreases by <span class="s1">5% </span>each year.</p>
</div>
<div class="specification">
<p class="p1">Lina will sell her car when its price reaches \(10\,000\)<span class="s1"> </span>dollars.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Find the correlation coefficient.</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Write down the value of \(a\) and of \(b\).</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 class="p1">Use the regression equation to estimate the price of Lina’s car, giving your answer to the nearest <span class="s1">100 </span>dollars.</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">Calculate the price of Lina’s car after <span class="s1">6 </span>years.</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">Find the year when Lina sells her car.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>A discrete random variable \(X\) has the following probability distribution.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2018-02-12_om_10.34.18.png" alt="N17/5/MATME/SP2/ENG/TZ0/04"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of \(k\).</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>Write down \({\text{P}}(X = 2)\).</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>Find \({\text{P}}(X = 2|X > 0)\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The maximum temperature \(T\), in degrees Celsius, in a park on six randomly selected days is shown in the following table. The table also shows the number of visitors, \(N\), to the park on each of those six days.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-14_om_17.34.22.png" alt="M17/5/MATME/SP2/ENG/TZ2/02"></p>
<p>The relationship between the variables can be modelled by the regression equation \(N = aT + b\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of \(a\) and of \(b\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the value of \(r\).</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>Use the regression equation to estimate the number of visitors on a day when the maximum temperature is 15 °C.</p>
<div class="marks">[3]</div>
<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 test has five questions. To pass the test, at least three of the questions must be </span><span style="font-family: times new roman,times; font-size: medium;">answered correctly.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The probability that Mark answers a question correctly is \(\frac{1}{5}\) </span><span style="font-family: times new roman,times; font-size: medium;">. Let <em>X</em> be the number of </span><span style="font-family: times new roman,times; font-size: medium;">questions that Mark answers correctly.</span></p>
</div>
<div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Bill also takes the test. Let <em>Y</em> be the number of questions that Bill answers correctly.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The following table is the probability distribution for <em>Y</em> .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/molly.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) Find E(<em>X</em> ) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Find the probability that Mark passes the test.</span></p>
<div class="marks">[6]</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;">(i) Show that \(4a + 2b = 0.24\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Given that \({\rm{E}}(Y) = 1\) , find <em>a</em> and <em>b</em> .</span></p>
<div class="marks">[8]</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;">Find which student is more likely to pass the test.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The following table shows a probability distribution for the random variable \(X\), where \({\text{E}}(X) = 1.2\).</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-15_om_06.18.09.png" alt="M17/5/MATME/SP2/ENG/TZ2/10"></p>
</div>
<div class="specification">
<p>A bag contains white and blue marbles, with at least three of each colour. Three marbles are drawn from the bag, without replacement. The number of blue marbles drawn is given by the random variable \(X\).</p>
</div>
<div class="specification">
<p>A game is played in which three marbles are drawn from the bag of ten marbles, without replacement. A player wins a prize if three white marbles are drawn.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \(q\).</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 \(p\).</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>Write down the probability of drawing three blue marbles.</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>Explain why the probability of drawing three white marbles is \(\frac{1}{6}\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The bag contains a total of ten marbles of which \(w\) are white. Find \(w\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Grant plays the game until he wins two prizes. Find the probability that he wins his second prize on his eighth attempt.</p>
<div class="marks">[4]</div>
<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;">In a school with 125 girls, each student is tested to see how many sit-up exercises (sit-ups) she can do in one minute. The results are given in the table below. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/margam.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;">(i) Write down the value of <em>p</em>. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> (ii) Find the value of <em>q</em>.</span></p>
<div class="marks">[3]</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 the median number of sit-ups.</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 mean number of sit-ups.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the following frequency table.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-14_om_09.52.57.png" alt="M17/5/MATME/SP2/ENG/TZ1/01"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the mode.</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>Find the value of the range.</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>Find the mean.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the variance.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Paula goes to work three days a week. On any day, the probability that she goes on a red bus is \(\frac{1}{4}\) .</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 expected number of times that Paula goes to work on a red bus in one week.</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;">In one week, find the probability that she goes to work on a red bus on exactly two days.</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;">In one week, find the probability that she goes to work on a red bus on at least one day.</span></p>
<div class="marks">[3]</div>
<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;">Two fair 4-sided dice, one red and one green, are thrown. For each die, the faces are </span><span style="font-family: times new roman,times; font-size: medium;">labelled 1, 2, 3, 4. The score for each die is the number which lands face 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;">List the pairs of scores that give a sum of 6.</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;">The probability distribution for the sum of the scores on the two dice is shown below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/bbw.png" alt></span></p>
<p><span style="font-size: medium;"><span style="font-family: times new roman,times;">Find the value of <em>p</em></span> , </span><span style="font-family: times new roman,times; font-size: medium;">of <em>q</em> , and of <em>r</em> .</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;">Fred plays a game. He throws two fair 4-sided dice four times. He wins a prize if the </span><span style="font-family: times new roman,times; font-size: medium;">sum is 5 on three or more throws.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the probability that Fred wins a prize.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The weights, \(W\), of newborn babies in Australia are normally distributed with a mean <span class="s1">3.41 kg </span>and standard deviation <span class="s1">0.57 kg</span>. A newborn baby has a low birth weight if it weighs less than \(w\) <span class="s1">kg</span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">Given that </span><span class="s2">5.3% </span>of newborn babies have a low birth weight, find \(w\).</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">A newborn baby has a low birth weight.</p>
<p class="p2">Find the probability that the baby weighs at least <span class="s1">2.15 kg</span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1"><span class="s1">Let \(C\) and \(D\) </span>be independent events, with \({\text{P}}(C) = 2k\) and \({\text{P}}(D) = 3{k^2}\), where \(0 < k < 0.5\)<span class="s1">.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Write down an expression for \({\text{P}}(C \cap D)\) <span class="s1">in terms of \(k\).</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 class="p1">Given that \({\text{P}}(C \cap D) = 0.162\) <span class="s1">find \(k\).</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 class="p1">Find \({\text{P}}(C'|D)\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">A factory has two machines, <span class="s1">A </span>and <span class="s1">B</span>. The number of breakdowns of each machine is independent from day to day.</p>
<p class="p1">Let \(A\) be the number of breakdowns of Machine <span class="s1">A </span>on any given day. The probability distribution for \(A\) can be modelled by the following table.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-02-01_om_15.41.35.png" alt="M16/5/MATME/SP2/ENG/TZ1/08"></p>
</div>
<div class="specification">
<p class="p1">Let \(B\) be the number of breakdowns of Machine <span class="s1">B </span>on any given day. The probability distribution for \(B\) can be modelled by the following table.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-02-01_om_15.43.25.png" alt="M16/5/MATME/SP2/ENG/TZ1/08.c+d"></p>
</div>
<div class="specification">
<p class="p1">On Tuesday, the factory uses both Machine <span class="s1">A </span>and Machine <span class="s1">B</span>. The variables \(A\) and \(B\) are independent.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find \(k\).</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) A day is chosen at random. Write down the probability that Machine <span class="s1">A </span>has no breakdowns.</p>
<p class="p1">(ii) Five days are chosen at random. Find the probability that Machine <span class="s1">A </span>has no breakdowns on exactly four of these days.</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">Find \({\text{E}}(B)\).</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 class="p1">(i) Find the probability that there are exactly two breakdowns on Tuesday.</p>
<p class="p1">(ii) Given that there are exactly two breakdowns on Tuesday, find the probability that both breakdowns are of Machine <span class="s1">A</span>.</p>
<div class="marks">[8]</div>
<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;">A fisherman catches 200 fish to sell. He measures the lengths, <em>l</em> cm of these fish, and the results are shown in the frequency table below.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/despic.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 an estimate for the standard deviation of the lengths of the 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;">A cumulative frequency diagram is given below for the lengths of the fish.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/flags.png" alt></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Use the graph to answer the following.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(i) Estimate the interquartile range.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Given that \(40\% \) of the fish have a length more than \(k{\text{ cm}}\), find the value of <em>k</em>.</span></p>
<p> </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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">In order to sell the fish, the fisherman classifies them as small, medium or large.</span></p>
<p style="margin-left: 60px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Small fish have a length less than \(20{\text{ cm}}\).</span></p>
<p style="margin-left: 60px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Medium fish have a length greater than or equal to </span><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: times new roman,times; font-size: medium;">\(20{\text{ cm}}\)</span> but less than </span><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: times new roman,times; font-size: medium;">\(60{\text{ cm}}\)</span>.</span></p>
<p style="margin-left: 60px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Large fish have a length greater than or equal to </span><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: times new roman,times; font-size: medium;">\(60{\text{ cm}}\)</span>.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the probability that a fish is small.</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The cost of a small fish is \(\$ 4\), a medium fish \(\$ 10\), and a large fish \(\$ 12\).</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Copy and complete the following table, which gives a probability distribution </span><span style="font-family: times new roman,times; font-size: medium;">for the cost \(\$ X\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/kirk.png" alt></span></p>
<p align="LEFT"> </p>
<div class="marks">[2]</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;">Find \({\text{E}}(X)\) .</span></p>
<div class="marks">[2]</div>
<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;">A random variable <em>X</em> is distributed normally with a mean of 20 and variance 9.</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 \({\rm{P}}(X \le 24.5)\) .</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;">Let \({\rm{P}}(X \le k) = 0.85\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(i) Represent this information on the following diagram.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/lowe.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Find the value of <em>k</em> .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<br><hr><br><div class="specification">
<p>Adam is a beekeeper who collected data about monthly honey production in his bee hives. The data for six of his hives is shown in the following table.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2018-02-12_om_10.46.13.png" alt="N17/5/MATME/SP2/ENG/TZ0/08"></p>
<p>The relationship between the variables is modelled by the regression line with equation \(P = aN + b\).</p>
</div>
<div class="specification">
<p>Adam has 200 hives in total. He collects data on the monthly honey production of all the hives. This data is shown in the following cumulative frequency graph.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2018-02-12_om_10.49.33.png" alt="N17/5/MATME/SP2/ENG/TZ0/08.c.d.e"></p>
<p>Adam’s hives are labelled as low, regular or high production, as defined in the following table.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2018-02-12_om_10.51.25.png" alt="N17/5/MATME/SP2/ENG/TZ0/08.c.d.e_02"></p>
</div>
<div class="specification">
<p>Adam knows that 128 of his hives have a regular production.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the value of \(a\) and of \(b\).</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>Use this regression line to estimate the monthly honey production from a hive that has 270 bees.</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>Write down the number of low production hives.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of \(k\);</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the number of hives that have a high production.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Adam decides to increase the number of bees in each low production hive. Research suggests that there is a probability of 0.75 that a low production hive becomes a regular production hive. Calculate the probability that 30 low production hives become regular production hives.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The following table shows the sales, \(y\) millions of dollars, of a company, \(x\) <span class="s1">years after it opened.</span></p>
<p class="p2" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-14_om_10.36.15.png" alt></p>
<p class="p2">The relationship between the variables is modelled by the regression line with equation \(y = ax + b\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Find the value of \(a\) and of \(b\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Write down the value of \(r\).</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 class="p1">Hence estimate the sales in millions of dollars after seven years.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</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;">The heights of a group of seven-year-old children are normally distributed with mean \(117{\text{ cm}}\)</span><span style="font-family: times new roman,times;"><span style="font-size: medium;"> and standard deviation \(5{\text{ cm}}\). A child is chosen at random from the group.</span></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the probability that this child is taller than \(122.5{\text{ cm}}\).</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;"><span style="font-size: medium;">The heights of a group of seven-year-old children are normally distributed with mean \(117{\text{ cm}}\)</span><span style="font-size: medium;"> and standard deviation \(5{\text{ cm}}\). A child is chosen at random from the group.</span></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The probability that this child is shorter than \(k{\text{ cm}}\) is \(0.65\). Find the value of k .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>In a large university the probability that a student is left handed is 0.08. A sample of 150 students is randomly selected from the university. Let \(k\) be the expected number of left-handed students in this sample.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \(k\).</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>Hence, find the probability that exactly \(k\) students are left handed;</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, find the probability that fewer than \(k\) students are left handed.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Ten students were surveyed about the number of hours, \(x\), they spent browsing the Internet during week <span class="s1">1 </span>of the school year. The results of the survey are given below.</p>
<p class="p1">\[\sum\limits_{i = 1}^{10} {{x_i} = 252,{\text{ }}\sigma = 5{\text{ and median}} = 27.} \]</p>
</div>
<div class="specification">
<p class="p1"><span class="s1">During week </span><span class="s2">4</span>, the survey was extended to all <span class="s2">200 </span>students in the school. The results are shown in the cumulative frequency graph:</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-03-03_om_16.35.16.png" alt="N16/5/MATME/SP2/ENG/TZ0/08.d"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the mean number of hours spent browsing the Internet.</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"><span class="s1">During week </span><span class="s2">2</span>, the students worked on a major project and they each spent an additional five hours browsing the Internet. For week <span class="s2">2</span><span class="s3">, write down</span></p>
<p class="p2">(i) <span class="Apple-converted-space"> </span>the mean;</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>the standard deviation.</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"><span class="s1">During week </span><span class="s2">3 </span>each student spent <span class="s2">5% </span>less time browsing the Internet than during week <span class="s2">1</span>. For week <span class="s2">3</span><span class="s3">, find</span></p>
<p class="p2">(i) <span class="Apple-converted-space"> </span>the median;</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>the variance.</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Find the number of students who spent between <span class="s1">25 </span>and <span class="s1">30 </span><span class="s2">hours browsing the Internet.</span></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Given that <span class="s1">10% </span>of the students spent more than <span class="s1"><em>k </em></span>hours browsing the Internet, find the maximum value of \(k\).</p>
<div class="marks">[6]</div>
<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;">A random variable <strong><em>X</em></strong> is distributed normally with mean 450. It is known that </span><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(X > a) = 0.27\) .</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;">Represent all this information on the following diagram.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/camel.png" alt></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;">Given that the standard deviation is 20, find <em>a</em> . Give your answer correct to the </span><span style="font-family: times new roman,times; font-size: medium;">nearest whole number.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The weights of fish in a lake are normally distributed with a mean of \(760\) g and standard deviation \(\sigma \). It is known that \(78.87\% \) of the fish have weights between \(705\) g and \(815\) g.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) Write down the probability that a fish weighs more than \(760\) g.</p>
<p class="p1">(ii) Find the probability that a fish weighs less than \(815\) g.</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 class="p1">(i) <span class="Apple-converted-space"> </span>Write down the standardized value for \(815\) g.</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Hence or otherwise, find <span class="s1">\(\sigma \)</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 class="p1">A fishing contest takes place in the lake. Small fish, called tiddlers, are thrown back into the lake. The maximum weight of a tiddler is \(1.5\) standard deviations below the mean.</p>
<p class="p1">Find the maximum weight of a tiddler.</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 class="p1">A fish is caught at random. Find the probability that it is a tiddler.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">\(25\% \) of the fish in the lake are salmon. \(10\% \) of the salmon are tiddlers. Given that a fish caught at random is a tiddler, find the probability that it is a salmon.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">A competition consists of two independent events, shooting at 100 <span class="s1">targets and running for one hour.</span></p>
<p class="p2">The number of targets a contestant hits is the \(S\) score. The \(S\) <span class="s2">scores are normally distributed with mean 65 and standard deviation 10</span>.</p>
</div>
<div class="specification">
<p class="p1"><span class="s1">The distance in km </span>that a contestant runs in one hour is the \(R\) score. The \(R\) <span class="s1">scores are normally distributed with mean 12 and standard deviation 2.5</span>. The \(R\) score is independent of the \(S\) score.</p>
<p class="p1">Contestants are disqualified if their \(S\) <span class="s1">score is less than 50 </span><strong>and </strong>their \(R\) score is less than \(x\) <span class="s1">km</span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">A contestant is chosen at random. Find the probability that their \(S\) <span class="s1">score is less than 50</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 class="p1"><span class="s1">Given that 1% </span>of the contestants are disqualified, find the value of \(x\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The weights, in grams, of oranges grown in an orchard, are normally distributed with a mean of 297 g. It is known that 79 % of the oranges weigh more than 289 g and 9.5 % of the oranges weigh more than 310 g.</p>
</div>
<div class="specification">
<p>The weights of the oranges have a standard deviation of σ.</p>
</div>
<div class="specification">
<p>The grocer at a local grocery store will buy the oranges whose weights exceed the 35th percentile.</p>
</div>
<div class="specification">
<p>The orchard packs oranges in boxes of 36.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that an orange weighs between 289 g and 310 g.</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 the standardized value for 289 g.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, find the value of σ.</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>To the nearest gram, find the minimum weight of an orange that the grocer will buy.</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>Find the probability that the grocer buys more than half the oranges in a box selected at random.</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>The grocer selects two boxes at random.</p>
<p>Find the probability that the grocer buys more than half the oranges in each box.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</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 weights in grams of 80 rats are shown in the following cumulative frequency diagram.</span></p>
<p style="font: normal normal normal 21px/normal 'Times New Roman'; text-align: center; margin: 0px;"><br><img src="images/maths_8.png" alt></p>
<p style="font: normal normal normal 21px/normal 'Times New Roman'; text-align: center; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="font: normal normal normal 21px/normal 'Times New Roman'; text-align: left; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>Do </em><strong><em>NOT </em></strong><em>write solutions on this page.</em></span></p>
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<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;">Write down the median weight of the rats.</span></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 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 percentage of rats that weigh 70 grams or less.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a(ii).</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;">The same data is presented in the following table.</span></p>
<table class="block_black_border" style="height: 72px; width: 802px;" border="0">
<tbody>
<tr>
<td><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Weights </strong>\(w\)<strong> grams<br></strong></span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">\(0 \leqslant w \leqslant 30\)</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">\(30 < w \leqslant 60\)</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">\(60 < w \leqslant 90\)</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">\(90 < w \leqslant 120\)</span></td>
</tr>
<tr>
<td><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Frequency</strong></span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">\(p\)</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">\(45\)</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">\(q\)</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">\(5\)</span></td>
</tr>
</tbody>
</table>
<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;">Write down the value of \(p\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b(i).</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;">The same data is presented in the following table.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"> </p>
<table class="block_black_border" style="height: 72px; width: 802px;" border="0">
<tbody>
<tr>
<td><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Weights </strong>\(w\)<strong> grams<br></strong></span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">\(0 \leqslant w \leqslant 30\)</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">\(30 < w \leqslant 60\)</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">\(60 < w \leqslant 90\)</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">\(90 < w \leqslant 120\)</span></td>
</tr>
<tr>
<td><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Frequency</strong></span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">\(p\)</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">\(45\)</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">\(q\)</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">\(5\)</span></td>
</tr>
</tbody>
</table>
<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 value of \(q\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b(ii).</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-size: medium; font-family: 'times new roman', times;">The same data is presented in the following table.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"> </p>
<table class="block_black_border" style="height: 72px; width: 802px;" border="0">
<tbody>
<tr>
<td><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Weights </strong>\(w\)<strong> grams<br></strong></span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">\(0 \leqslant w \leqslant 30\)</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">\(30 < w \leqslant 60\)</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">\(60 < w \leqslant 90\)</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">\(90 < w \leqslant 120\)</span></td>
</tr>
<tr>
<td><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Frequency</strong></span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">\(p\)</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">\(45\)</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">\(q\)</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">\(5\)</span></td>
</tr>
</tbody>
</table>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-size: medium; font-family: 'times new roman', times;">Use the values from the table to estimate the mean and standard deviation of the weights.</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 17.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Assume that the weights of these rats are normally distributed with the mean and standard deviation estimated in part (c).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 17.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the percentage of rats that weigh 70 grams or less.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</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;">Assume that the weights of these rats are normally distributed with the mean and standard deviation estimated in part (c).</span></p>
<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;">A sample of five rats is chosen at random. Find the probability that at most three rats weigh 70 grams or less.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
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<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;">A factory makes lamps. The probability that a lamp is defective is 0.05. A random </span><span style="font-family: times new roman,times; font-size: medium;">sample of 30 lamps is tested.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the probability that there is at least one defective lamp in the sample.</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 factory makes lamps. The probability that a lamp is defective is 0.05. A random </span><span style="font-family: times new roman,times; font-size: medium;">sample of 30 lamps is tested.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Given that there is at least one defective lamp in the sample, find the probability </span><span style="font-family: times new roman,times; font-size: medium;">that there are at most two defective lamps.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
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<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">A random variable <em>X</em> is distributed normally with mean 450 and standard deviation 20.</span></p>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \({\rm{P}}(X \le 475)\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Given that \({\rm{P}}(X > a) = 0.27\) , find \(a\).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
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<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;">A forest has a large number of tall trees. The heights of the trees are normally distributed with a mean of \(53\) metres and a standard deviation of \(8\) metres. Trees are classified as giant trees if they are more than \(60\) metres tall.</span></p>
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<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;">A tree is selected at random from the forest.</span></p>
<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 probability that this tree is a giant.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a(i).</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;">A tree is selected at random from the forest.</span></p>
<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 this tree is a giant, find the probability that it is taller than \(70\) metres.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a(ii).</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;">Two trees are selected at random. Find the probability that they are both giants.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
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<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;">\(100\) trees are selected at random.</span></p>
<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 expected number of these trees that are giants.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c(i).</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;">\(100\) trees are selected at random.</span></p>
<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 probability that at least \(25\) of these trees are giants.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c(ii).</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Evan likes to play two games of chance, A and B.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">For game A, the probability that Evan wins is 0.9. He plays game A seven times.</span></p>
</div>
<div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">For game B, the probability that Evan wins is <em>p</em> . He plays game B seven times.</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 probability that he wins exactly four games.</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;">Write down an expression, in terms of <em>p</em> , for the probability that he wins exactly </span><span style="font-family: times new roman,times; font-size: medium;">four games.</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Hence, find the values of <em>p</em> such that the probability that he wins exactly four </span><span style="font-family: times new roman,times; font-size: medium;">games is 0.15.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">A company makes containers of yogurt. The volume of yogurt in the containers is normally distributed with a mean of \(260\) ml and standard deviation of \(6\) ml.</p>
<p class="p1">A container which contains less than \(250\) ml of yogurt is <strong>underfilled</strong>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">A container is chosen at random. Find the probability that it is underfilled.</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">The company decides that the probability of a container being underfilled should be reduced to \(0.02\). It decreases the standard deviation to \(\sigma \) and leaves the mean unchanged.</p>
<p class="p1">Find \(\sigma \).</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">The company changes to the new standard deviation, \(\sigma \), and leaves the mean unchanged.</p>
<p class="p1">A container is chosen at random for inspection. It passes inspection if its volume of yogurt is between \(250\) and \(271\) ml.</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>Find the probability that it passes inspection.</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Given that the container is <strong>not</strong> underfilled, find the probability that it passes inspection.</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">A sample of \(50\) containers is chosen at random. Find the probability that \(48\) or more of the containers pass inspection.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</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;">Samantha goes to school five days a week. When it rains, the probability that she goes to school by bus is 0.5. When it does not rain, the probability that she goes to school by bus is 0.3. The probability that it rains on any given day is 0.2.</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: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">On a randomly selected school day, find the probability that Samantha goes to school by bus.</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 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 Samantha went to school by bus on Monday, find the probability that it was raining.</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;">In a randomly chosen school week, find the probability that Samantha goes to school by bus on exactly three days.</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 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;">After \(n\) school days, the probability that Samantha goes to school by bus at least once is greater than \(0.95\). Find the smallest value of \(n\).</span></p>
<div class="marks">[5]</div>
<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;">A standard die is rolled 36 times. The results are shown in the following table.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/bedtime.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 standard deviation.</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;">Write down the median score.</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><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">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The following table shows the average number of hours per day spent watching television by seven mothers and each mother’s youngest child.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-13_om_17.56.59.png" alt></p>
<p class="p1">The relationship can be modelled by the regression line with equation \(y = ax + b\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Find the correlation coefficient.</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Write down the value of \(a\) and of \(b\).</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 class="p1">Elizabeth watches television for an average of \(3.7\) hours per day.</p>
<p class="p1">Use your regression line to predict the average number of hours of television watched per day by Elizabeth’s youngest child. Give your answer correct to one decimal place.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</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;">The probability of obtaining “tails” when a biased coin is tossed is \(0.57\). The coin is </span><span style="font-family: times new roman,times; font-size: medium;">tossed ten times. Find the probability of obtaining </span><span style="font-family: times new roman,times; font-size: medium;"><strong>at least</strong> four tails.</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;">The probability of obtaining “tails” when a biased coin is tossed is 0.57. The coin is </span><span style="font-family: times new roman,times; font-size: medium;">tossed ten times. Find the probability of obtaining </span><span style="font-family: times new roman,times; font-size: medium;">the fourth tail on the tenth toss.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The following table shows the mean weight, <em>y</em> kg , of children who are <em>x</em> years old.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p style="text-align: left;">The relationship between the variables is modelled by the regression line with equation \(y = ax + b\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of<em> a</em> and of <em>b</em>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the correlation coefficient.</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>Use your equation to estimate the mean weight of a child that is 1.95 years old.</p>
<div class="marks">[2]</div>
<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;">The weights of players in a sports league are normally distributed with a mean of \(76.6{\text{ kg}}\), </span><span style="font-family: times new roman,times; font-size: medium;">(correct to three significant figures). It is known that \(80\% \) of the players have weights </span><span style="font-family: times new roman,times; font-size: medium;">between \(68{\text{ kg}}\) and \(82{\text{ kg}}\). The probability that a player weighs less than \(68{\text{ kg}}\) is 0.05.</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 probability that a player weighs more than \(82{\text{ kg}}\).</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;">(i) Write down the standardized value, <em>z</em>, for \(68{\text{ kg}}\).</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Hence, find the standard deviation of weights.</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;">To take part in a tournament, a player’s weight must be within 1.5 standard deviations </span><span style="font-family: times new roman,times; font-size: medium;">of the mean.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Find the set of all possible weights of players that take part in the </span><span style="font-family: times new roman,times; font-size: medium;">tournament.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) A player is selected at random. Find the probability that the player takes </span><span style="font-family: times new roman,times; font-size: medium;">part in the tournament.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">c(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;">Of the players in the league, \(25\% \) are women. Of the women, \(70\% \) take part in </span><span style="font-family: times new roman,times; font-size: medium;">the tournament.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Given that a player selected at random takes part in the tournament, find the </span><span style="font-family: times new roman,times; font-size: medium;">probability that the selected player is a woman.</span></p>
<div class="marks">[4]</div>
<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;">The probability of obtaining heads on a biased coin is 0.18. The coin is tossed </span><span style="font-family: times new roman,times; font-size: medium;">seven times.</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 probability of obtaining <strong>exactly</strong> two heads.</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 probability of obtaining <strong>at least</strong> two heads.</span></p>
<div class="marks">[3]</div>
<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;">The probability of obtaining heads on a biased coin is 0.4. The coin is tossed </span><span style="font-family: times new roman,times; font-size: medium;">600 times.</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 mean number of heads.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Find the standard deviation of the number of heads.</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 the number of heads obtained is less than one standard </span><span style="font-family: times new roman,times; font-size: medium;">deviation away from the mean.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A random variable \(X\) is normally distributed with \(\mu = 150\) and \(\sigma = 10\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the interquartile range of \(X\) .</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">The mass \(M\) <span class="s1">of a decaying substance is measured at one minute intervals. The points \((t,{\text{ }}\ln M)\) are plotted for \(0 \leqslant t \leqslant 10\)</span>, where \(t\) is in minutes. The line of best fit is drawn. This is shown in the following diagram.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-02-01_om_15.21.06.png" alt="M16/5/MATME/SP2/ENG/TZ1/05"></p>
<p class="p1">The correlation coefficient for this linear model is \(r = - 0.998\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">State <strong>two </strong></span>words that describe the linear correlation between \(\ln M\) <span class="s1">and \(t\).</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 class="p1">The equation of the line of best fit is \(\ln M = - 0.12t + 4.67\)<span class="s1">. Given that \(M = a \times {b^t}\), find the value of \(b\).</span></p>
<div class="marks">[4]</div>
<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;">In any given season, a soccer team plays 65 % of their games at home.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">When the team plays at home, they win 83 % of their games.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">When they play away from home, they win 26 % of their games.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The team plays one game.</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 probability that the team wins the game.</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;">If the team does not win the game, find the probability that the game was played </span><span style="font-family: times new roman,times; font-size: medium;">at home.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>A biased four-sided die is rolled. The following table gives the probability of each score.</p>
<p style="text-align: center;"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAeQAAABLCAYAAAChk/R2AAAYoUlEQVR4Ae2dD3BTx53HvzIc4RLzpxRfeCI5OOzD7Y1pEuyBlnITGYJMhivJ2YWktJZT+1q4QMKEw3JNQ462CQbkmiEtN/lzEsZcufBHHpIcTWxHxs2RUntkGmrmjBxDTWJL9JwD/yvBgLQ3+6RnPT3LYNl61hP8NOPR2/d2f7v72dX+dn/722cdY4yBPkSACBABIkAEiEBMCSTENHfKnAgQASJABIgAERAJjFdy0Ol0ylsUJgJEgAgQASJABFQiIBmqBylknp/0UKW8SWwcEOATM+oH6jQUsSWu6hBQTyr1WXXYKrmSyVodziSVCBABIkAEiEBEBEghR4SLIhMBIkAEiAARUIcAKWR1uJJUIkAEiAARIAIRESCFHBEuikwEiAARIAJEQB0CpJDV4UpSiQARIAJEgAhERIAUckS4KDIRIAJEgAgQAXUIkEJWhytJJQJEgAgQASIQEQFSyBHhoshEgAgQASJABNQhQApZHa4klQgQASJABIhARARIIUeEiyITASJABIgAEVCHQNwrZJ+nEZWledDrdOCvIdPp9MgssqG2pUcdYiRVQwR60FL7No6W5iGlqBbU4tFoGh/6WqpQmjcv8HvKQtH+Bnh80ZBNMnyej7BbYpu5BZU0TqnQKa6jo3ID9Fk2tMRZv41rhezz1ODFNd9CTmEFPAPN6kHdrgIsTV2D0saugbt0cacRuIYW2wv4yf5yPF9Ygat3WvViVB9fx/t48ziwcu8ZMNYPd/1KXCp+EmvKGtAXozLdMdn2ncGx6nF4el8TmNeN+pUd2GDYgdqeONMaGm8QX8d/4aUNe2U6QeMFlhUvjhXyNbS+9yZK6jyAkA9rc7f4zxC87moUGwQAx1G4pTLuZkiytqHLWxKYiLn5VvxqXwleNvL2ps/oCVzDHz+ZhtUbszA3kQ8NEyAsKMCPX16MurJjaCDFMQrE13De+QUW5n4dAkebIGBBQR5yUY0q5+VRyKWkIQT6GlC25UNMX5YWcjteAnGskG+i93Knn/O9X8VDKZPF6wTh7/FM7mL//aZWtPcFZ5+DzNv6PJTWtChm/twMekhmsgtnAv8MlXkp0OkysaPyKLZk6kVTeZbtHHzgJr9a2IqyyIQeL78CKmeAwEQkG76OmSGjwgTMmJ0CmvKMtpMMZuu71IaPU7+L1QumjVY4pRcJdKHxtYPA888ia8Y98cmEKT7+/76ouKnJ4BfMZV3FeHkBgRnMbzC7w8V6hyprbz2zGIRAfJ5G+ktj+fY25hXTdbNmaz4TBp5Jcfj3CmZxXglI/5TZTckyGfx5OjM7Opm33c7yBXm6wLWQz6zN3UOVTnP346YfeJuZ1Sgwwexg8UI3btiKvdLLuh3FLDnfztr9PxLN9VWpQPHDtZu5HFZmNr3MHO5+qfia/tY+Wy/rde5hJks962WdzGFOZzBamSvO+mzIXDi+phQTMXf1S7CauGmC7xv/EDlLUzFJp4M+rxRH326UOaJcQ8vhUhRy87ZhGxzufv/+mGMbDDgL24bXUdfjg6/lKDYW2OBBGkzWJvQyBuZth6PY6DeBb96HRtmKW+RlKIOz1wvm/RBbv+5D3avbYfPI0rNuNFvzIXhseHGfkxyP4quTUWlxGc6qDqx7doli5UxoRkSgpxZF+ilIXVqAXRUfourUBYWFbkRSKVGfE6+9Cjy/LgOJ8UxDOS3T/kxIUWKvmzmPvcHMYVa/gqmcNffyKZK0ok1mJvunCgFSULbiVs6suh3MLK56Dczi5GtwSZ7AjNbmwOqaMXbDySzJYVbH0opbKGaObo1P2QI44qYf0ApZ6sAqfMtXHSqIj7LIuOmzjDGvu56Vm40MkFvoogwkiuK0zfYKc5aVMHu7ZG2gFXLs5iQJAtKf+AF2nnCDsW64HHZYzXxFC3gqfoF9DZeBm/+LCx+dv00Zg3vSycsewhy57eDeKUi6lyf/DE0Xr8jk6PHw7OkYiNp5EU23ysZzGV1/Du5pywTRJRHQHoE+J948/CUUx/uqQ3tkkSAsQF7JHliN/4f36s/TKnnEbeRDX+NbsM/6Dp6cOWHEUrSScECXaKVAwy6HaPrxnzv2O1PxlJMxd0k28l/ZDksyDzfCfvoibo7/K8z5Jr9xFZe6/ozwKnE8Jk1LErM/X3MGF+SRrnajUzxX8yDmzfqSrIiTkDRlYjB831TMEL1fDLA4e0Wvb8bN3gN/ryFbGB+MT1dEQKsEfBfx9pt/xONbv4uviB7XWi1oHJcrYTYWPxVwQI3jasS26JfRcPjfUZIzG+MG3kWRhKW7GoHqAqSO0yOoH2Jb0uHkHr8KefLXkJWbLu4fV79Ygj0NnoCi7cG5in9DmbhSTUfO/FkYjySkPTrfH/fAIdR5rgPcG/rcfuTpuVJfDVsLkLJ4OcS1dfVuvFJ+1j9r9XWgdsdO7OIHnQ0rkZkqLpXDsx0oUx3KXrXjHN9v5um3+D2u9fTyivDc6K62CPA+u+c9THr6H4PKuO8s9ts+Ih+IqLZUH9pd3Xh8YXJ873tGlUmkwqZjyU6nbNHDF0CdcJjTAaMVLq8bVflfCVoxIxU/1vGV2wza3isILe2QHs2BPdvgHjJjbEgva4EZiquZW9zavcKclhUK72lpTzicl7W0pyyVy8t6m8uZibysJSDqf9MecnQZ9zYzu7i3KfV76VvhLxHdXKMiTdNjl7eN2fPTmcFcwZyiZ3U/czu2MeOAn0tUEKgmRNNsB9U6fveQ+cwi5BNf4LlzhJMds5qZQXKc4t+CiVmOnGAu0aErWD0e124xBY818Xh2Z0AZS/H4kYS3mMWUFlDM/EiVlTlc8kM1klOXUiFzGV7W63Iwq2xQE0xlrDokvZSXdr+13w/8x3FCj6itYlbXF9qFGiiZZtmKSkPq95Iilr61z1azXMV2VxypDDv2aLfraputklv8KmQdr4p8Vc7fB624JX9M13cJAeoH6jU0sVWHLXFVhyuXSmzVYavkGr97yOrwIalEgAgQASJABGJCgBRyTLBTpkSACBABIkAEQgmQQg7lQSEiQASIABEgAjEhQAo5JtgpUyJABIgAESACoQRIIYfyoBARIAJEgAgQgZgQIIUcE+yUKREgAkSACBCBUAKkkEN5UIgIEAEiQASIQEwIkEKOCXbKlAgQASJABIhAKAFSyKE8KEQEiAARIAJEICYESCHHBDtlSgSIABEgAkQglEDYV2eGRqEQESACRIAIEAEioBYB6XXVYf85r/RQrcxJrvYJKN+xqv0Sx08Jia06bUVc1eHKpRJbddgquZLJWh3OJJUIEAEiQASIQEQESCFHhIsiEwEiQASIABFQhwApZHW4klQiQASIABEgAhERIIUcES6KTASIABEgAkRAHQKkkNXhSlKJABEgAkSACEREgBRyRLgoMhEgAkSACBABdQiQQlaHK0klAkSACBABIhARAVLIEeGiyESACBABIkAE1CFAClkdriSVCBABIkAEiEBEBEghR4SLIhOBu4FAHxpLM8W3M+l061DpuXk3VJrqSARiTmAMFbL8R64L/Nil73nIK61Eo+d69IDcbERpCpefgrzKz0Yt92ZjKVJ0XN7tBqjPUJmXEpqvpxJ5IWlvwlO5zs8grxIeqXR9Lag9Wop1uxtxdw6B1+FpPICiTD10unnI2/0RPD4JzlDfPWip2Y08faAvZRZhf6MHg5L1taCmNA96sR100OftRk1Lz1BC7/L7iUjf/C6cFgNgXIC0+8O+YfcuZxSsvs/TgP1FWeLvWZ+3Fw3DGsd86GupQ+XRUuSlbEFtz6AeG8jgc9QWZcjGSz2ybOcG9+9gce7oq5Gxvg5Pw97AGKFHZlElWvqG4h1bfGOokG9V0bOoKMxBxpq9aNQoqFuVPjrP+tD42losXVWID7zRkRhfUnzoa9yLNZsvIOtgG5j3feR1lmBNWQP6hqzIdXS8fQDHsQJ73QzM60b9yksozvgByhq7gql8F1G5MQd5TUtQ2+sFY/1ozOvBdsOOWwyEweR35ZWvHWdqXEhe9hDmaGSU0GQ79DWgbM1P4cqywSv2q89RNIxxzNdSjqd/sg/25wtRcXXomvk6PsSvDjTKIizGU4tn465skhGx9qHv9++hGtnY52bwuo9g5aV/heFnddDkdJwpPgAUd6IV7GVOi4EBycxk/1QmtJ+563/JTAIYkM7Mjk7Zs1Fc3nAySzKXqcxvZDJvOC0sGVzeWmZ334hMiNvOTLdNK/EBS7Y4WYQ5RFaeYcRWrx8Mkbm3mVmNi0Lbv9vBzMIqZnV9ET6R9wI7ceJT5pU/FeUITDA7WHfgvtdlZUZl3wqXn1yOitdjznYkdRH7bBR/jyMpQ4Rpxp7rF8xlXRXS1xjrZA7zIma0Nof2y7B18aeHUMwc3SG9OBD7CnNaTKG/ibBy1L859myVdRoh60FjhJd1O4qZMCRzZb7qhpVcNTDRmgBhwUo8sSwZQBcudX0BIGjSTdlxFFVb/OYgXZYNLaKloQcttYdQmjcvYMrhZggbaocyQfo60LBbMldy83iVwmTBzUdVMnk66PR5KK1sDGsy9blPYreUN49X0yJbxYUxWcvmt/7LYP10osmap3kYGYV14uPzhRn4C10mdthKkSWaWFfD1nJNJuUaWmyr/SayolptzvRkpR3Opa/1tzhUPROpDyQGo0/+GrJyO3DoZFt4E13C38BgeDB0tZAwHbMf1gdlAEiYMRsPC240nL4QbKc+N1znH0VWxrSQuBTgBHzoaXaiBnNC24PghBLwteHkodOYl6pHsNdOQ0bWo2g69Fu0jtYq2nMah8sqsOtnFtgq6xRjVmhR7vjQSFkPGiOu41LbRaRuehILJmtA/SkaTgMl4vb9d/B2zXkAD2LerC+FFPF88SosL6kW7wkPz8aMhB6cs70Aw9KnUVhxNhDXg7pdBViaugalclOl+PQ8KlZ9Aws3VQT2arl5fDkM6w/gXMA87us4ho2G5TJ5ADwVKMx5Bi8eu6hQBq9jVUYmNkl583jGHKy3nQ0O9iE1GHlg/EM5KDKnAzgZqpTEznkSQDpys76GySPPQiMpr6H15PuoFlIwe8YERZn6UR3x4PZlPL4wOThITp6P1Zvmo66wwN9Ovg7Uln6AGQefg0GDP0oFgBgEr+KT06fgMS7H4vt+HfB/kPw9budDEYPixihL/yRyKh6ePT10UsjLU/0+TrbKJ9GRFvIaWo6+hl3cwaRuFwpyMpH6rRdROdSiI1LxcRY/Kqy5j47tp3il9Rkc3LQgOD5oiEUMFPJ5VOT8tcxJ4R7oF25AhQcQ8p/Ddx4JzjX9nFbA4rwCxm6gZesiJLYcxcYCGzxIg8nahF7G9w7b4Sg2AjiOws37Bu9DG7bB4e4X9w7djm0wiPr2F9jXcBnANbRWvQWbR4DBUh+Q1wZ7fhqAs/jNhc8VClmAobgabq8837OoePEgGoZ0zLhdiz+I7P0f+51oACRbnLjBTmBz+ixkZBkhwBOilPyd0wMYn0WBYfrthMfB8z60uy4A81LwQOIou2TPH1D18XI8a5SvnKcifdObqC+bj5qCeZg07gW0fe8lvLBAGDyQxgEt1YsY2D8WJ8AzH0OxdS2MZjtc4v77a8gWyMmLWxH62lvRpJoVYSLm5h/2j1nOd2E1G4G6EuSstQ4e31TvELHOYLSsfeip3QL9pFQsLShBxakTONWqyR1krYxHRpitDtTteRIzleOxMRsrH5kKYDwSE+FfSfH+YXwBP34mzT/LSZiJJT8qglngs8l3cMIl95JIhum5fCwR+MprAoQl67FVXHU2wn76Im5C6vhtOJj5Oaorj8JW/EPk2Pyr76ud3ZBLA1biuecyIfByyvP1nMLpT0Jjjr4bJ2ByxmPI5fUamHF34ffvVKIaAoxPLUKKktfoM41jCV1ofOMdJG3/PtKVij3hPkydlY5Nlk0w4AgK1u5E7bC8YeMYx0iL/qf/wW+q9ch97Ms4t3sLLNgA+85szFUyHal8ShcBgQkQ0v8B+TvfhbiYqPsVDosLiQhE3PVREzB5yXa4WT/czgqYUY4cwxZUdkTxVE+UGMdgOE+Gyf4p9xyT/VVhZ/6SsD94v5laqu1N9F7uFAODvD/vnYKke/mjz9B08YqUIIwZfCKmJE2SPfeh79x+5OnvgT5jBXJyVqFgl99EziPdmzQFolgpRXIKZiXJVggD+UoRovwtmltXBM3W4r7ScUDIxtqsOVqZUY2y0ol4IHUO0NSK9hF72XMv7bdweNo/YV06n8DJP3yb4yWU4Qls2vxzONzVKMbrWDoMb1i5lLvjWto/7kLD9gIs3PQ7TJ89Q5Pmvdi2RwISH0jBPFyAq33ocwDRK6O0mAAOVP3hjvAbGT6baLHmk5tclLz+Moye36Hepb1VcgwU8vCbgccMVYjjMWlakijgfM0ZXJA7TVztRqe4QFXuQ7tQc6ZdZna+hu7O3mAhfC04vLEYFdxkbX4D9mNOuL29A+bjYMTA1fkGnLkg2xsayHdQzCjdmIpHVmbDKJqt/xvO+g9wgJv3c7Px2EzlfmuUshxzMRORsng5+KZDyMf3Odo+7hqWJcDX8T7ePLsYW/MDVhOZIJ+4zeHGgrT7xQlMgrAM2989Bgv2YsvhFlnfkCW6ay8D+8fCo8jd+0tYje67UAEMr/ETUhbhKeM9isjcaajVv/+eMlHxbLRBPnFNVTiRjVZmfKSPJuvwsrTBQfMKORSTbOCu3o1XygOOVNxJZ8dOvwOEYSUyU+VrWg+qD1TgmOgMcR2e2r342S5+ri8dOfNnYTz3tm3inhNfxpyFRjz5RDru/9Mp2I+7QrMeCJ3EAeuv/R6P8nyFb2D+38rzHUgw6gt/BxKA6tdRtP0wPHeMM1cQjVjHeb9BlZPv6wc+YtvMv+25S5+nFnsOT8TTuZIy5laPo7DVfS56DPv3+iShge/EOZi/4H7FTQpC2j/O/S6+/XdpeGhZKjz20/jk7nxTza07RMJsLH5qpmLCwv0hOoY1iby18HBP+9DeOg9F3557h1jGwtVxiHvRZC2esHgEC1M16A6rPGWlPBelfD7ysHTOdjjngm8wt30t42UZfCaXn81bIT7jz0P/VjCL84q/iAPnkJVx/GEh387a+dE/bxuz56cp5AhskWERE4CBM4bBc8jh5KWxfHtb4Nzhp8xuSg49/zzoHHKwfjDZmVssceBM4kCdDMzi7A3g7mft9vViecT6Gq3MFe7Y4sgbZ1BKns/Yfrys11nGDIZtzOHuZ8zbzhzFK5jBUs8kCowFOBjKmLNXbDzW67Izs0FQtB9vI9n55d56ZjEITDCVs2YpXXM5MyWvZ/b2/rGtJt+rGXO2w6+i/8y29BsNnNnkLJtbmWPrVmZtlk53D1/mWMWMCVexb61gxY525mX9zO3YxgwD/ZPXXNln5TRucQ7Z62bOY8eZk/8W+MfrZvVlxWyrmI9cxthcx4Stsmq3Zc2HDTvLF4J6wB82MnN5PXOLQwYfV1Yxk7VJNq4oMxq7sJJrnK2Q+expKtI3H4TL8RYsJu4JzT/c3GyFw3UQmwftHybDdOQUnOVm0bsa3Dvb8n7QgSxhFp582Ypy7sUofowwlx/D0f/4Fyzj3tgHPoAzxHt6LY44TwbjCyZYqu3Ykz1rlLPWiUh5fCPKBup0n3ge21+mCZj5WLbfueuOdeZKQGL6ehzcOR370++Bblw+qlJfuuXxBP9xtRzsqht4+WigDbnT33IslkyGiQuw6eAxlMw4iK9OGgedbhzmllzG9369Hdl3jNk/WPWRX0nerPPxaBrfGpKcCo+g4J+t6PreZuR/RYOripFXePQpxb5VjKT9yzFONxtrquag9OD6wU6Fypx6alGk/0ukFhwBPCVYOuUBxSsxfeiu34MM/T3+18iWfYi+FT/CT5fMHOU4oyxIHIVHwDphylx8c5kbu55ZCP04HfTf/0905ezBvjBbW1ogoeNzAXlBdDqd6Gwlv0fXsSbATbAHsH7JM6jwrILVVYH8udHenwqtI/WDUB7RDBHbaNIMyiKuQRbRviK20Sbql6fkKnMXVidDkjoaAvyNXhugz3l9QIhgXodvq6yMBzKjCyJABIgAERgzAnFosh4zNhrIaDySZqWAv1TUb5a3o26r4Q54M5cG0FIRiAARIAIaI0Ama401iFaKozSlaKVcd0I5iK06rUhc1eHKpRJbddgqudIKWR3OJJUIEAEiQASIQEQESCFHhIsiEwEiQASIABFQhwApZHW4klQiQASIABEgAhERIIUcES6KTASIABEgAkRAHQKkkNXhSlKJABEgAkSACEREgBRyRLgoMhEgAkSACBABdQiEPfakTlYklQgQASJABIgAEVASkF6YOehNXdIDZQIKEwEiQASIABEgAuoRIJO1emxJMhEgAkSACBCBYRMghTxsVBSRCBABIkAEiIB6BP4f9Hr8chWJwKgAAAAASUVORK5CYII="></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of<em> k</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>Calculate the expected value of the score.</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>The die is rolled 80 times. On how many rolls would you expect to obtain a three?</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>A random variable \(X\) is normally distributed with mean, \(\mu \). In the following diagram, the shaded region between 9 and \(\mu \) represents 30% of the distribution.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-14_om_10.15.49.png" alt="M17/5/MATME/SP2/ENG/TZ1/09"></p>
</div>
<div class="specification">
<p>The standard deviation of \(X\) is 2.1.</p>
</div>
<div class="specification">
<p>The random variable \(Y\) is normally distributed with mean \(\lambda \) and standard deviation 3.5. The events \(X > 9\) and \(Y > 9\) are independent, and \(P\left( {(X > 9) \cap (Y > 9)} \right) = 0.4\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \({\text{P}}(X < 9)\).</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 the value of \(\mu \).</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 \(\lambda \).</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that \(Y > 9\), find \({\text{P}}(Y < 13)\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Consider the independent events <em>A</em> and <em>B</em> . Given that \({\rm{P}}(B) = 2{\rm{P}}(A)\) , and \({\rm{P}}(A \cup B) = 0.52\) , </span><span style="font-family: times new roman,times; font-size: medium;">find \({\rm{P}}(B)\) .</span></p>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A company uses two machines, A and B, to make boxes. Machine A makes \(60\% \) of </span><span style="font-family: times new roman,times; font-size: medium;">the boxes.</span></p>
<p style="margin-left: 60px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(80\% \) of the boxes made by machine A pass inspection.</span></p>
<p style="margin-left: 60px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(90\% \) of the boxes made by machine B pass inspection.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">A box is selected at random.</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 probability that it passes inspection.</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;">The company would like the probability that a box passes inspection to be 0.87.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the percentage of boxes that should be made by machine B to achieve this.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The following table shows a probability distribution for the random variable \(X\), where \({\text{E}}(X) = 1.2\).</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-15_om_06.18.09.png" alt="M17/5/MATME/SP2/ENG/TZ2/10"></p>
</div>
<div class="specification">
<p>A bag contains white and blue marbles, with at least three of each colour. Three marbles are drawn from the bag, without replacement. The number of blue marbles drawn is given by the random variable \(X\).</p>
</div>
<div class="specification">
<p>A game is played in which three marbles are drawn from the bag of ten marbles, without replacement. A player wins a prize if three white marbles are drawn.</p>
</div>
<div class="question">
<p>Jill plays the game nine times. Find the probability that she wins exactly two prizes.</p>
</div>
<br><hr><br><div class="specification">
<p class="p1">A random variable \(X\) is distributed normally with a mean of <span class="s1">20 </span>and standard deviation of <span class="s1">4</span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">On the following diagram, shade the region representing \({\text{P}}(X \leqslant 25)\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-02-01_om_16.44.01.png" alt="M16/5/MATME/SP2/ENG/TZ1/01"></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"><span class="s1">Write down \({\text{P}}(X \leqslant 25)\)</span>, correct to two decimal places.</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"><span class="s1">Let \({\text{P}}(X \leqslant c) = 0.7\)</span>. Write down the value of \(c\).</p>
<div class="marks">[2]</div>
<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 van can take either Route A or Route B for a particular journey.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">If Route A is taken, the journey time may be assumed to be normally distributed </span><span style="font-family: times new roman,times; font-size: medium;">with mean 46 minutes and a standard deviation 10 minutes.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">If Route B is taken, the journey time may be assumed to be normally distributed </span><span style="font-family: times new roman,times; font-size: medium;">with mean \(\mu \) minutes and standard deviation 12 minutes.</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;">For Route A, find the probability that the journey takes <strong>more</strong> than \(60\) 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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">For Route B, the probability that the journey takes <strong>less</strong> than \(60\) minutes is \(0.85\).</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the value of \(\mu \) .</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;">The van sets out at 06:00 and needs to arrive before 07:00.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Which route should it take?</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Justify your answer.</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;">On five consecutive days the van sets out at 06:00 and takes Route B. Find the </span><span style="font-family: times new roman,times; font-size: medium;">probability that</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) it arrives before 07:00 on all five days;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) it arrives before 07:00 on at least three days.</span></p>
<div class="marks">[5]</div>
<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;">Jan plays a game where she tosses two fair six-sided dice. She wins a prize if the </span><span style="font-family: times new roman,times; font-size: medium;">sum of her scores is 5.</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;">Jan tosses the two dice once. Find the probability that she wins a prize.</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;">Jan tosses the two dice 8 times. Find the probability that she wins 3 prizes.</span></p>
<div class="marks">[2]</div>
<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 following table shows the probability distribution of a discrete random variable <em>X</em>.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/alice.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 <em>k</em>.</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><span style="font-family: times new roman,times; font-size: medium;">Find the expected value of <em>X</em>.</span></p>
<div class="marks">[3]</div>
<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;"><span style="font-size: medium;">The scores of a test given to students are normally distributed with a mean of 21. \(80\% \)</span><span style="font-size: medium;"> of the students have scores less than 23.7.</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 the standard deviation of the scores.</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;">A student is chosen at random. This student has the same probability of having a score </span><span style="font-family: times new roman,times; font-size: medium;">less than 25.4 as having a score greater than <em>b</em>.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Find the probability the student has a score less than 25.4.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) Find the value of <em>b</em>.</span></p>
<p align="LEFT"> </p>
<div class="marks">[4]</div>
<div class="question_part_label">b(i) and (ii).</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;">At a large school, students are required to learn at least one language, Spanish </span><span style="font-family: times new roman,times; font-size: medium;">or French. It is known that \(75\% \) of the students learn Spanish, and \(40\% \) learn French.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the percentage of students who learn <strong>both</strong> Spanish and French.</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;">At a large school, students are required to learn at least one language, Spanish </span><span style="font-family: times new roman,times; font-size: medium;">or French. It is known that \(75\% \) of the students learn Spanish, and \(40\% \) learn French.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the percentage of students who learn Spanish, but not French.</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">At a large school, students are required to learn at least one language, Spanish </span><span style="font-family: times new roman,times; font-size: medium;">or French. It is known that \(75\% \) of the students learn Spanish, and \(40\% \) learn French.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">At this school, \(52\% \) of the students are girls, and \(85\% \) of the girls learn Spanish.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A student is chosen at random. Let <em>G</em> be the event that the student is a girl, and </span><span style="font-family: times new roman,times; font-size: medium;">let <em>S</em> be the event that the student learns Spanish.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(i) Find \({\rm{P}}(G \cap S)\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Show that <em>G</em> and <em>S</em> are <strong>not</strong> independent.</span></p>
<p> </p>
<div class="marks">[5]</div>
<div class="question_part_label">c(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;">At a large school, students are required to learn at least one language, Spanish </span><span style="font-family: times new roman,times; font-size: medium;">or French. It is known that \(75\% \) of the students learn Spanish, and \(40\% \) learn French.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">At this school, \(52\% \) of the students are girls, and \(85\% \) of the girls learn Spanish.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">A boy is chosen at random. Find the probability that he learns Spanish.</span></p>
<div class="marks">[6]</div>
<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;">A bag contains four gold balls and six silver balls.</span></p>
</div>
<div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Two balls are drawn at random from the bag, with replacement. Let \(X\) be the </span><span style="font-family: times new roman,times; font-size: medium;">number of gold balls drawn from the bag.</span></p>
</div>
<div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Fourteen balls are drawn from the bag, with replacement.</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;">(i) Find \({\rm{P}}(X = 0)\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) Find \({\rm{P}}(X = 1)\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iii) Hence, find \({\rm{E}}(X)\) .</span></p>
<div class="marks">[8]</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;">Hence, find \({\rm{E}}(X)\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.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 probability that exactly five of the balls are gold.</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 at most five of the balls are gold.</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Given that at most five of the balls are gold, find the probability that exactly five </span><span style="font-family: times new roman,times; font-size: medium;">of the balls are gold. Give the answer correct to two decimal places.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</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;">Two events \(A\) and \(B\) are such that \({\text{P}}(A) = 0.2\) and \({\text{P}}(A \cup B) = 0.5\).</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: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Given that \(A\) and \(B\) are mutually exclusive, find \({\text{P}}(B)\).</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;">Given that \(A\) and \(B\) are independent, find \({\text{P}}(B)\).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The following table shows the Diploma score \(x\) and university entrance mark \(y\) for seven IB Diploma students.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2015-12-14_om_08.02.07.png" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the correlation coefficient.</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">The relationship can be modelled by the regression line with equation \(y = ax + b\).</p>
<p class="p1">Write down the value of \(a\) and of \(b\).</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">Rita scored a total of \(26\) in her IB Diploma.</p>
<p class="p1">Use your regression line to estimate Rita’s university entrance mark.</p>
<div class="marks">[2]</div>
<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 cumulative frequency curve below represents the heights of 200 sixteen-year-old boys.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/donald.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Use the graph to answer the following.</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 median value.</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;">A boy is chosen at random. Find the probability that he is shorter than \(161{\text{ cm}}\).</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 \(82\% \) of the boys are taller than \(h{\text{ cm}}\), find <em>h</em> .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The following cumulative frequency graph shows the monthly income, \(I\) dollars, of \(2000\) families.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2015-12-14_om_10.22.12.png" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the median monthly income.</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>Write down the number of families who have a monthly income of \(2000\) dollars or less.</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Find the number of families who have a monthly income of more than \(4000\) dollars.</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">The \(2000\) families live in two different types of housing. The following table gives information about the number of families living in each type of housing and their monthly income \(I\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2015-12-14_om_11.02.20.png" alt></p>
<p class="p1">Find the value of \(p\).</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 class="p1">A family is chosen at random.</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>Find the probability that this family lives in an apartment.</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Find the probability that this family lives in an apartment, given that its monthly income is greater than \(4000\) dollars.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Estimate the mean monthly income for families living in a villa.</p>
<div class="marks">[2]</div>
<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 following table gives the examination grades for 120 students.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/easley.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 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><span style="font-family: times new roman,times; font-size: medium;">(ii) <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 the mean grade.</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;">Write down the standard deviation.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The masses of watermelons grown on a farm are normally distributed with a mean of \(10\) kg.</p>
<p class="p1">The watermelons are classified as small, medium or large.</p>
<p class="p2"><span class="s1">A watermelon is small if its mass is less than \(4\) kg</span>. Five percent of the watermelons are classified as small.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the standard deviation of the masses of the watermelons.</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 class="p1">The following table shows the percentages of small, medium and large watermelons grown on the farm.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-25_om_09.47.19.png" alt></p>
<p class="p1">A watermelon is large if its mass is greater than \(w\) <span class="s1">kg</span>.</p>
<p class="p1">Find the value of \(w\).</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">All the medium and large watermelons are delivered to a grocer.</p>
<p class="p1">The grocer selects a watermelon at random from <strong>this </strong>delivery. Find the probability that it is medium.</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">All the medium and large watermelons are delivered to a grocer.</p>
<p class="p1">The grocer sells all the medium watermelons for $1.75 each, and all the large watermelons for $3.00 each. His costs on this delivery are $300, and his total profit is $150<span class="s1">. Find the number of watermelons in the delivery.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</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 table shows the amount of fuel (\(y\)<em> </em>litres) used by a car to travel certain distances (\(x\)<em> </em>km).</span></p>
<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;"> </span></p>
<table class="block_black_border" border="0" align="center">
<tbody>
<tr>
<td><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Distance</strong> (<em>x</em> km)</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">40</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">75</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">120</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">150</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">195</span></td>
</tr>
<tr>
<td><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Amount of fuel</strong> (<em>y</em> litres)</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">3.6</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">6.5</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">9.9</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">13.1</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">16.2</span></td>
</tr>
</tbody>
</table>
<p> </p>
<p>This data can be modelled by the regression line with equation \(y = ax + b\).</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;">Write down the value of \(a\) and of \(b\).</span></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 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;">Explain what the gradient \(a\) represents.</span></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 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;">Use the model to estimate the amount of fuel the car would use if it is driven \(110\) km.</span></p>
<div class="marks">[2]</div>
<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 \(X\) is normally distributed with mean \(20\) and standard deviation \(5\).</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 \({\rm{P}}(X \le 22.9)\) .</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;">Given that \({\rm{P}}(X < k) = 0.55\) , find the value of \(k\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The following table shows values of ln <em>x</em> and ln <em>y</em>.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p style="text-align: left;">The relationship between ln <em>x</em> and ln <em>y</em> can be modelled by the regression equation ln <em>y</em> = <em>a</em> ln <em>x</em> + <em>b</em>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <em>a</em> and of <em>b</em>.</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>Use the regression equation to estimate the value of <em>y</em> when<em> x</em> = 3.57.</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>The relationship between <em>x</em> and <em>y</em> can be modelled using the formula <em>y</em> = <em>kx<sup>n</sup></em>, where <em>k</em> ≠ 0 , <em>n</em> ≠ 0 , <em>n</em> ≠ 1.</p>
<p>By expressing ln <em>y</em> in terms of ln <em>x</em>, find the value of <em>n</em> and of <em>k</em>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">An environmental group records the numbers of coyotes and foxes in a wildlife reserve after \(t\) <span class="s1">years, starting on 1 January 1995</span>.</p>
<p class="p1">Let \(c\) be the number of coyotes in the reserve after \(t\) years. The following table shows the number of coyotes after \(t\) years.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-25_om_08.53.25.png" alt></p>
<p class="p1">The relationship between the variables can be modelled by the regression equation \(c = at + b\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the value of \(a\) and of \(b\).</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">Use the regression equation to estimate the number of coyotes in the reserve when \(t = 7\).</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">Let \(f\) be the number of foxes in the reserve after \(t\) years. The number of foxes can be modelled by the equation \(f = \frac{{2000}}{{1 + 99{{\text{e}}^{ - kt}}}}\), where \(k\) <span class="s1">is a constant.</span></p>
<p class="p2">Find the number of foxes in the reserve on 1 January 1995.</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"><span class="s1">After five years, there were 64 </span>foxes in the reserve. Find \(k\).</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">During which year were the number of coyotes the same as the number of foxes?</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Let \(f(x) = {{\text{e}}^{0.5x}} - 2\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">For the graph of <em>f</em>:</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>write down the \(y\)-intercept;</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>find the \(x\)-intercept;</p>
<p class="p1">(iii) <span class="Apple-converted-space"> </span>write down the equation of the horizontal asymptote.</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 class="p1">On the following grid, sketch the graph of \(f\)<span class="s1">, for \( - 4 \leqslant x \leqslant 4\).</span></p>
<p class="p1" style="text-align: center;"><span class="s1"><img src="images/Schermafbeelding_2017-02-03_om_06.14.24.png" alt="M16/5/MATME/SP2/ENG/TZ2/03.b"></span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The following table shows the probability distribution of a discrete random variable \(X\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-22_om_16.09.18.png" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the value of \(k\).</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">[2]</div>
<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 following diagram is a box and whisker plot for a set of data.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/sharpay.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The interquartile range is 20 and the range is 40.</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 median value.</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;">Find the value of</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) \(a\) ;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \(b\) .<br></span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">A jar contains <span class="s1">5 </span>red discs, <span class="s1">10 </span>blue discs and \(m\) green discs. A disc is selected at random and replaced. This process is performed four times.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Write down the probability that the first disc selected is red.</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="p1"><span class="s1">Let \(X\) </span>be the number of red discs selected. Find the smallest value of \(m\) for which \({\text{Var}}(X{\text{ }}) < 0.6\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Two events <em>A</em> and <em>B</em> are such that P(<em>A</em>) = 0.62 and P\(\left( {A \cap B} \right)\) = 0.18.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find P(<em>A</em> ∩ <em>B′ </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>Given that P((<em>A</em> ∪ <em>B</em>)′<em> </em>) = 0.19, find P(<em>A </em>|<em> </em><em>B</em>′<em> </em>).</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>The heights of adult males in a country are normally distributed with a mean of 180 cm and a standard deviation of \(\sigma {\text{ cm}}\). 17% of these men are shorter than 168 cm. 80% of them have heights between \((192 - h){\text{ cm}}\) and 192 cm.</p>
<p>Find the value of \(h\).</p>
</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 time taken for a student to complete a task is normally distributed with a mean of \(20\) minutes and a standard deviation of \(1.25\) minutes.</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: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">A student is selected at random. Find the probability that the student completes the task in less than \(21.8\) 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;">The probability that a student takes between \(k\) and \(21.8\) minutes is \(0.3\). Find the value of \(k\).</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">In a large city, the time taken to travel to work is normally distributed with </span><span style="font-family: times new roman,times; font-size: medium;">mean \(\mu \) and standard deviation \(\sigma \) . It is found that \(4\% \) of the population take less </span><span style="font-family: times new roman,times; font-size: medium;">than 5 minutes to get to work, and \(70\% \) take less than 25 minutes.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the value of \(\mu \) and of \(\sigma \) .</span></p>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">A box holds 240 eggs. The probability that an egg is brown is 0.05.</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 expected number of brown eggs in the box.</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 probability that there are 15 brown eggs in the box.</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 there are at least 10 brown eggs in the box.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The mass <em>M</em> of apples in grams is normally distributed with mean <em>μ</em>. The following table shows probabilities for values of <em>M</em>.</p>
<p style="text-align: center;"><img 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"></p>
</div>
<div class="specification">
<p>The apples are packed in bags of ten.</p>
<p>Any apples with a mass less than 95 g are classified as small.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the value of <em>k</em>.</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>Show that <em>μ</em> = 106.</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>Find <em>P</em>(M < 95) .</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>Find the probability that a bag of apples selected at random contains at most one small apple.</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>Find the expected number of bags in this crate that contain at most one small apple.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that at least 48 bags in this crate contain at most one small apple.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.ii.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">A Ferris wheel with diameter \(122\) metres rotates clockwise at a constant speed. </span><span style="font-family: times new roman,times; font-size: medium;">The wheel completes \(2.4\) rotations every hour. The bottom of the wheel is \(13\) metres </span><span style="font-family: times new roman,times; font-size: medium;">above the ground.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img style="display: block; margin-left: auto; margin-right: auto;" src="images/open.png" alt></span></p>
<p> <span style="font-family: times new roman,times; font-size: medium;">A seat starts at the bottom of the wheel.</span></p>
</div>
<div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">After <em><strong>t</strong> </em>minutes, the height \(h\) metres above the ground of the seat is given by\[h = 74 + a\cos bt {\rm{ .}}\]<br></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 maximum height above the ground of the seat.</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-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;">(i) Show that the period of \(h\) is \(25\) minutes.</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;">(ii) Write down the <strong>exact</strong> value of \(b\) .</span></p>
<p> </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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(b) (i) Show that the period of \(h\) is \(25\) minutes.</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (ii) Write down the <strong>exact</strong> value of \(b\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(c) </span><span style="font-family: times new roman,times; font-size: medium;">Find the value of \(a\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(d) </span><span style="font-family: times new roman,times; font-size: medium;">Sketch the graph of \(h\) , for \(0 \le t \le 50\) .</span></p>
<div class="marks">[9]</div>
<div class="question_part_label">bcd.</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 \(a\) .</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;">Sketch the graph of \(h\) , for \(0 \le t \le 50\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</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;">In one rotation of the wheel, find the probability that a randomly selected seat is </span><span style="font-family: times new roman,times; font-size: medium;">at least \(105\) metres above the ground.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br>