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</div><h2>HL Paper 2</h2><div class="question">
<p>Find the Cartesian equation of plane <em>Π</em> containing the points \({\text{A}}\left( {6,{\text{ }}2,{\text{ }}1} \right)\) and \({\text{B}}\left( {3,{\text{ }} - 1,{\text{ }}1} \right)\) and perpendicular to the plane \(x + 2y - z - 6 = 0\).</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><strong>METHOD 1</strong></p>
<p>\(\overrightarrow {{\text{AB}}} = \left( {\begin{array}{*{20}{c}} { - 3} \\ { - 3} \\ 0 \end{array}} \right)\) <strong><em>(A1)</em></strong></p>
<p>\(\left( {\begin{array}{*{20}{c}} { - 3} \\ { - 3} \\ 0 \end{array}} \right) \times \left( {\begin{array}{*{20}{c}} 1 \\ 2 \\ { - 1} \end{array}} \right)\) <strong><em>M1A1</em></strong></p>
<p>\( = \left( {\begin{array}{*{20}{c}} 3 \\ { - 3} \\ { - 3} \end{array}} \right)\) <strong><em>A1</em></strong></p>
<p>\(x - y - z = k\) <strong><em>M1</em></strong></p>
<p>\(k = 3\) equation of plane <em>Π</em> is \(x - y - z = 3\) or equivalent <strong><em>A1</em></strong></p>
<p><strong>METHOD 2</strong></p>
<p>let plane <em>Π</em> be \(ax + by + cz = d\)</p>
<p>attempt to form one or more simultaneous equations: <strong><em>M1</em></strong></p>
<p>\(a + 2b - c = 0\) (1) <strong><em>A1</em></strong></p>
<p>\(6a + 2b + c = d\) (2)</p>
<p>\(3a - b + c = d\) (3) <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award second <strong><em>A1 </em></strong>for equations (2) and (3).</p>
<p> </p>
<p>attempt to solve <strong><em>M1</em></strong></p>
<p><strong>EITHER</strong></p>
<p>using GDC gives \(a = \frac{d}{3},{\text{ }}b = - \frac{d}{3},{\text{ }}c = - \frac{d}{3}\) <strong><em>(A1)</em></strong></p>
<p>equation of plane <em>Π</em> is \(x - y - z = 3\) or equivalent <strong><em>A1</em></strong></p>
<p><strong>OR</strong></p>
<p>row reduction <strong><em>M1</em></strong></p>
<p>equation of plane <em>Π</em> is \(x - y - z = 3\) or equivalent <strong><em>A1</em></strong></p>
<p><strong><em>[6 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p>Given that <strong><em>a</em></strong> \( \times \) <strong><em>b</em></strong> \( = \) <strong><em>b</em></strong> \( \times \) <strong><em>c</em></strong> \( \ne \) <strong>0 </strong>prove that <strong><em>a</em></strong> \( + \) <strong><em>c</em></strong> \( = \) <em>s<strong>b </strong></em>where <em>s </em>is a scalar.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><strong>METHOD 1</strong></p>
<p><strong><em>a</em></strong> \( \times \) <strong><em>b</em></strong> = <strong><em>b</em></strong> \( \times \) <strong><em>c</em></strong></p>
<p>(<strong><em>a</em></strong> \( \times \) <strong><em>b</em></strong>) \( - \) (<strong><em>b</em></strong> \( \times \) <strong><em>c</em></strong>) = 0</p>
<p>(<strong><em>a</em></strong> \( \times \) <strong><em>b</em></strong>) + (<strong><em>c</em></strong> \( \times \) <strong><em>b</em></strong>) = 0 <strong><em>M1A1</em></strong></p>
<p>(<strong><em>a</em></strong> + <strong><em>c</em></strong>) \( \times \) <strong><em>b</em></strong> = 0 <strong><em>A1</em></strong></p>
<p>(<strong><em>a</em></strong> + <strong><em>c</em></strong>) is parallel to <strong><em>b</em></strong> \( \Rightarrow \) <strong><em>a</em></strong> + <strong><em>c</em></strong> = <em>s<strong>b</strong></em> <strong><em>R1AG</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Condone absence of arrows, underlining, or other otherwise “correct” vector notation throughout this question.</p>
<p> </p>
<p><strong>Note:</strong> Allow “is in the same direction to”, for the final <strong><em>R </em></strong>mark.</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><strong><em>a</em></strong> \( \times \) <strong><em>b</em></strong> = <strong><em>b</em></strong> \( \times \) <strong><em>c</em></strong> \( \Rightarrow \left( {\begin{array}{*{20}{c}} {{a_2}{b_3} - {a_3}{b_2}} \\ {{a_3}{b_1} - {a_1}{b_3}} \\ {{a_1}{b_2} - {a_2}{b_1}} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {{b_2}{c_3} - {b_3}{c_2}} \\ {{b_3}{c_1} - {b_1}{c_3}} \\ {{b_1}{c_2} - {b_2}{c_1}} \end{array}} \right)\) <strong><em>M1A1</em></strong></p>
<p>\({a_2}{b_3} - {a_3}{b_2} = {b_2}{c_3} - {b_3}{c_2} \Rightarrow {b_3}({a_2} + {c_2}) = {b_2}({a_3} + {c_3})\)</p>
<p>\({a_3}{b_1} - {a_1}{b_3} = {b_3}{c_1} - {b_1}{c_3} \Rightarrow {b_1}({a_3} + {c_3}) = {b_3}({a_1} + {c_1})\)</p>
<p>\({a_1}{b_2} - {a_2}{b_1} = {b_1}{c_2} - {b_2}{c_1} \Rightarrow {b_2}({a_1} + {c_1}) = {b_1}({a_2} + {c_2})\)</p>
<p>\(\frac{{({a_1} + {c_1})}}{{{b_1}}} = \frac{{({a_2} + {c_2})}}{{{b_2}}} = \frac{{({a_3} + {c_3})}}{{{b_3}}} = s\) <strong><em>A1</em></strong></p>
<p>\( \Rightarrow {a_1} + {c_1} = s{b_1}\)</p>
<p>\( \Rightarrow {a_2} + {c_2} = s{b_2}\)</p>
<p>\( \Rightarrow {a_3} + {c_3} = s{b_3}\)</p>
<p>\( \Rightarrow \left( {\begin{array}{*{20}{c}} {{a_1}} \\ {{a_2}} \\ {{a_3}} \end{array}} \right) + \left( {\begin{array}{*{20}{c}} {{c_1}} \\ {{c_2}} \\ {{c_3}} \end{array}} \right) = s\left( {\begin{array}{*{20}{c}} {{b_1}} \\ {{b_2}} \\ {{b_3}} \end{array}} \right)\) <strong><em>A1</em></strong></p>
<p>\( \Rightarrow \) <strong><em>a</em></strong> + <strong><em>c</em></strong> = <em>s<strong>b</strong></em> <strong><em>AG</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the points P(−3, −1, 2) and Q(5, 5, 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: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find a vector equation for the line, \({L_1}\), which passes through the points P and Q.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The line \({L_2}\) has equation</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[r = \left( {\begin{array}{*{20}{c}}<br> { - 4} \\ <br> 0 \\ <br> 4 <br>\end{array}} \right) + s\left( {\begin{array}{*{20}{c}}<br> 5 \\ <br> 2 \\ <br> 0 <br>\end{array}} \right){\text{.}}\]</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that \({L_1}\) and \({L_2}\) intersect at the point R(1, 2, 4).</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the acute angle between \({L_1}\) and \({L_2}\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Let S be a point on \({L_2}\) such that \(\left| {\overrightarrow {{\text{RP}}} } \right| = \left| {\overrightarrow {{\text{RS}}} } \right|\).</span><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that one of the possible positions for S is \({{\text{S}}_1}\)(−4, 0, 4) and find the coordinates of the other possible position, \({{\text{S}}_2}\).</span></p>
<div class="marks">[6]</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: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Let S be a point on \({L_2}\) such that \(\left| {\overrightarrow {{\text{RP}}} } \right| = \left| {\overrightarrow {{\text{RS}}} } \right|\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find a vector equation of the line which passes through R and bisects \({\rm{P\hat R}}{{\text{S}}_1}\).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\overrightarrow {{\text{PQ}}} = \left( {\begin{array}{*{20}{c}}<br> 8 \\ <br> 6 \\ <br> 4 <br>\end{array}} \right)\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">equation of line: \(r = \left( {\begin{array}{*{20}{c}}<br> { - 3} \\ <br> { - 1} \\ <br> 2 <br>\end{array}} \right) + t\left( {\begin{array}{*{20}{c}}<br> 8 \\ <br> 6 \\ <br> 4 <br>\end{array}} \right)\) (or equivalent) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>M1A0</em></strong> if <strong><em>r</em></strong> = is omitted.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 1</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(x:{\text{ }}- 4 + 5s = - 3 + 8t\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(y:{\text{ }}2s = - 1 + 6t\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(z:{\text{ }}4 = 2 + 4t\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">solving any two simultaneously <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>t</em> = 0.5, <em>s</em> = 1 (or equivalent) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">verification that these values give R when substituted into <strong>both</strong> equations (or that the three equations are consistent and that one gives R) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 2</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(1, 2, 4) is given by <em>t</em> = 0.5 for \({L_1}\) and <em>s</em> = 1 for \({L_2}\) <strong><em>M1A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">because (1, 2, 4) is on both lines it is the point of intersection of the two lines <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left( {\begin{array}{*{20}{c}}<br> 5 \\ <br> 2 \\ <br> 0 <br>\end{array}} \right) \cdot \left( {\begin{array}{*{20}{c}}<br> 4 \\ <br> 3 \\ <br> 2 <br>\end{array}} \right) = 26 = \sqrt {29} \times \sqrt {29} \cos \theta \) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\cos \theta = \frac{{26}}{{29}}\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\theta = 0.459\) or 26.3° <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\overrightarrow {{\text{RP}}} = \left( {\begin{array}{*{20}{c}}<br> { - 3} \\ <br> { - 1} \\ <br> 2 <br>\end{array}} \right) - \left( {\begin{array}{*{20}{c}}<br> 1 \\ <br> 2 \\ <br> 4 <br>\end{array}} \right) = \left( {\begin{array}{*{20}{c}}<br> { - 4} \\ <br> { - 3} \\ <br> { - 2} <br>\end{array}} \right)\), \(\left| {\overrightarrow {{\text{RP}}} } \right| = \sqrt {29} \) <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> This could also be obtained from \(\left| {0.5\left( {\begin{array}{*{20}{c}}<br> 8 \\ <br> 6 \\ <br> 4 <br>\end{array}} \right)} \right|\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span><strong style="font-family: 'times new roman', times; font-size: medium;">EITHER</strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\overrightarrow {{\text{R}}{{\text{S}}_{\text{1}}}} = \left( {\begin{array}{*{20}{c}}<br> { - 4} \\ <br> 0 \\ <br> 4 <br>\end{array}} \right) - \left( {\begin{array}{*{20}{c}}<br> 1 \\ <br> 2 \\ <br> 4 <br>\end{array}} \right) = \left( {\begin{array}{*{20}{c}}<br> { - 5} \\ <br> { - 2} \\ <br> 0 <br>\end{array}} \right)\), \(\left| {\overrightarrow {{\text{R}}{{\text{S}}_{\text{1}}}} } \right| = \sqrt {29} \) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\therefore \overrightarrow {{\text{O}}{{\text{S}}_2}} = \overrightarrow {{\text{O}}{{\text{S}}_{\text{1}}}} + 2\overrightarrow {{{\text{S}}_{\text{1}}}{\text{R}}} = \left( {\begin{array}{*{20}{c}}<br> { - 4} \\ <br> 0 \\ <br> 4 <br>\end{array}} \right) + 2\left( {\begin{array}{*{20}{c}}<br> 5 \\ <br> 2 \\ <br> 0 <br>\end{array}} \right)\)</span><span style="font-family: 'times new roman', times; font-size: medium;"> <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left( {{\text{or }}\overrightarrow {{\text{O}}{{\text{S}}_2}} = \overrightarrow {{\text{OR}}} + \overrightarrow {{{\text{S}}_{\text{1}}}{\text{R}}} = \left( {\begin{array}{*{20}{c}}<br> 1 \\ <br> 2 \\ <br> 4 <br>\end{array}} \right) + \left( {\begin{array}{*{20}{c}}<br> 5 \\ <br> 2 \\ <br> 0 <br>\end{array}} \right)} \right)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \left( {\begin{array}{*{20}{c}}<br> 6 \\ <br> 4 \\ <br> 4 <br>\end{array}} \right)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({{\text{S}}_2}\) is (6, 4, 4) <strong><em>A1</em></strong></span><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left( {\begin{array}{*{20}{c}}<br> { - 4 + 5s} \\ <br> {2s} \\ <br> 4 <br>\end{array}} \right) - \left( {\begin{array}{*{20}{c}}<br> 1 \\ <br> 2 \\ <br> 4 <br>\end{array}} \right) = \left( {\begin{array}{*{20}{c}}<br> {5s - 5} \\ <br> {2s - 2} \\ <br> 0 <br>\end{array}} \right)\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({(5s - 5)^2} + {(2s - 2)^2} = 29\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(29{s^2} - 58s + 29 = 29\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(s(s - 2) = 0,{\text{ }}s = 0,{\text{ 2}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\((6,{\text{ }}4,{\text{ 4}}){\text{ }}\left( {{\text{and (}} - 4,{\text{ }}0,{\text{ }}4{\text{)}}} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> There are several geometrical arguments possible using information obtained in previous parts, depending on what forms the previous answers had been given.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>EITHER</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">midpoint of \([{\text{P}}{{\text{S}}_1}]\) is M(–3.5, –0.5, 3) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\overrightarrow {{\text{RM}}} = \left( {\begin{array}{*{20}{c}}<br> { - 4.5} \\ <br> { - 2.5} \\ <br> { - 1} <br>\end{array}} \right)\) <strong><em>A1</em></strong></span><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\overrightarrow {{\text{R}}{{\text{S}}_1}} = \left( {\begin{array}{*{20}{c}}<br> { - 5} \\ <br> { - 2} \\ <br> 0 <br>\end{array}} \right)\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">the direction of the line is \({\overrightarrow {{\text{RS}}} _1} + \overrightarrow {{\text{RP}}} \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left( {\begin{array}{*{20}{c}}<br> { - 5} \\ <br> { - 2} \\ <br> 0 <br>\end{array}} \right) + \left( {\begin{array}{*{20}{c}}<br> { - 4} \\ <br> { - 3} \\ <br> { - 2} <br>\end{array}} \right) = \left( {\begin{array}{*{20}{c}}<br> { - 9} \\ <br> { - 5} \\ <br> { - 2} <br>\end{array}} \right)\) <strong><em>M1A1</em></strong></span><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>THEN</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">the equation of the line is:</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(r = \left( {\begin{array}{*{20}{c}}<br> 1 \\ <br> 2 \\ <br> 4 <br>\end{array}} \right) + t\left( {\begin{array}{*{20}{c}}<br> 9 \\ <br> 5 \\ <br> 2 <br>\end{array}} \right)\) or equivalent <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Marks cannot be awarded for methods involving halving the angle, unless it is clear that the candidate considers also the equation of the plane of \({L_1}\) and \({L_2}\) to reduce the number of parameters involved to one (to obtain the vector equation of the required line).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">There were many good answers to part (a) showing a clear understanding of finding the vector equation of a line. Unfortunately this understanding was marred by many students failing to write the equation properly resulting in just 2 marks out of the 3. The most common response was of the form \({L_1} = \left( {\begin{array}{*{20}{c}}<br> { - 3} \\ <br> { - 1} \\ <br> 2 <br>\end{array}} \right) + \lambda \left( {\begin{array}{*{20}{c}}<br> 4 \\ <br> 3 \\ <br> 2 <br>\end{array}} \right)\)</span><span style="font-family: times new roman,times; font-size: medium;"> which seemed a waste of a mark.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In part (b) many students failed to verify that the lines do indeed intersect.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (c) was very well done. </span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In part (d) most candidates were able to obtain the first three marks, but few were able to find the second point.</span></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">There were few correct answers to part (e).</span></p>
<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: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The points P(−1, 2, − 3), Q(−2, 1, 0), R(0, 5, 1) and S form a parallelogram, where S is diagonally opposite Q.</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: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the coordinates of S.</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: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The vector product \(\overrightarrow {{\text{PQ}}} \times \overrightarrow {{\text{PS}}} = \left( {\begin{array}{*{20}{c}}<br> { - 13} \\ <br> 7 \\ <br> m <br>\end{array}} \right)\). Find the value of <em>m</em> .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Hence calculate the area of parallelogram PQRS.</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: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the Cartesian equation of the plane, \({\prod _1}\) , containing the parallelogram PQRS.</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Write down the vector equation of the line through the origin (0, 0, 0) that is perpendicular to the plane \({\prod _1}\) .</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Hence find the point on the plane that is closest to the origin.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A second plane, \({\prod _2}\) , has equation <em>x</em> − 2<em>y</em> + <em>z</em> = 3. Calculate the angle between the two planes.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">g.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\overrightarrow {{\text{PQ}}} = \left( {\begin{array}{*{20}{c}}<br> { - 1} \\ <br> { - 1} \\ <br> 3 <br>\end{array}} \right)\) , \(\overrightarrow {{\text{SR}}} = \left( {\begin{array}{*{20}{c}}<br> {0 - x} \\ <br> {5 - y} \\ <br> {1 - z} <br>\end{array}} \right)\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">point S = (1, 6, −2) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\overrightarrow {{\text{PQ}}} = \left( {\begin{array}{*{20}{c}}<br> { - 1} \\ <br> { - 1} \\ <br> 3 <br>\end{array}} \right)\)\(\overrightarrow {{\text{PS}}} = \left( {\begin{array}{*{20}{c}}<br> 2 \\ <br> 4 \\ <br> 1 <br>\end{array}} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\overrightarrow {{\text{PQ}}} \times \overrightarrow {{\text{PS}}} = \left( {\begin{array}{*{20}{c}}<br> { - 13} \\ <br> 7 \\ <br> { - 2} <br>\end{array}} \right)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>m</em> = −2 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">area of parallelogram PQRS \( = \left| {\overrightarrow {{\text{PQ}}} \times \overrightarrow {{\text{PS}}} } \right| = \sqrt {{{( - 13)}^2} + {7^2} + {{( - 2)}^2}} \) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \sqrt {222} = 14.9\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">equation of plane is −13<em>x</em> + 7<em>y</em> − 2<em>z</em> = <em>d</em> <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">substituting any of the points given gives <em>d</em> = 33</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">−13<em>x</em> + 7<em>y</em> − 2<em>z</em> = 33 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">equation of line is \(\boldsymbol{r} = \left( {\begin{array}{*{20}{c}}<br> 0 \\ <br> 0 \\ <br> 0 <br>\end{array}} \right) + \lambda \left( {\begin{array}{*{20}{c}}<br> { - 13} \\ <br> 7 \\ <br> { - 2} <br>\end{array}} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> To get the <strong><em>A1</em></strong> must have \(\boldsymbol{r} =\) or equivalent.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[1 mark]</em></strong></span></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(169\lambda + 49\lambda + 4\lambda = 33\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\lambda = \frac{{33}}{{222}}{\text{ }}( = 0.149…)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">closest point is \(\left( { - \frac{{143}}{{74}},\frac{{77}}{{74}}, - \frac{{11}}{{37}}} \right){\text{ }}\left( { = ( - 1.93,{\text{ 1.04, - 0.297)}}} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">angle between planes is the same as the angle between the normals <strong><em>(R1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\cos \theta = \frac{{ - 13 \times 1 + 7 \times - 2 - 2 \times 1}}{{\sqrt {222} \times \sqrt 6 }}\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\theta = 143^\circ \) (accept \(\theta = 37.4^\circ \) or 2.49 radians or 0.652 radians) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">g.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">This was a multi-part question that was well answered by many candidates. Wrong answers to part (a) were mainly the result of failing to draw a diagram. Follow through benefitted many candidates. A high proportion of candidates lost the mark in (e) by not writing their answer as an equation in the form <em>r</em> = ...</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">This was a multi-part question that was well answered by many candidates. Wrong answers to part (a) were mainly the result of failing to draw a diagram. Follow through benefitted many candidates. A high proportion of candidates lost the mark in (e) by not writing their answer as an equation in the form <em>r</em> = ...</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">This was a multi-part question that was well answered by many candidates. Wrong answers to part (a) were mainly the result of failing to draw a diagram. Follow through benefitted many candidates. A high proportion of candidates lost the mark in (e) by not writing their answer as an equation in the form <em>r</em> = ...</span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">This was a multi-part question that was well answered by many candidates. Wrong answers to part (a) were mainly the result of failing to draw a diagram. Follow through benefitted many candidates. A high proportion of candidates lost the mark in (e) by not writing their answer as an equation in the form <em>r</em> = ...</span></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">This was a multi-part question that was well answered by many candidates. Wrong answers to part (a) were mainly the result of failing to draw a diagram. Follow through benefitted many candidates. A high proportion of candidates lost the mark in (e) by not writing their answer as an equation in the form \(\boldsymbol{r} = \) ...</span></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">This was a multi-part question that was well answered by many candidates. Wrong answers to part (a) were mainly the result of failing to draw a diagram. Follow through benefitted many candidates. A high proportion of candidates lost the mark in (e) by not writing their answer as an equation in the form <em>r</em> = ...</span></p>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">This was a multi-part question that was well answered by many candidates. Wrong answers to part (a) were mainly the result of failing to draw a diagram. Follow through benefitted many candidates. A high proportion of candidates lost the mark in (e) by not writing their answer as an equation in the form <em>r</em> = ...</span></p>
<div class="question_part_label">g.</div>
</div>
<br><hr><br><div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">The vector equation of line \(l\) is given as \(\left( {\begin{array}{*{20}{c}}<br> x \\ <br> y \\ <br> z <br>\end{array}} \right) = \left( {\begin{array}{*{20}{c}}<br> 1 \\ <br> 3 \\ <br> 6 <br>\end{array}} \right) + \lambda \left( {\begin{array}{*{20}{c}}<br> { - 1} \\ <br> 2 \\ <br> { - 1} <br>\end{array}} \right)\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the Cartesian equation of the plane containing the line \(l\) and the point A(4, − 2, 5) .</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><strong><span style="font-family: times new roman,times; font-size: medium;">EITHER</span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(l\) goes through the point (1, 3, 6) , and the plane contains A(4, –2, 5)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">the vector containing these two points is on the plane, <em>i.e.</em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\left( {\begin{array}{*{20}{c}}<br> 1 \\ <br> 3 \\ <br> 6 <br>\end{array}} \right) - \left( {\begin{array}{*{20}{c}}<br> 4 \\ <br> { - 2} \\ <br> 5 <br>\end{array}} \right) = \left( {\begin{array}{*{20}{c}}<br> { - 3} \\ <br> 5 \\ <br> 1 <br>\end{array}} \right)\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">(M1)A1</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\left( {\begin{array}{*{20}{c}}<br> { - 1} \\ <br> 2 \\ <br> { - 1} <br>\end{array}} \right) \times \left( {\begin{array}{*{20}{c}}<br> { - 3} \\ <br> 5 \\ <br> 1 <br>\end{array}} \right) = \left| {\begin{array}{*{20}{c}}<br> {\boldsymbol{i}}&{\boldsymbol{j}}&{\boldsymbol{k}} \\ <br> { - 1}&2&{ - 1} \\ <br> { - 3}&5&1 <br>\end{array}} \right| = 7{\boldsymbol{i}} + 4{\boldsymbol{j}} + {\boldsymbol{k}}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;"> M1A1</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\left( {\begin{array}{*{20}{c}}<br> 4 \\ <br> { - 2} \\ <br> 5 <br>\end{array}} \right) \cdot \left( {\begin{array}{*{20}{c}}<br> 7 \\ <br> 4 \\ <br> 1 <br>\end{array}} \right) = 25\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">(M1)</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">hence, Cartesian equation of the plane is \(7x + 4y + z = 25\) <em><strong>A1</strong></em></span></p>
<p><strong><span style="font-family: times new roman,times; font-size: medium;">OR</span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">finding a third point <em><strong>M1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>e.g.</em> (0, 5, 5) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">three points are (1, 3, 6), (4, –2, 5), (0, 5, 5)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">equation is \(ax + by + cz = 1\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">system of equations <em><strong>M1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(a + 3b + 6c = 1\)</span><br><span style="font-family: times new roman,times; font-size: medium;">\(4a - 2b + 5c = 1\)</span><br><span style="font-family: times new roman,times; font-size: medium;">\(5b + 5c = 1\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(a = \frac{7}{{25}}\)</span><span style="font-family: times new roman,times; font-size: medium;"> , </span><span style="font-family: times new roman,times; font-size: medium;">\(b = \frac{4}{{25}}\)</span><span style="font-family: times new roman,times; font-size: medium;"> , </span><span style="font-family: times new roman,times; font-size: medium;">\(c = \frac{1}{{25}}\)</span><span style="font-family: times new roman,times; font-size: medium;"> , from GDC <em><strong>M1A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">so</span> <span style="font-family: times new roman,times; font-size: medium;">\(\frac{7}{{25}}x + \frac{4}{{25}}y + \frac{1}{{25}}z = 1\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;"> A1</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">or \(7x + 4y + z = 25\)</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[6 marks]</span></strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">There were many successful answers to this question, as would be expected. There seemed to be some students, however, that had not been taught the vector geometry section</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the vectors <strong><em>a</em></strong> \( = \sin (2\alpha )\)<strong><em>i</em></strong> \( - \cos (2\alpha )\)<strong><em>j</em></strong> + <strong><em>k</em></strong> and <strong><em>b</em></strong> \( = \cos \alpha \)<strong><em>i</em></strong> \( - \sin \alpha \)<strong><em>j</em></strong> − <strong><em>k</em></strong>, </span><span style="font-family: 'times new roman', times; font-size: medium;">where \(0 < \alpha < 2\pi \).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(\theta \) be the angle between the vectors <strong><em>a</em></strong> and <strong><em>b</em></strong>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Express \(\cos \theta \) in terms of \(\alpha \).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Find the acute angle \(\alpha \) for which the two vectors are perpendicular.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) For \(\alpha = \frac{{7\pi }}{6}\), determine the vector product of <strong><em>a</em></strong> and <strong><em>b</em></strong> and comment on the geometrical significance of this result.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) \(\cos \theta = \frac{{\boldsymbol{ab}}}{{\left| \boldsymbol{a} \right|\left| \boldsymbol{b} \right|}} = \frac{{\sin 2\alpha \cos \alpha + \sin \alpha \cos 2\alpha - 1}}{{\sqrt 2 \times \sqrt 2 }}{\text{ }}\left( { = \frac{{\sin 3\alpha - 1}}{2}} \right)\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) \({\boldsymbol{a}} \bot {\boldsymbol{b}} \Rightarrow \cos \theta = 0\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\sin 2\alpha \cos \alpha + \sin \alpha \cos 2\alpha - 1 = 0\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\alpha = 0.524{\text{ }}\left( { = \frac{\pi }{6}} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 1</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left| {\begin{array}{*{20}{c}}<br> {\boldsymbol{i}}&{\boldsymbol{j}}&{\boldsymbol{k}} \\ <br> {\sin 2\alpha }&{ - \cos 2\alpha }&1 \\ <br> {\cos \alpha }&{ - \sin \alpha }&{ - 1} <br>\end{array}} \right|\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">assuming \(\alpha = \frac{{7\pi }}{6}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Allow substitution at any stage.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left| {\begin{array}{*{20}{c}}<br> \boldsymbol{i}&\boldsymbol{j}&\boldsymbol{k} \\ <br> {\frac{{\sqrt 3 }}{2}}&{ - \frac{1}{2}}&1 \\ <br> { - \frac{{\sqrt 3 }}{2}}&{\frac{1}{2}}&{ - 1} <br>\end{array}} \right|\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(= \boldsymbol{i} \left( {\frac{1}{2} - \frac{1}{2}} \right) - \boldsymbol{j} \left( { - \frac{{\sqrt 3 }}{2} + \frac{{\sqrt 3 }}{2}} \right) + \boldsymbol{k}\left( {\frac{{\sqrt 3 }}{2} \times \frac{1}{2} - \frac{1}{2} \times \frac{{\sqrt 3 }}{2}} \right)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">= 0 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>a</em></strong> and <strong><em>b</em></strong> are parallel <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Accept decimal equivalents.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 2</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">from (a) \(\cos \theta = - 1{\text{ (and }}\sin \theta = 0)\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\boldsymbol{a} \times \boldsymbol{b}\) = 0 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>a</em></strong> and <strong><em>b</em></strong> are parallel <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[8 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">This question was attempted by most candidates who in general were able to find the dot product of the vectors in part (a). However the simplification of the expression caused difficulties which affected the performance in part (b). Many candidates had difficulties in interpreting the meaning of <strong><em>a</em></strong> \( \times \) <strong><em>b</em></strong> = 0 in part (c).</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">The lines \({l_1}\) and \({l_2}\) are defined as</p>
<p class="p1"><span class="Apple-converted-space"> </span>\({l_1}:\frac{{x - 1}}{3} = \frac{{y - 5}}{2} = \frac{{z - 12}}{{ - 2}}\)</p>
<p class="p1"><span class="Apple-converted-space"> </span>\({l_2}:\frac{{x - 1}}{8} = \frac{{y - 5}}{{11}} = \frac{{z - 12}}{6}\).</p>
<p class="p1">The plane \(\pi \) contains both \({l_1}\) and \({l_2}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the Cartesian equation of \(\pi \).</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 line \({l_3}\) passing through the point \((4,{\text{ }}0,{\text{ }}8)\) is perpendicular to \(\pi \).</p>
<p class="p1">Find the coordinates of the point where \({l_3}\) meets \(\pi \).</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">attempting to find a normal to \(\pi {\text{ }}eg{\text{ }}\left( {\begin{array}{*{20}{c}} 3 \\ 2 \\ { - 2} \end{array}} \right) \times \left( {\begin{array}{*{20}{c}} 8 \\ {11} \\ 6 \end{array}} \right)\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1">\(\left( {\begin{array}{*{20}{c}} 3 \\ 2 \\ { - 2} \end{array}} \right) \times \left( {\begin{array}{*{20}{c}} 8 \\ {11} \\ 6 \end{array}} \right) = 17\left( {\begin{array}{*{20}{c}} 2 \\ { - 2} \\ 1 \end{array}} \right)\) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1">\({{r}} \bullet \left( {\begin{array}{*{20}{c}} 2 \\ { - 2} \\ 1 \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 1 \\ 5 \\ {12} \end{array}} \right) \bullet \left( {\begin{array}{*{20}{c}} 2 \\ { - 2} \\ 1 \end{array}} \right)\) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1">\(2x - 2y + z = 4\) (or equivalent) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><strong><em>[4 marks]<br></em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\({l_3}:{{r}} = \left( {\begin{array}{*{20}{c}} 4 \\ 0 \\ 8 \end{array}} \right) + t\left( {\begin{array}{*{20}{c}} 2 \\ { - 2} \\ 1 \end{array}} \right),\;\;\;t \in \mathbb{R}\) (<strong><em>A1)</em></strong></p>
<p>attempting to solve \(\left( {\begin{array}{*{20}{c}} {4 + 2t} \\ { - 2t} \\ {8 + t} \end{array}} \right) \bullet \left( {\begin{array}{*{20}{c}} 2 \\ { - 2} \\ 1 \end{array}} \right) = 4\;\;\;{\text{for }}t\;\;\;ie{\text{ }}9t + 16 = 4\;\;\;{\text{for }}t\) <strong><em>M1</em></strong></p>
<p>\(t = - \frac{4}{3}\) <strong><em>A1</em></strong></p>
<p>\(\left( {\frac{4}{3},{\text{ }}\frac{8}{3},{\text{ }}\frac{{20}}{3}} \right)\) <strong><em>A1</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<p><strong><em>Total [8 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Part (a) was reasonably well done. Some candidates made numerical errors when attempting to find a normal to \(\pi \).</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">In part (b), a number of candidates were awarded follow through marks from numerical errors committed in part (a).</p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">(a) If \(a = 4\) find the coordinates of the point of intersection of the three planes.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(b) (i) Find the value of \(a\) for which the planes do not meet at a unique point.</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (ii) For this value of \(a\) show that the three planes do not have any common </span><span style="font-family: times new roman,times; font-size: medium;">point.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">(a) let </span><span style="font-family: times new roman,times; font-size: medium;">\({\boldsymbol{A}} = \left( {\begin{array}{*{20}{c}}<br> 1&1&2 \\ <br> 2&{ - 1}&3 \\ <br> 5&{ - 1}&4 <br>\end{array}} \right)\)</span><span style="font-family: times new roman,times; font-size: medium;"> , \({\boldsymbol{X}} = \left( {\begin{array}{*{20}{c}}<br> x \\ <br> y \\ <br> z <br>\end{array}} \right)\) , and \({\boldsymbol{B}} = \left( {\begin{array}{*{20}{c}}<br> 2 \\ <br> 2 \\ <br> 5 <br>\end{array}} \right)\)</span><span style="font-family: times new roman,times; font-size: medium;"> </span><em><strong><span style="font-family: times new roman,times; font-size: medium;"> (M1)</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">point of intersection is \(\left( {\frac{{11}}{{12}},\frac{7}{{12}},\frac{1}{4}} \right)\)</span><span style="font-family: times new roman,times; font-size: medium;"> or \(\left( {{\text{or }}\left( {{\text{0}}{\text{.917, 0}}{\text{.583, 0}}{\text{.25}}} \right)} \right)\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1</span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> </span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(b) <strong>METHOD 1</strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(i) \(\det \left( {\begin{array}{*{20}{c}}<br> 1&1&2 \\ <br> 2&{ - 1}&3 \\ <br> 5&{ - 1}&4 <br>\end{array}} \right) = 0\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">M1</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( - 3a + 24 = 0\) <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(a = 8\) <em><strong>A1 N1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) consider the augmented matrix \(\left( {\begin{array}{*{20}{ccc|c}}<br> 1&1&2&2 \\ <br> 2&{ - 1}&3&2 \\ <br> 5&{ - 1}&4&5 <br>\end{array}} \right)\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">M1</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">use row reduction to obtain \(\left( {\begin{array}{*{20}{ccc|c}}<br> 1&1&2&2 \\ <br> 0&{ - 3}&{ - 1}&{ - 2} \\ <br> 0&0&0&{ - 1} <br>\end{array}} \right)\)</span><span style="font-family: times new roman,times; font-size: medium;"> or \(\left( {\begin{array}{*{20}{ccc|c}}<br> 1&0&{\frac{5}{3}}&0 \\ <br> 0&1&{\frac{1}{3}}&0 \\ <br> 0&0&0&1 <br>\end{array}} \right)\) </span><span style="font-family: times new roman,times; font-size: medium;">(or equivalent) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">any valid reason <em><strong> R1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(<em>e.g.</em> as the last row is not all zeros, the planes do not meet) <em><strong>N0</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 2</span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">use of row reduction (or equivalent manipulation of equations) <em><strong>M1</strong></em></span></p>
<p><em><span style="font-family: times new roman,times; font-size: medium;">e.g.</span></em> <span style="font-family: times new roman,times; font-size: medium;">\(\left( {\begin{array}{*{20}{c}}<br> 1&1&2&2 \\ <br> 2&{ - 1}&3&2 \\ <br> 5&{ - 1}&a&5 <br>\end{array}} \right) \Rightarrow \left( {\begin{array}{*{20}{c}}<br> 1&1&2&2 \\ <br> 0&{ - 3}&{ - 1}&{ - 2} \\ <br> 0&{ - 6}&{a - 10}&{ - 5} <br>\end{array}} \right)\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1A1</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note:</strong> Award an <em><strong>A1</strong></em> for each correctly reduced row.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(i) \(a -10 = -2 \Rightarrow a = 8\) <em><strong>M1A1 N1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) when \(a = 8\) , row 3 \( \ne \) 2 \( \times \) row 2 <em><strong>R1 N0</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[8 marks]</span></strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">Few students were able to do this question efficiently. Many students were able to do part (a) by manipulating equations, whereas calculator methods would yield the solution quickly and easily. Part (b) was poorly attempted and it was apparent that many students used a lot of time manipulating equations without real understanding of what they were looking for.</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">The points A and B have position vectors \(\overrightarrow {{\text{OA}}} = \left\{ {\begin{array}{*{20}{c}} 1 \\ 2 \\ { - 2} \end{array}} \right\}\) and \(\overrightarrow {{\text{OB}}} = \left\{ {\begin{array}{*{20}{c}} 1 \\ 0 \\ 2 \end{array}} \right\}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find \(\overrightarrow {{\text{OA}}} Â \times \overrightarrow {{\text{OB}}} \).</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">Hence find the area of the triangle <span class="s1">OAB</span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">\(\overrightarrow {{\text{OA}}} \times \overrightarrow {{\text{OB}}} = \left( {\begin{array}{*{20}{c}} 4 \\ { - 4} \\ { - 2} \end{array}} \right)\)<strong>Â <span class="Apple-converted-space">Â Â </span></strong><span class="s1"><strong><em>(M1)A1</em></strong></span></p>
<p class="p3"><strong>Note: <span class="Apple-converted-space">Â Â </span><em>M1A0 </em></strong>can be awarded for attempt at a correct method <strong>shown</strong><span class="s2">, or correct method implied by the digits 4, 4, 2 </span>found in the correct order.</p>
<p class="p3"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">\({\text{area}} = \frac{1}{2}\sqrt {{4^2} + {4^2} + {2^2}} Â = 3\)Â <span class="Apple-converted-space">Â Â </span><span class="s1"><strong><em>M1A1</em></strong></span></p>
<p class="p2"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Generally well done. Most students were able to obtain full marks on this question. Most of the errors made were due to careless mistakes.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Generally well done. Most students were able to obtain full marks on this question. Most of the errors made were due to careless mistakes. A few students did not take notice of the “hence” in part (b) and were consequently not able to obtain the marks.</p>
<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: 25.0px Helvetica;"><span style="font-size: medium; font-family: 'times new roman', times;">The points A and B have coordinates (1, 2, 3) and (3, 1, 2) relative to an origin O.</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: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Find \(\overrightarrow {{\text{OA}}} \times \overrightarrow {{\text{OB}}} \) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Determine the area of the triangle OAB.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) Find the Cartesian equation of the plane OAB.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-size: medium; font-family: 'times new roman', times;">(i) Find the vector equation of the line \({L_1}\) containing the points A and B.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-size: medium; font-family: 'times new roman', times;">(ii) The line \({L_2}\) has vector equation \(\left( {\begin{array}{*{20}{c}}<br> x \\ <br> y \\ <br> z <br>\end{array}} \right) = \left( {\begin{array}{*{20}{c}}<br> 2 \\ <br> 4 \\ <br> 3 <br>\end{array}} \right) + \mu \left( {\begin{array}{*{20}{c}}<br> 1 \\ <br> 3 \\ <br> 2 <br>\end{array}} \right)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-size: medium; font-family: 'times new roman', times;">Determine whether or not \({L_1}\) and \({L_2}\) are skew.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) \(\overrightarrow {{\text{OA}}} \times \overrightarrow {{\text{OB}}} = \) <strong><em>i</em></strong> + 7<strong><em>j</em></strong> – 5<strong><em>k</em></strong> <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) area \( = \frac{1}{2}|\)<strong><em>i</em></strong> + 7<strong><em>j</em></strong> – 5<strong><em>k</em></strong>\(| = \frac{{5\sqrt 3 }}{2}{\text{(4.33)}}\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) equation of plane is \(x + 7y - 5z = k\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(x + 7y - 5z = 0\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) direction of line = (3<strong><em>i</em></strong> + <strong><em>j</em></strong> + 2<strong><em>k</em></strong>) – (<strong><em>i</em></strong> + 2<strong><em>j</em></strong> + 3<strong><em>k</em></strong>) = 2<strong><em>i</em></strong> – <strong><em>j</em></strong> – <strong><em>k</em></strong> <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">equation of line is</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>r</em></strong> = (<strong><em>i</em></strong> + 2<strong><em>j</em></strong> + 3<strong><em>k</em></strong>) + \(\lambda \)(2<strong><em>i</em></strong> – <strong><em>j</em></strong> – <strong><em>k</em></strong>) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) at a point of intersection,</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(1 + 2\lambda = 2 + \mu \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(2 - \lambda = 4 + 3\mu \) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(3 - \lambda = 3 + 2\mu \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">solving the \({2^{{\text{nd}}}}\) and \({3^{{\text{rd}}}}\) equations, \(\lambda = 4{\text{, }}\mu = - 2\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">these values do not satisfy the \({1^{{\text{st}}}}\) equation so the lines are skew <strong><em>R1</em></strong></span><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[7 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the planes \({\pi _1}:x - 2y - 3z = 2{\text{ and }}{\pi _2}:2x - y - z = k\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the angle between the planes \({\pi _1}\)and \({\pi _2}\) .</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: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">The planes \({\pi _1}\) and \({\pi _2}\) intersect in the line \({L_1}\) . Show that the vector equation of</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\({L_1}\) is \(r = \left( {\begin{array}{*{20}{c}}<br>0\\<br>{2 - 3k}\\<br>{2k - 2}<br>\end{array}} \right) + t\left( {\begin{array}{*{20}{c}}<br>1\\<br>5\\<br>{ - 3}<br>\end{array}} \right)\)</span></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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">The line \({L_2}\) has Cartesian equation \(5 - x = y + 3 = 2 - 2z\) . The lines \({L_1}\) and \({L_2}\) intersect at a point X. Find the coordinates of X.</span></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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Determine a Cartesian equation of the plane \({\pi _3}\) containing both lines \({L_1}\) and \({L_2}\) .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Let Y be a point on \({L_1}\) and Z be a point on \({L_2}\) such that XY is perpendicular to YZ and the area of the triangle XYZ is 3. Find the perimeter of the triangle XYZ.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Accept alternative notation for vectors (<em>eg</em> \(\langle a{\text{, }}b{\text{, }}c\rangle {\text{ or }}\left( {a{\text{, }}b{\text{, }}c} \right)\)).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\(\boldsymbol{n} = \left( {\begin{array}{*{20}{c}}<br> 1 \\ <br> { - 2} \\ <br> { - 3} <br>\end{array}} \right)\)</span><span style="font-family: 'times new roman', times; font-size: medium;"> and \(\boldsymbol{m} = \left( {\begin{array}{*{20}{c}}<br> 2 \\ <br> { - 1} \\ <br> { - 1} <br>\end{array}} \right)\)</span><span style="font-family: 'times new roman', times; font-size: medium;"> <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\cos \theta = \frac{{\boldsymbol{n} \cdot \boldsymbol{m}}}{{\left| \boldsymbol{n} \right|\left| \boldsymbol{m} \right|}}\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\cos \theta = \frac{{2 + 2 + 3}}{{\sqrt {1 + 4 + 9} \sqrt {4 + 1 + 1} }} = \frac{7}{{\sqrt {14} \sqrt 6 }}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\theta = 40.2^\circ \,\,\,\,\,(0.702{\text{ rad}})\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note: </strong>Accept alternative notation for vectors (<em>eg</em> \(\langle a{\text{, }}b{\text{, }}c\rangle {\text{ or }}\left( {a{\text{, }}b{\text{, }}c} \right)\)).</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 1</strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">eliminate <em>z </em>from <em>x</em> – 2<em>y</em> – 3<em>z</em> = 2 and 2<em>x</em> – <em>y</em> – <em>z</em> = <em>k</em></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\(5x - y = 3k - 2 \Rightarrow x = \frac{{y - (2 - 3k)}}{5}\) <strong><em>M1A1</em></strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">eliminate <em>y </em>from <em>x</em> – 2<em>y</em> – 3<em>z</em> = 2 and 2<em>x</em> – <em>y</em> – <em>z</em> = <em>k</em></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\(3x + z = 2k - 2 \Rightarrow x = \frac{{z - (2k - 2)}}{{ - 3}}\) <strong><em>A1</em></strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;"><em>x</em> =<em> t</em>,<em> y </em>= (2 − 3<em>k</em>) + 5t and <em>z</em> = (2<em>k </em>− 2) − 3<em>t <strong>A1A1</strong></em></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\(r = \left( {\begin{array}{*{20}{c}}<br>0\\<br>{2 - 3k}\\<br>{2k - 2}<br>\end{array}} \right) + t\left( {\begin{array}{*{20}{c}}<br>1\\<br>5\\<br>{ - 3}<br>\end{array}} \right)\) <strong><em>AG</em></strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
<p><strong style="font-family: 'times new roman', times; font-size: medium;">METHOD 2</strong></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\(\left( {\begin{array}{*{20}{c}}<br>1\\<br>{ - 2}\\<br>{ - 3}<br>\end{array}} \right) \times \left( {\begin{array}{*{20}{c}}<br>2\\<br>{ - 1}\\<br>{ - 1}<br>\end{array}} \right) = \left( {\begin{array}{*{20}{c}}<br>{ - 1}\\<br>{ - 5}\\<br>3<br>\end{array}} \right) \Rightarrow {\text{direction is }}\left( {\begin{array}{*{20}{c}}<br>1\\<br>5\\<br>{ - 3}<br>\end{array}} \right)\) <em><strong> M1A1</strong></em></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">Let <em>x</em> = 0</span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\(0 - 2y - 3z = 2{\text{ and }}2 \times 0 - y - z = k\) <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">solve simultaneously <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\(y = 2 - 3k{\text{ and }}z = 2k - 2\) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">therefore <strong>r</strong> \( = \left( {\begin{array}{*{20}{c}}<br>0\\<br>{2 - 3k}\\<br>{2k - 2}<br>\end{array}} \right) + t\left( {\begin{array}{*{20}{c}}<br>1\\<br>5\\<br>{ - 3}<br>\end{array}} \right)\) <em><strong>AG</strong></em></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;"><em><strong>[5 marks]</strong></em></span></p>
<p><strong style="font-family: 'times new roman', times; font-size: medium;">METHOD 3</strong></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">substitute \(x = t,{\text{ }}y = (2 - 3k) + 5t{\text{ and }}z = (2k - 2) - 3t{\text{ into }}{\pi _1}{\text{ and }}{\pi _2}\) <em><strong> M1</strong></em></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">for \({\pi _1}:t - 2(2 - 3k + 5t) - 3(2k - 2 - 3t) = 2\) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">for \({\pi _2}:2t - (2 - 3k + 5t) - (2k - 2 - 3t) = k\) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">the planes have a unique line of intersection <em><strong>R2</strong></em></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">therefore the line is \(r = \left( {\begin{array}{*{20}{c}}<br>0\\<br>{2 - 3k}\\<br>{2k - 2}<br>\end{array}} \right) + t\left( {\begin{array}{*{20}{c}}<br>1\\<br>5\\<br>{ - 3}<br>\end{array}} \right)\) <em><strong>AG</strong></em></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;"><em><strong>[5 marks]</strong></em></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Accept alternative notation for vectors (<em>eg</em> \(\langle a{\text{, }}b{\text{, }}c\rangle {\text{ or }}\left( {a{\text{, }}b{\text{, }}c} \right)\)).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px Times; color: #3f3f3f;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(5 - t = (2 - 3k + 5t) + 3 = 2 - 2(2k - 2 - 3t)\) <em><strong>M1A1</strong></em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note: </strong><span style="font-family: 'times new roman', times; font-size: medium;">Award </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>M1A1 </em></strong><span style="font-family: 'times new roman', times; font-size: medium;">if candidates use vector or parametric equations of \({L_2}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><em><span style="font-family: 'times new roman',times; font-size: medium;">eg </span></em><span style="font-family: 'times new roman',times; font-size: medium;">\(\left( {\begin{array}{*{20}{c}}<br>0\\<br>{2 - 3k}\\<br>{2k - 2}<br>\end{array}} \right) + t\left( {\begin{array}{*{20}{c}}<br>1\\<br>5\\<br>{ - 3}<br>\end{array}} \right) = \left( {\begin{array}{*{20}{c}}<br>5\\<br>{ - 3}\\<br>1<br>\end{array}} \right) + s\left( {\begin{array}{*{20}{c}}<br>{ - 2}\\<br>2\\<br>{ - 1}<br>\end{array}} \right)\) or \( \Rightarrow \left\{ {\begin{array}{*{20}{l}}<br>{t = 5 - 2s}\\<br>{2 - 3k + 5t = - 3 + 2s}\\<br>{2k - 2 - 3t = 1 + s}<br>\end{array}} \right.\)<br></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">solve simultaneously <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(k = 2,{\text{ }}t = 1{\text{ }}(s = 2)\) <em><strong>A1</strong></em><br></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">intersection point (\(1\), \(1\), \( - 1\)) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Accept alternative notation for vectors (<em>eg</em> \(\langle a{\text{, }}b{\text{, }}c\rangle {\text{ or }}\left( {a{\text{, }}b{\text{, }}c} \right)\)).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px Times; color: #3f3f3f;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\overrightarrow l _2} = \left( {\begin{array}{*{20}{c}}<br>2\\<br>{ - 2}\\<br>1<br>\end{array}} \right)\) <em><strong>A1</strong></em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\overrightarrow l _1} \times {\overrightarrow l _2} = \left| {\begin{array}{*{20}{c}}<br>\boldsymbol{i}&\boldsymbol{j}&\boldsymbol{k}\\<br>1&5&{ - 3}\\<br>2&{ - 2}&1<br>\end{array}} \right| = \left( {\begin{array}{*{20}{c}}<br>{ - 1}\\<br>{ - 7}\\<br>{ - 12}<br>\end{array}} \right)\) <em><strong>(M1)A1</strong></em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\boldsymbol{r} \cdot \left( {\begin{array}{*{20}{c}}<br>1\\<br>7\\<br>{12}<br>\end{array}} \right) = \left( {\begin{array}{*{20}{c}}<br>1\\<br>1\\<br>{ - 1}<br>\end{array}} \right) \cdot \left( {\begin{array}{*{20}{c}}<br>1\\<br>7\\<br>{12}<br>\end{array}} \right)\) <em><strong>(M1)</strong></em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\(x + 7y + 12z = - 4\) <em><strong>A1</strong></em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><em><strong>[5 marks]</strong></em></span></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Accept alternative notation for vectors (<em>eg</em> \(\langle a{\text{, }}b{\text{, }}c\rangle {\text{ or }}\left( {a{\text{, }}b{\text{, }}c} \right)\)).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px Times; color: #3f3f3f;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(\theta \) be the angle between the lines \({\overrightarrow l _1} = \left( {\begin{array}{*{20}{c}}<br>1\\<br>5\\<br>{ - 3}<br>\end{array}} \right)\) and \({\overrightarrow l _2} = \left( {\begin{array}{*{20}{c}}<br>2\\<br>{ - 2}\\<br>1<br>\end{array}} \right)\)<br></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\cos \theta = \frac{{\left| {2 - 10 - 3} \right|}}{{\sqrt {35} \sqrt 9 }} \Rightarrow \theta = 0.902334...{\text{ }}51.699...^\circ )\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">as the triangle XYZ has a right angle at Y,</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{XZ}} = a \Rightarrow {\text{YZ}} = a\sin \theta {\text{ and XY}} = a\cos \theta \) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{area = 3}} \Rightarrow \frac{{{a^2}\sin \theta \cos \theta }}{2} = 3\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(a = 3.5122...\) <em><strong>(A1)</strong></em><br></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">perimeter \( = a + a\sin \theta + a\cos \theta = 8.44537... = 8.45\) <em><strong>A1</strong></em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note: </strong><span style="font-family: 'times new roman', times; font-size: medium;">If candidates attempt to find coordinates of Y and Z award </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>M1 </em></strong><span style="font-family: 'times new roman', times; font-size: medium;">for expression of vector YZ in terms of two parameters, </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>M1 </em></strong><span style="font-family: 'times new roman', times; font-size: medium;">for attempt to use perpendicular condition to determine relation between parameters, </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>M1 </em></strong><span style="font-family: 'times new roman', times; font-size: medium;">for attempt to use the area to find the parameters and </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A2 </em></strong><span style="font-family: 'times new roman', times; font-size: medium;">for final answer.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Although this was the last question in part B, it was answered surprisingly well by many candidates, except for part (e). Even those who had not done so well elsewhere often gained a number of marks in some parts of the question. Nevertheless the presence of parameters seemed to have blocked the abilities of weaker candidates to solve situations in which vectors were involved. Mathematical skills for this particular question were sometimes remarkable, however, calculations proved incomplete due to the way that planes were presented. Most candidates found a correct angle in part (a). Occasional arithmetic errors in calculating the magnitude of a vector and dot product occurred. In part (b) the vector product approach was popular. In some case candidates simply verified the result by substitution. There was a lot of simultaneous equation solving, much of which was not very pretty. In part (c), a number of candidates made errors when attempting to solve a system of equations involving parameters. Many of the results for the point were found in terms of <em>k</em>. It was notorious that candidates did not use their GDC to try to find the coordinates of the intersection point between lines. In part (d), a number of candidates used an incorrect point but this part was often done well. </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Very few excellent answers to part (e) were seen using an efficient method. Most candidates attempted methods involving heavy algebraic manipulation and had little success in this part of the question.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Although this was the last question in part B, it was answered surprisingly well by many candidates, except for part (e). Even those who had not done so well elsewhere often gained a number of marks in some parts of the question. Nevertheless the presence of parameters seemed to have blocked the abilities of weaker candidates to solve situations in which vectors were involved. Mathematical skills for this particular question were sometimes remarkable, however, calculations proved incomplete due to the way that planes were presented. Most candidates found a correct angle in part (a). Occasional arithmetic errors in calculating the magnitude of a vector and dot product occurred. In part (b) the vector product approach was popular. In some case candidates simply verified the result by substitution. There was a lot of simultaneous equation solving, much of which was not very pretty. In part (c), a number of candidates made errors when attempting to solve a system of equations involving parameters. Many of the results for the point were found in terms of <em>k</em>. It was notorious that candidates did not use their GDC to try to find the coordinates of the intersection point between lines. In part (d), a number of candidates used an incorrect point but this part was often done well. </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Very few excellent answers to part (e) were seen using an efficient method. Most candidates attempted methods involving heavy algebraic manipulation and had little success in this part of the question.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Although this was the last question in part B, it was answered surprisingly well by many candidates, except for part (e). Even those who had not done so well elsewhere often gained a number of marks in some parts of the question. Nevertheless the presence of parameters seemed to have blocked the abilities of weaker candidates to solve situations in which vectors were involved. Mathematical skills for this particular question were sometimes remarkable, however, calculations proved incomplete due to the way that planes were presented. Most candidates found a correct angle in part (a). Occasional arithmetic errors in calculating the magnitude of a vector and dot product occurred. In part (b) the vector product approach was popular. In some case candidates simply verified the result by substitution. There was a lot of simultaneous equation solving, much of which was not very pretty. In part (c), a number of candidates made errors when attempting to solve a system of equations involving parameters. Many of the results for the point were found in terms of <em>k</em>. It was notorious that candidates did not use their GDC to try to find the coordinates of the intersection point between lines. In part (d), a number of candidates used an incorrect point but this part was often done well. </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Very few excellent answers to part (e) were seen using an efficient method. Most candidates attempted methods involving heavy algebraic manipulation and had little success in this part of the question.</span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Although this was the last question in part B, it was answered surprisingly well by many candidates, except for part (e). Even those who had not done so well elsewhere often gained a number of marks in some parts of the question. Nevertheless the presence of parameters seemed to have blocked the abilities of weaker candidates to solve situations in which vectors were involved. Mathematical skills for this particular question were sometimes remarkable, however, calculations proved incomplete due to the way that planes were presented. Most candidates found a correct angle in part (a). Occasional arithmetic errors in calculating the magnitude of a vector and dot product occurred. In part (b) the vector product approach was popular. In some case candidates simply verified the result by substitution. There was a lot of simultaneous equation solving, much of which was not very pretty. In part (c), a number of candidates made errors when attempting to solve a system of equations involving parameters. Many of the results for the point were found in terms of <em>k</em>. It was notorious that candidates did not use their GDC to try to find the coordinates of the intersection point between lines. In part (d), a number of candidates used an incorrect point but this part was often done well. </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Very few excellent answers to part (e) were seen using an efficient method. Most candidates attempted methods involving heavy algebraic manipulation and had little success in this part of the question.</span></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Although this was the last question in part B, it was answered surprisingly well by many candidates, except for part (e). Even those who had not done so well elsewhere often gained a number of marks in some parts of the question. Nevertheless the presence of parameters seemed to have blocked the abilities of weaker candidates to solve situations in which vectors were involved. Mathematical skills for this particular question were sometimes remarkable, however, calculations proved incomplete due to the way that planes were presented. Most candidates found a correct angle in part (a). Occasional arithmetic errors in calculating the magnitude of a vector and dot product occurred. In part (b) the vector product approach was popular. In some case candidates simply verified the result by substitution. There was a lot of simultaneous equation solving, much of which was not very pretty. In part (c), a number of candidates made errors when attempting to solve a system of equations involving parameters. Many of the results for the point were found in terms of <em>k</em>. It was notorious that candidates did not use their GDC to try to find the coordinates of the intersection point between lines. In part (d), a number of candidates used an incorrect point but this part was often done well. </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Very few excellent answers to part (e) were seen using an efficient method. Most candidates attempted methods involving heavy algebraic manipulation and had little success in this part of the question.</span></p>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the vector equation of the line of intersection of the three planes represented by the following system of equations.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[2x - 7y + 5z = 1\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[6x + 3y - z = - 1\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[ - 14x - 23y + 13z = 5\]</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 1</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(from GDC)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left( {\begin{array}{*{20}{ccc|c}}<br> 1&0&{\frac{1}{6}}&{ - \frac{1}{{12}}} \\ <br> 0&1&{ - \frac{2}{3}}&{ - \frac{1}{6}} \\ <br> 0&0&0&0 <br>\end{array}} \right)\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(x + \frac{1}{6}\lambda = - \frac{1}{{12}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(y - \frac{2}{3}\lambda = - \frac{1}{6}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\boldsymbol{r} = \left( { - \frac{1}{{12}}\boldsymbol{i} - \frac{1}{6}\boldsymbol{j}} \right) + \lambda \left( { - \frac{1}{6}\boldsymbol{i} + \frac{2}{3}\boldsymbol{j} + \boldsymbol{k}} \right)\) <strong><em>A1A1A1</em></strong> <strong><em>N3</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 2</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(Elimination method either for equations or row reduction of matrix)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Eliminating one of the variables <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Finding a point on the line <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Finding the direction of the line <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The <strong>vector</strong> equation of the line <strong><em>A1</em></strong> <strong><em>N3</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A large number of candidates did not use their GDC in this question. Some candidates who attempted analytical solutions looked for a point solution although the question specifically states that the planes intersect in a line. Other candidates eliminated one variable and then had no clear strategy for proceeding with the solution.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Some candidates failed to write ‘<em>r</em> =’, and others did not give the equation in vector form.</span></p>
</div>
<br><hr><br><div class="specification">
<p>Two submarines A and B have their routes planned so that their positions at time <em>t</em> hours, 0 ≤ <em>t</em> < 20 , would be defined by the position vectors <em><strong>r</strong><sub>A</sub></em> \( = \left( \begin{gathered}<br> \,2 \hfill \\<br> \,4 \hfill \\<br> - 1 \hfill \\ <br>\end{gathered} \right) + t\left( \begin{gathered}<br> - 1 \hfill \\<br> \,1 \hfill \\<br> - 0.15 \hfill \\ <br>\end{gathered} \right)\) and <em><strong>r</strong><sub>B</sub></em> \( = \left( \begin{gathered}<br> \,0 \hfill \\<br> \,3.2 \hfill \\<br> - 2 \hfill \\ <br>\end{gathered} \right) + t\left( \begin{gathered}<br> - 0.5 \hfill \\<br> \,1.2 \hfill \\<br> \,0.1 \hfill \\ <br>\end{gathered} \right)\) relative to a fixed point on the surface of the ocean (all lengths are in kilometres).</p>
</div>
<div class="specification">
<p>To avoid the collision submarine B adjusts its velocity so that its position vector is now given by</p>
<p style="padding-left: 120px;"><em><strong>r</strong><sub>B</sub></em> \( = \left( \begin{gathered}<br> \,0 \hfill \\<br> \,3.2 \hfill \\<br> - 2 \hfill \\ <br>\end{gathered} \right) + t\left( \begin{gathered}<br> - 0.45 \hfill \\<br> \,1.08 \hfill \\<br> \,0.09 \hfill \\ <br>\end{gathered} \right)\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the two submarines would collide at a point P and write down the coordinates of P.</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>Show that submarine B travels in the same direction as originally planned.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <em>t</em> when submarine B passes through P.</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>Find an expression for the distance between the two submarines in terms of <em>t</em>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <em>t</em> when the two submarines are closest together.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the distance between the two submarines at this time.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.iii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><em><strong>r</strong><sub>A </sub>=Â <strong>r</strong><sub>B Â Â Â Â <strong>(M1)</strong></sub></em></p>
<p>2 − <em>t</em> = − 0.5t ⇒ <em>t</em> = 4    <strong>A1</strong></p>
<p>checking <em>t</em> = 4 satisfies 4 + <em>t</em> = 3.2 + 1.2<em>t</em> and − 1 − 0.15<em>t</em> = − 2 + 0.1<em>t   <strong>R1</strong></em></p>
<p>P(−2, 8, −1.6)   <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Do not award final <em><strong>A1</strong></em> if answer given as column vector.</p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(0.9 \times \left( \begin{gathered}<br> - 0.5 \hfill \\<br> \,1.2 \hfill \\<br> \,0.1 \hfill \\ <br>\end{gathered} \right) = \left( \begin{gathered}<br> - 0.45 \hfill \\<br> \,1.08 \hfill \\<br> \,0.09 \hfill \\ <br>\end{gathered} \right)\)Â Â Â <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Accept use of cross product equalling zero.</p>
<p>hence in the same direction   <em><strong>AG</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(\left( \begin{gathered}<br> \, - 0.45t \hfill \\<br> 3.2 + 1.08t \hfill \\<br> - 2 + 0.09t \hfill \\ <br>\end{gathered} \right) = \left( \begin{gathered}<br> - 2 \hfill \\<br> \,8 \hfill \\<br> - 1.6 \hfill \\ <br>\end{gathered} \right)\)Â Â Â <em><strong>M1</strong></em></p>
<p><strong>Note:</strong> The <strong><em>M1</em></strong> can be awarded for any one of the resultant equations.</p>
<p>\( \Rightarrow t = \frac{{40}}{9} = 4.44 \ldots \)Â Â Â <em><strong>A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><em><strong>r</strong><sub>A</sub></em> − <em><strong>r</strong><sub>B</sub></em> = \(\left( \begin{gathered}<br> \,2 - t \hfill \\<br> \,4 + t \hfill \\<br> - 1 - 0.15t \hfill \\ <br>\end{gathered} \right) - \left( \begin{gathered}<br> \, - 0.45t \hfill \\<br> 3.2 + 1.08t \hfill \\<br> - 2 + 0.09t \hfill \\ <br>\end{gathered} \right)\)   <em><strong> (M1)(A1)</strong></em></p>
<p>\( = \left( \begin{gathered}<br> \,2 - 0.55t \hfill \\<br> \,0.8 - 0.08t \hfill \\<br> 1 - 0.24t \hfill \\ <br>\end{gathered} \right)\)Â Â Â <em><strong>(A1)</strong></em></p>
<p><strong>Note</strong>: Accept <em><strong>r</strong><sub>A</sub></em> − <em><strong>r</strong><sub>B</sub></em>.</p>
<p>distance \(D = \sqrt {{{\left( {2 - 0.55t} \right)}^2} + {{\left( {0.8 - 0.08t} \right)}^2} + {{\left( {1 - 0.24t} \right)}^2}} \)   <em><strong> M1A1</strong></em></p>
<p>\(\left( { = \sqrt {8.64 - 2.688t + 0.317{t^2}} } \right)\)</p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>minimum when \(\frac{{{\text{d}}D}}{{{\text{d}}t}} = 0\)    <em><strong>(M1)</strong></em></p>
<p><em>t</em>Â = 3.83Â Â Â <em><strong>A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>0.511 (km)Â Â Â <em><strong>A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">c.iii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.iii.</div>
</div>
<br><hr><br><div class="specification">
<p>The points A, B and C have the following position vectors with respect to an origin O.</p>
<p style="text-align: center;">\(\overrightarrow {{\rm{OA}}} = 2\)<strong><em>i</em></strong> + <strong><em>j</em></strong> – 2<strong><em>k</em></strong></p>
<p style="text-align: center;">\(\overrightarrow {{\rm{OB}}} = 2\)<strong><em>i</em></strong> – <strong><em>j</em></strong> + 2<strong><em>k</em></strong></p>
<p style="text-align: center;">\(\overrightarrow {{\rm{OC}}} = \) <strong><em>i</em></strong> + 3<strong><em>j</em></strong> + 3<strong><em>k</em></strong></p>
</div>
<div class="specification">
<p>The plane <em>Π</em>\(_2\) contains the points O, A and B and the plane <em>Π</em>\(_3\) contains the points O, A and C.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the vector equation of the line (BC).</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>Determine whether or not the lines (OA) and (BC) intersect.</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>Find the Cartesian equation of the plane <em>Î </em>\(_1\), which passes through C and is perpendicular to \(\overrightarrow {{\rm{OA}}} \).</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>Show that the line (BC) lies in the plane <em>Π</em>\(_1\).</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>Verify that 2<strong><em>j </em></strong>+ <strong><em>k </em></strong>is perpendicular to the plane <em>Î </em>\(_2\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find a vector perpendicular to the plane <em>Î </em>\(_3\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the acute angle between the planes <em>Î </em>\(_2\) and <em>Î </em>\(_3\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">g.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>\(\overrightarrow {{\rm{BC}}} \) = (<strong><em>i</em></strong> + 3<strong><em>j</em></strong> + 3<strong><em>k</em></strong>) \( - \) (2<strong><em>i</em></strong> \( - \)Â <strong><em>j</em></strong> + 2<strong><em>k</em></strong>) = \( - \)<strong><em>i</em></strong> + 4<strong><em>j</em></strong> + <strong><em>k</em></strong> Â Â Â <strong><em>(A1)</em></strong></p>
<p><strong><em>r</em></strong> = (2<strong><em>i</em></strong> \( - \) <strong><em>j</em></strong> + 2<strong><em>k</em></strong>) + \(\lambda \)(\( - \)<strong><em>i</em></strong> + 4<strong><em>j</em></strong> + <strong><em>k</em></strong>)</p>
<p>(or <strong><em>r</em></strong> = (<strong><em>i</em></strong> + 3<strong><em>j</em></strong> + 3<strong><em>k</em></strong>) + \(\lambda \)(\( - \)<strong><em>i</em></strong> + 4<strong><em>j </em></strong>+ <strong><em>k</em></strong>)Â Â Â Â <strong><em>(M1)A1</em></strong></p>
<p>Â </p>
<p><strong>Note:</strong>Â Â Â Â Do not award <strong><em>A1 </em></strong>unless <strong><em>r </em></strong>= or equivalent correct notation seen.</p>
<p>Â </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to write in parametric form using two different parameters <strong>AND </strong>equate    <strong><em>M1</em></strong></p>
<p>\(2\mu = 2 - \lambda \)</p>
<p>\(\mu = - 1 + 4\lambda \)</p>
<p>\( - 2\mu = 2 + \lambda \)Â Â Â Â <strong><em>A1</em></strong></p>
<p>attempt to solve first pair of simultaneous equations for two parameters    <strong><em>M1</em></strong></p>
<p>solving first two equations gives \(\lambda = \frac{4}{9},{\text{ }}\mu = \frac{7}{9}\)Â Â Â Â <strong><em>(A1)</em></strong></p>
<p>substitution of these two values in third equation    <strong><em>(M1)</em></strong></p>
<p>since the values do not fit, the lines do not intersect    <strong><em>R1</em></strong></p>
<p>Â </p>
<p><strong>Note:</strong>Â Â Â Â Candidates may note that adding the first and third equations immediately leads to a contradiction and hence they can immediately deduce that the lines do not intersect.</p>
<p>Â </p>
<p><strong><em>[6 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>plane is of the form <strong><em>r</em></strong> \( \bullet \) (2<strong><em>i</em></strong> + <strong><em>j</em></strong> \( - \) 2<strong><em>k</em></strong>) = <em>d</em>Â Â Â Â <strong><em>(A1)</em></strong></p>
<p><em>d </em>= (<strong><em>i</em></strong> + 3<strong><em>j</em></strong> + 3<strong><em>k</em></strong>) \( \bullet \) (2<strong><em>i</em></strong> + <strong><em>j</em></strong> \( - \) 2<strong><em>k</em></strong>) = \( - \)1Â Â Â Â <strong><em>(M1)</em></strong></p>
<p>hence Cartesian form of plane is \(2x + y - 2z = - 1\)Â Â Â Â <strong><em>A1</em></strong></p>
<p><strong>METHOD 2</strong></p>
<p>plane is of the form \(2x + y - 2z = d\)Â Â Â Â <strong><em>(A1)</em></strong></p>
<p>substituting \((1,{\text{ }}3,{\text{ }}3)\) (to find gives \(2 + 3 - 6 = - 1\))Â Â Â Â <strong><em>(M1)</em></strong></p>
<p>hence Cartesian form of plane is \(2x + y - 2z = - 1\)Â Â Â Â <strong><em>A1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>attempt scalar product of direction vector BC with normal to plane    <strong><em>M1</em></strong></p>
<p>(\( - \)<strong><em>i</em></strong> + 4<strong><em>j</em></strong> + <strong><em>k</em></strong>) \( \bullet \) (2<strong><em>i</em></strong> + <strong><em>j</em></strong> \( - \) 2<strong><em>k</em></strong>) \( = - 2 + 4 - 2\)</p>
<p>\( = 0\)Â Â Â Â <strong><em>A1</em></strong></p>
<p>hence BC lies in <em>Î </em>\(_1\)Â Â Â Â <strong><em>AG</em></strong></p>
<p><strong>METHOD 2</strong></p>
<p>substitute eqn of line into plane    <strong><em>M1</em></strong></p>
<p>\({\text{line }}r = \left( {\begin{array}{*{20}{c}} 2 \\ { - 1} \\ 2 \end{array}} \right) + \lambda \left( {\begin{array}{*{20}{c}} { - 1} \\ 4 \\ 1 \end{array}} \right).{\text{ Plane }}{\pi _1}:2x + y - 2z = - 1\)</p>
<p>\(2(2 - \lambda ) + ( - 1 + 4\lambda ) - 2(2 + \lambda )\)</p>
<p>\( = - 1\)Â Â Â Â <strong><em>A1</em></strong></p>
<p>hence BC lies in <em>Î </em>\(_1\)Â Â Â Â <strong><em>AG</em></strong></p>
<p>Â </p>
<p><strong>Note:</strong>Â Â Â Â Candidates may also just substitute \(2i - j + 2k\) into the plane since they are told C lies on \({\pi _1}\).</p>
<p>Â </p>
<p><strong>Note:</strong>Â Â Â Â Do not award <strong><em>A1FT</em></strong>.</p>
<p>Â </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>applying scalar product to \(\overrightarrow {{\rm{OA}}} \) and \(\overrightarrow {{\rm{OB}}} \)Â Â Â Â <strong><em>M1</em></strong></p>
<p>(2<strong><em>j</em></strong> + <strong><em>k</em></strong>) \( \bullet \) (2<strong><em>i</em></strong> + <strong><em>j </em></strong>\( - \) 2<strong><em>k</em></strong>) = 0Â Â Â Â <strong><em>A1</em></strong></p>
<p>(2<strong><em>j</em></strong> + <strong><em>k</em></strong>) \( \bullet \) (2<strong><em>i</em></strong> \( - \) <strong><em>j</em></strong> + 2<strong><em>k</em></strong>) =0Â Â Â Â <strong><em>A1</em></strong></p>
<p><strong>METHOD 2</strong></p>
<p>attempt to find cross product of \(\overrightarrow {{\rm{OA}}} \) and \(\overrightarrow {{\rm{OB}}} \)Â Â Â Â <strong><em>M1</em></strong></p>
<p>plane <em>Î </em>\(_2\) has normal \(\overrightarrow {{\text{OA}}} \times \overrightarrow {{\text{OB}}} \) = \( - \) 8<strong><em>j </em></strong>\( - \) 4<strong><em>k</em></strong>Â Â Â Â <strong><em>A1</em></strong></p>
<p>since \( - \)8<strong><em>j </em></strong>\( - \) 4<strong><em>k </em></strong>= \( - \)4(2<strong><em>j</em></strong> + <strong><em>k</em></strong>), 2<strong><em>j </em></strong>+ <strong><em>k </em></strong>is perpendicular to the plane <em>Î </em>\(_2\)Â Â Â Â <strong><em>R1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>plane <em>Î </em>\(_3\) has normal \(\overrightarrow {{\text{OA}}} \times \overrightarrow {{\text{OC}}} \) = 9<strong><em>i</em></strong> \( - \) 8<strong><em>j</em></strong> + 5<strong><em>k</em></strong>Â Â Â Â <strong><em>A1</em></strong></p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to use dot product of normal vectors    <strong><em>(M1)</em></strong></p>
<p>\(\cos \theta = \frac{{(2j + k) \bullet (9i - 8j + 5k)}}{{\left| {2j + k} \right|\left| {9i - 8j + 5k} \right|}}\)Â Â Â Â <strong><em>(M1)</em></strong></p>
<p>\( = \frac{{ - 11}}{{\sqrt 5 \sqrt {170} }}\,\,\,( = - 0.377 \ldots )\)Â Â Â Â <strong><em>(A1)</em></strong></p>
<p>Â </p>
<p><strong>Note:</strong>Â Â Â Â Accept \(\frac{{11}}{{\sqrt 5 \sqrt {170} }}\). Â acute angle between planes \( = 67.8^\circ \,\,\,{\text{(}} = 1.18^\circ )\)Â Â Â Â <strong><em>A1</em></strong></p>
<p>Â </p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">g.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">g.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">OACB is a parallelogram with \(\overrightarrow {{\text{OA}}} = \) <strong><em>a </em></strong>and \(\overrightarrow {{\text{OB}}} = \) <strong><em>b</em></strong><span class="s1">, where </span><strong><em>a </em></strong><span class="s1">and </span><strong><em>b </em></strong><span class="s1">are non-zero vectors.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that</p>
<p class="p1">(i) <span class="Apple-converted-space">Â Â \({\left| {\overrightarrow {{\text{OC}}} } \right|^2} = |\)</span><strong><em>a</em></strong>\({|^2} + 2\)<strong><em>a</em></strong>Â \( \bullet \) <strong><em>b</em></strong>Â \( + |\)<strong><em>b</em></strong>\({|^2}\);</p>
<p class="p1">(ii) <span class="Apple-converted-space">Â Â \({\left| {\overrightarrow {{\text{AB}}} } \right|^2} = |\)</span><strong><em>a</em></strong>\({|^2} - 2\)<strong><em>a</em></strong>Â \( \bullet \) <strong><em>b</em></strong>Â \( + |\)<strong><em>b</em></strong>\({|^2}\).</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">Given that \(\left| {\overrightarrow {{\text{OC}}} } \right| = \left| {\overrightarrow {{\text{AB}}} } \right|\), prove that OACB <span class="s1">is a rectangle.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1"><strong>METHOD 1</strong></p>
<p class="p2">\({\left| {\overrightarrow {{\text{OC}}} } \right|^2} = \overrightarrow {{\text{OC}}} Â \bullet \overrightarrow {{\text{OC}}} \)</p>
<p class="p2">= (<strong><em>a </em></strong>+ <strong><em>b</em></strong>) \( \bullet \) (<strong><em>a </em></strong>+ <strong><em>b</em></strong>)    <span class="s2"><strong><em>A1</em></strong></span></p>
<p class="p3">= <strong><em>a</em></strong> \( \bullet \)<span class="s1"> </span><strong><em>a </em></strong>+ <strong><em>a</em></strong> \( \bullet \)<span class="s1"> </span><strong><em>b </em></strong>+ <strong><em>b</em></strong> \( \bullet \)<span class="s1"> </span><strong><em>a </em></strong>+ <strong><em>b</em></strong> \( \bullet \)<span class="s1"> </span><strong><em>b <span class="Apple-converted-space">  </span></em></strong><span class="s2"><strong><em>A1</em></strong></span></p>
<p class="p3">= \(|\)<strong><em>a</em></strong>\({|^2}\)<strong><em> </em></strong>+ 2<strong><em>a</em></strong> \( \bullet \)<span class="s1"> </span><strong><em>b </em></strong>+ \(|\)<strong><em>b</em></strong>\({|^2}\)<strong><em> <span class="Apple-converted-space">  </span></em></strong><span class="s2"><strong><em>AG</em></strong></span></p>
<p class="p4"><strong>METHOD 2</strong></p>
<p class="p5"><span class="Apple-converted-space">\({\left| {\overrightarrow {{\text{OC}}} } \right|^2} = {\left| {\overrightarrow {{\text{OA}}} } \right|^2} + {\left| {\overrightarrow {{\text{OB}}} } \right|^2} - 2\left| {\overrightarrow {{\text{OA}}} } \right|\left| {\overrightarrow {{\text{OB}}} } \right|\cos ({\rm{O\hat AC}})\) Â Â </span><span class="s3"><strong><em>A1</em></strong></span></p>
<p class="p5">\(\left| {\overrightarrow {{\text{OA}}} } \right|\left| {\overrightarrow {{\text{OB}}} } \right|\cos ({\rm{O\hat AC}}) = Â - \)(<strong><em>a</em></strong>Â \( \bullet \)<span class="s1">Â </span><strong><em>b</em></strong>)<strong><em> <span class="Apple-converted-space">Â Â </span></em></strong><span class="s3"><strong><em>A1</em></strong></span></p>
<p class="p6">\({\left| {\overrightarrow {{\text{OC}}} } \right|^2} = |\)<strong><em>a</em></strong>\({|^2}\) + 2<strong><em>a</em></strong> \( \bullet \)<span class="s4"> </span><strong><em>b </em></strong>+ \(|\)<strong><em>b</em></strong>\({|^2}\)<strong><em> <span class="Apple-converted-space">  </span></em></strong><span class="s3"><strong><em>AG</em></strong></span></p>
<p class="p4">(ii) <span class="Apple-converted-space">Â Â </span><strong>METHOD 1</strong></p>
<p class="p5">\({\left| {\overrightarrow {{\text{AB}}} } \right|^2} = \overrightarrow {{\text{AB}}} Â \bullet \overrightarrow {{\text{AB}}} \)</p>
<p class="p5">= (<strong><em>b </em></strong>− <strong><em>a</em></strong>) \( \bullet \)<span class="s1"> </span>(<strong><em>b </em></strong>− <strong><em>a</em></strong>) <span class="Apple-converted-space">  </span><span class="s3"><strong><em>A1</em></strong></span></p>
<p class="p5">= <strong><em>b </em></strong>\( \bullet \)<span class="s1"> </span><strong><em>b </em></strong>− <strong><em>b </em></strong>\( \bullet \)<span class="s1"> </span><strong><em>a </em></strong>− <strong><em>a </em></strong><span class="s1">\( \bullet \) </span><strong><em>b </em></strong>+ <strong><em>a </em></strong><span class="s1">\( \bullet \) </span><strong><em>a <span class="Apple-converted-space">  </span></em></strong><span class="s3"><strong><em>A1</em></strong></span></p>
<p class="p5">= \(|\)<strong><em>a</em></strong>\({|^2}\) – 2<strong><em>a</em></strong> \( \bullet \) <strong><em>b </em></strong>+ \(|\)<strong><em>b</em></strong>\({|^2}\) <span class="Apple-converted-space">  </span><span class="s3"><strong><em>AG</em></strong></span></p>
<p class="p4"><strong>METHOD 2</strong></p>
<p class="p5"><span class="Apple-converted-space">\({\left| {\overrightarrow {{\text{AB}}} } \right|^2} = {\left| {\overrightarrow {{\text{AC}}} } \right|^2} + {\left| {\overrightarrow {{\text{BC}}} } \right|^2} - 2\left| {\overrightarrow {{\text{AC}}} } \right|\left| {\overrightarrow {{\text{BC}}} } \right|\cos ({\rm{A\hat CB}})\) Â Â </span><strong><em>A1</em></strong></p>
<p class="p5">\(\left| {\overrightarrow {{\text{AC}}} } \right|\left| {\overrightarrow {{\text{BC}}} } \right|\cos ({\rm{A\hat CB}}) = \)Â <strong><em>a</em></strong>Â \( \bullet \)Â <strong><em>b <span class="Apple-converted-space">Â Â </span>A1</em></strong></p>
<p class="p6">\({\left| {\overrightarrow {{\text{AB}}} } \right|^2} = |\)<strong><em>a </em></strong>\({|^2} - \)Â 2<strong><em>a </em></strong><span class="s4">\( \bullet \)Â </span><strong><em>b </em></strong>+Â \(|\)<strong><em>b</em></strong>\({|^2}\) Â Â Â <span class="s5"><strong><em>AG</em></strong></span></p>
<p class="p4"><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">\(\left| {\overrightarrow {{\text{OC}}} } \right| = \left| {\overrightarrow {{\text{AB}}} } \right| \Rightarrow {\left| {\overrightarrow {{\text{OC}}} } \right|^2} = {\left| {\overrightarrow {{\text{AB}}} } \right|^2} \Rightarrow |\)<span class="s1"><strong><em>a</em></strong>\({|^2} + {\text{2}}\)<strong><em>a</em></strong> \( \bullet \) <strong><em>b </em></strong>\( + |\)<strong><em>b</em></strong>\({|^2} = |\)<strong><em>a</em></strong>\({|^2} - 2\)<strong><em>a</em></strong> \( \bullet \) <strong><em>b </em></strong>\( + |\)<strong><em>b</em></strong>\({|^2}\) <span class="Apple-converted-space">Â Â </span></span><strong><em>R1(M1)</em></strong></p>
<p class="p1"><strong>Note: <span class="Apple-converted-space">Â Â </span></strong>Award <strong><em>R1 </em></strong>for \(\left| {\overrightarrow {{\text{OC}}} } \right| = \left| {\overrightarrow {{\text{AB}}} } \right| \Rightarrow {\left| {\overrightarrow {{\text{OC}}} } \right|^2} = {\left| {\overrightarrow {{\text{AB}}} } \right|^2}\) and <strong>(<em>M1</em></strong><span class="s2">) </span>for \(|\)<strong><em>a</em></strong>\({|^2} + {\text{2}}\)<strong><em>a</em></strong> \( \bullet \) <strong><em>b </em></strong>\( + |\)<strong><em>b</em></strong>\({|^2} = |\)<strong><em>a</em></strong>\({|^2} - 2\)<strong><em>a</em></strong> \( \bullet \) <strong><em>b </em></strong>\( + |\)<strong><em>b</em></strong>\({|^2}\).</p>
<p class="p4"><strong><em>a</em></strong> \( \bullet \)<span class="s3">Â </span><strong><em>b</em></strong> \( = 0\)Â <span class="Apple-converted-space">Â Â </span><span class="s4"><strong><em>A1</em></strong></span></p>
<p class="p1"><span class="s5">hence OACB </span>is a rectangle (<span class="s5"><strong><em>a </em></strong></span>and <span class="s5"><strong><em>b </em></strong></span>both non-zero)</p>
<p class="p1">with adjacent sides at right angles <span class="Apple-converted-space">Â Â </span><strong><em>R1</em></strong></p>
<p class="p1"><strong>Note: <span class="Apple-converted-space">Â Â </span></strong>Award <strong><em>R1(M1)A0R1 </em></strong>if the dot product has not been used.</p>
<p class="p1"><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">In part (a), a significant number of candidates either did not use correct vector notation or simply did not use vector notation at all. A large number of candidates who appeared to adopt a scalar product approach, did not use the scalar product ‘dot’ and represented <em><strong>a</strong></em> \( \bullet \)<strong> </strong><em><strong>b</strong></em> as <em><strong>ab</strong></em>. A few candidates successfully used the cosine rule with correct vector notation. A small number of candidates expressed <em><strong>a</strong></em> and <em><strong>b</strong></em> in general component form. In part (a) (ii), quite a number of candidates expressed \({\overrightarrow {{\text{AB}}} }\) as <em><strong>a</strong></em> – <em><strong>b</strong></em> rather than as <em><strong>b</strong></em> – <em><strong>a</strong></em>.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">In part (b), some very well structured proofs were offered by a small number of candidates.</p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Ed walks in a straight line from point \({\text{P}}( - 1,{\text{ }}4)\) to point \({\text{Q}}(4,{\text{ }}16)\) with constant speed.</p>
<p class="p2"><span class="s1">Ed starts from point \(P\) </span>at time \(t = 0\) <span class="s1">and arrives at point \(Q\) </span>at time \(t = 3\), where \(t\) is measured in hours.</p>
<p class="p2">Given that, at time \(t\), Ed’s position vector, relative to the origin, can be given in the form, \({{r}} = {{a}} + t{{b}}\),</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">find the vectors \({{a}}\) and \({{b}}\).</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">Roderick is at a point \({\text{C}}(11,{\text{ }}9)\). During Ed’s walk from <span class="s1">\(P\)</span> to <span class="s1">\(Q\)</span> Roderick wishes to signal to Ed. He decides to signal when Ed is at the closest point to <span class="s1">\(C\)</span><span class="s1">.</span></p>
<p class="p2">Find the time when Roderick signals to Ed.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">\({{a}} = \left( {\begin{array}{*{20}{c}} { - 1} \\ 4 \end{array}} \right)\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1">\({{b}} = \frac{1}{3}\left( {\left( {\begin{array}{*{20}{c}} 4 \\ {16} \end{array}} \right) - \left( {\begin{array}{*{20}{c}} { - 1} \\ 4 \end{array}} \right)} \right) = \left( {\begin{array}{*{20}{c}} {\frac{5}{3}} \\ 4 \end{array}} \right)\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)A1</em></strong></span></p>
<p class="p1"><span class="s1"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>Roderick must signal in a direction vector perpendicular to Ed’s path. <strong><em>(M1)</em></strong></p>
<p>the equation of the signal is \({\mathbf{s}} = \left( {\begin{array}{*{20}{c}} {11} \\ 9 \end{array}} \right) + \lambda \left( {\begin{array}{*{20}{c}} { - 12} \\ 5 \end{array}} \right)\;\;\;\)(or equivalent) <strong><em>A1</em></strong></p>
<p>\(\left( {\begin{array}{*{20}{c}} { - 1} \\ 4 \end{array}} \right) + \frac{t}{3}\left( {\begin{array}{*{20}{c}} 5 \\ {12} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {11} \\ 9 \end{array}} \right) + \lambda \left( {\begin{array}{*{20}{c}} { - 12} \\ 5 \end{array}} \right)\) <strong><em>M1</em></strong></p>
<p>\(\frac{5}{3}t + 12\lambda = 12\) and \(4t - 5\lambda = 5\) <strong><em>M1</em></strong></p>
<p>\(t = 2.13\;\;\;\left( { = \frac{{360}}{{169}}} \right)\) <strong><em>A1</em></strong></p>
<p><strong>METHOD 2</strong></p>
<p>\(\left( {\begin{array}{*{20}{c}} 5 \\ {12} \end{array}} \right) \bullet \left( {\left( {\begin{array}{*{20}{c}} {11} \\ 9 \end{array}} \right) - \left( {\begin{array}{*{20}{c}} { - 1 + \frac{5}{3}t} \\ {4 + 4t} \end{array}} \right)} \right) = 0\;\;\;\)(or equivalent) <strong><em>M1A1A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award the <strong><em>M1 </em></strong>for an attempt at a scalar product equated to zero, <strong><em>A1 </em></strong>for the first factor and <strong><em>A1 </em></strong>for the complete second factor.</p>
<p> </p>
<p>attempting to solve for \(t\) <strong><em>(M1)</em></strong></p>
<p>\(t = 2.13\;\;\;\left( {\frac{{360}}{{169}}} \right)\) <strong><em>A1</em></strong></p>
<p><strong>METHOD 3</strong></p>
<p>\(x = \sqrt {{{\left( {12 - \frac{{5t}}{3}} \right)}^2} + {{(5 - 4t)}^2}} \;\;\;\)(or equivalent)\(\;\;\;\left( {{x^2} = {{\left( {12 - \frac{{5t}}{3}} \right)}^2} + {{(5 - 4t)}^2}} \right)\) <strong><em>M1A1A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>M1 </em></strong>for use of Pythagoras’ theorem, <strong><em>A1 </em></strong>for \({\left( {12 - \frac{{5t}}{3}} \right)^2}\) and <strong><em>A1</em></strong> for \({(5 - 4t)^2}\).</p>
<p> </p>
<p>attempting (graphically or analytically) to find \(t\) such that \(\frac{{{\text{d}}x}}{{{\text{d}}t}} = 0\left( {\frac{{{\text{d}}({x^2})}}{{{\text{d}}t}} = 0} \right)\) <strong><em>(M1)</em></strong></p>
<p>\(t = 2.13\;\;\;\left( { = \frac{{360}}{{169}}} \right)\) <strong><em>A1</em></strong></p>
<p><strong>METHOD 4</strong></p>
<p>\(\cos \theta = \frac{{\left( {\begin{array}{*{20}{c}} {12} \\ 5 \end{array}} \right) \bullet \left( {\begin{array}{*{20}{c}} 5 \\ {12} \end{array}} \right)}}{{\left| {\left( {\begin{array}{*{20}{c}} {12} \\ 5 \end{array}} \right)} \right|\left| {\left( {\begin{array}{*{20}{c}} 5 \\ {12} \end{array}} \right)} \right|}} = \frac{{120}}{{169}}\) <strong><em>M1A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>M1 </em></strong>for attempting to calculate the scalar product.</p>
<p> </p>
<p>\(\frac{{120}}{{13}} = \frac{t}{3}\left| {\left( {\begin{array}{*{20}{c}} 5 \\ {12} \end{array}} \right)} \right|\;\;\;\)(or equivalent) <strong><em>(A1)</em></strong></p>
<p>attempting to solve for \(t\) <strong><em>(M1)</em></strong></p>
<p>\(t = 2.13\;\;\;\left( { = \frac{{360}}{{169}}} \right)\) <strong><em>A1</em></strong></p>
<p><strong><em>[5 marks]</em></strong></p>
<p><strong><em>Total [8 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the values of <em>k </em>for which the following system of equations has no solutions and the value of <em>k </em>for the system to have an infinite number of solutions.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[x - 3y + z = 3\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[x + 5y - 2z = 1\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[16y - 6z = k\]</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Given that the system of equations can be solved, find the solutions in the form of a vector equation of a line, <strong><em>r </em></strong>= <strong><em>a </em></strong>+ <span style="font: 12.5px Helvetica;">λ</span><strong><em>b </em></strong>, where the components of <strong><em>b </em></strong>are integers.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The plane \( \div \) is parallel to both the line in part (b) and the line \(\frac{{x - 4}}{3} = \frac{{y - 6}}{{ - 2}} = \frac{{z - 2}}{0}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Given that </span><span style="font-family: 'times new roman', times; font-size: medium;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \div \)</span> contains the point (1, 2, 0) , show that the Cartesian equation of ÷ is 16<em>x </em>+ 24<em>y </em>− 11<em>z </em>= 64 .</span></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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">The <em>z-</em>axis meets the plane </span><span style="font-family: 'times new roman', times; font-size: medium;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \div \)</span> at the point P. Find the coordinates of P.</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: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the angle between the line \(\frac{{x - 2}}{3} = \frac{{y + 5}}{4} = \frac{z}{2}\) and the plane </span><span style="font-family: 'times new roman', times; font-size: medium;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \div \)</span> .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">in augmented matrix form \(\left| {\begin{array}{*{20}{c}}<br> 1&{ - 3}&1&3 \\ <br> 1&5&{ - 2}&1 \\ <br> 0&{16}&{ - 6}&k <br>\end{array}} \right|\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">attempt to find a line of zeros <strong><em>(M1)<br></em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\({r_2} - {r_1}\left| {\begin{array}{*{20}{c}}<br> 1&{ - 3}&1&3 \\ <br> 0&8&{ - 3}&{ - 2} \\ <br> 0&{16}&{ - 6}&k <br>\end{array}} \right|\) <em><strong>(A1)</strong></em><strong><em><br></em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\({r_3} - 2{r_2}\left| {\begin{array}{*{20}{c}}<br> 1&{ - 3}&1&3 \\ <br> 0&8&{ - 3}&{ - 2} \\ <br> 0&{0}&{0}&{k + 4} <br>\end{array}} \right|\) <strong> <em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">there is an infinite number of solutions when \(k = - 4\)</span><span style="font-family: 'times new roman', times; font-size: medium;"> </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>R1</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">there is no solution when</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(k \ne - 4,{\text{ }}(k \in \mathbb{R})\) <strong><em>R1</em></strong></span><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Approaches other than using the augmented matrix are acceptable. </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[5 marks]</span><br></em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">using \(\left| {\begin{array}{*{20}{c}}<br> 1&{ - 3}&1&3 \\ <br> 0&8&{ - 3}&{ - 2} \\ <br> 0&{0}&{ 0}&{k + 4} <br>\end{array}} \right|\) and letting \(\boldsymbol{z} = \lambda \) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(8y - 3\lambda = - 2\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow y = \frac{{3\lambda - 2}}{8}\) <strong> <em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(x - 3y + z = 3\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow x - \left( {\frac{{9\lambda - 6}}{8}} \right) + \lambda = 3\) <strong> <em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow 8x - 9\lambda + 6 + 8\lambda = 24\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow x = \frac{{18 + \lambda }}{8}\) <strong> <em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow \left( {\begin{array}{*{20}{c}}<br> x \\ <br> y \\ <br> z <br>\end{array}} \right) = \left( {\begin{array}{*{20}{c}}<br> {\frac{{18}}{8}} \\ <br> { - \frac{2}{8}} \\ <br> 0 <br>\end{array}} \right) + \lambda \left( {\begin{array}{*{20}{c}}<br> {\frac{1}{8}} \\ <br> {\frac{3}{8}} \\ <br> 1 <br>\end{array}} \right)\) <strong> <em>(M1)(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(r = \left( {\begin{array}{*{20}{c}}<br> {\frac{9}{4}} \\ <br> { - \frac{1}{4}} \\ <br> 0 <br>\end{array}} \right) + \lambda \left( {\begin{array}{*{20}{c}}<br> 1 \\ <br> 3 \\ <br> 8 <br>\end{array}} \right)\) <strong> <em>A1</em></strong></span><strong style="font-family: 'times new roman', times; font-size: medium;"> </strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Accept equivalent answers. </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[7 marks]</span><br></em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">recognition that \(\left( {\begin{array}{*{20}{c}}<br> 3 \\ <br> { - 2} \\ <br> 0 <br>\end{array}} \right)\) is parallel to the plane <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">direction normal of the plane is given by \(\left| {\begin{array}{*{20}{c}}<br> \boldsymbol{i}&\boldsymbol{j}&\boldsymbol{k} \\ <br> 1&3&8 \\ <br> 3&{ - 2}&0 <br>\end{array}} \right|\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">= 16<em><strong>i</strong></em> + 24<em><strong>j</strong></em> – 11<em><strong>k</strong></em> <strong> <em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Cartesian equation of the plane is given by 16<em>x</em> + 24<em>y</em> –11<em>z</em> = <em>d</em> and a point which fits this equation is (1, 2, 0) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow 16 + 48 = d\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>d</em> = 64 <strong> <em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">hence Cartesian equation of plane is 16<em style="font-style: italic;">x</em> + 24<em style="font-style: italic;">y</em> –11<em style="font-style: italic;">z</em> = 64 <strong><em>AG</em></strong></span><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Accept alternative methods using dot product.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]<br></em></strong></span></p>
<p> </p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">the plane crosses the <em>z</em>-axis when <em>x</em> = <em>y</em> = 0 <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">coordinates of P are \(\left( {0,\,0,\, - \frac{{64}}{{11}}} \right)\) <strong><em>A1</em></strong></span><strong style="font-family: 'times new roman', times; font-size: medium;"> </strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Award <strong><em>A1 </em></strong>for stating \(z = - \frac{{64}}{{11}}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong> </strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Accept. \(\left( {\begin{array}{*{20}{c}}<br> 0 \\ <br> 0 \\ <br> { - \frac{{64}}{{11}}} <br>\end{array}} \right)\)<br></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[2 marks]</span><br></em></strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">recognition that the angle between the line and the direction normal is given by:</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left( {\begin{array}{*{20}{c}}<br> 3 \\ <br> 4 \\ <br> 2 <br>\end{array}} \right)\left( {\begin{array}{*{20}{c}}<br> {16} \\ <br> {24} \\ <br> {-11} <br>\end{array}} \right) = \sqrt {29} \sqrt {953} \cos \theta \) where \(\theta \) is the angle between the line and the normal vector <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow 122 = \sqrt {29} \sqrt {953} \cos \theta \) <strong> <em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow \theta = 42.8^\circ {\text{ (0.747 radians)}}\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">hence the angle between the line and the plane is 90° – 42.8° = 47.2° (0.824 radians) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks] </em></strong></span><strong style="font-family: 'times new roman', times; font-size: medium;"> </strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Accept use of the formula <em><strong>a</strong></em>.<em><strong>b</strong></em> = \(\left| {} \right.\)<em><strong>a</strong></em>\(\left. {} \right|\)\(\left| {} \right.\)<em><strong>b</strong></em>\(\left| {\sin \theta } \right.\) .</span></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Many candidates were able to start this question, but only a few candidates gained full marks. Many candidates successfully used the augmented matrix in part (a) to find the correct answer. Part (b) was less successful with only a limited number of candidates using the calculator to its full effect here and with many candidates making arithmetic and algebraic errors. This was the hardest part of the question. Many candidates understood the vector techniques necessary to answer parts (c), (d) and (e) but a number made arithmetic and algebraic errors in the working.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Many candidates were able to start this question, but only a few candidates gained full marks. Many candidates successfully used the augmented matrix in part (a) to find the correct answer. Part (b) was less successful with only a limited number of candidates using the calculator to its full effect here and with many candidates making arithmetic and algebraic errors. This was the hardest part of the question. Many candidates understood the vector techniques necessary to answer parts (c), (d) and (e) but a number made arithmetic and algebraic errors in the working.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Many candidates were able to start this question, but only a few candidates gained full marks. Many candidates successfully used the augmented matrix in part (a) to find the correct answer. Part (b) was less successful with only a limited number of candidates using the calculator to its full effect here and with many candidates making arithmetic and algebraic errors. This was the hardest part of the question. Many candidates understood the vector techniques necessary to answer parts (c), (d) and (e) but a number made arithmetic and algebraic errors in the working.</span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Many candidates were able to start this question, but only a few candidates gained full marks. Many candidates successfully used the augmented matrix in part (a) to find the correct answer. Part (b) was less successful with only a limited number of candidates using the calculator to its full effect here and with many candidates making arithmetic and algebraic errors. This was the hardest part of the question. Many candidates understood the vector techniques necessary to answer parts (c), (d) and (e) but a number made arithmetic and algebraic errors in the working.</span></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Many candidates were able to start this question, but only a few candidates gained full marks. Many candidates successfully used the augmented matrix in part (a) to find the correct answer. Part (b) was less successful with only a limited number of candidates using the calculator to its full effect here and with many candidates making arithmetic and algebraic errors. This was the hardest part of the question. Many candidates understood the vector techniques necessary to answer parts (c), (d) and (e) but a number made arithmetic and algebraic errors in the working.</span></p>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">(a) Find the coordinates of the point \(A\) on \({l_1}\) and the point \(B\) on \({l_2}\) such that \(\overrightarrow {{\text{AB}}} \) </span><span style="font-family: times new roman,times; font-size: medium;">is </span><span style="font-family: times new roman,times; font-size: medium;">perpendicular to both </span><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: times new roman,times; font-size: medium;">\({l_1}\)</span> and </span><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: times new roman,times; font-size: medium;">\({l_2}\)</span> .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(b) Find \(\left| {{\text{AB}}} \right|\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(c) Find the Cartesian equation of the plane \(\prod \) which contains </span><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: times new roman,times; font-size: medium;">\({l_1}\)</span> and does not </span><span style="font-family: times new roman,times; font-size: medium;">intersect </span><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: times new roman,times; font-size: medium;">\({l_2}\)</span> .</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">(a) on </span><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: times new roman,times; font-size: medium;">\({l_1}\) </span></span> A(\( - 3 + 3\lambda \), \( - 4 + 2\lambda \), \(6 - 2\lambda \)) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">on \({l_2}\) \({l_2}:r = \left( {\begin{array}{*{20}{c}}<br> 4 \\ <br> { - 7} \\ <br> 3 <br>\end{array}} \right) + \mu \left( {\begin{array}{*{20}{c}}<br> { - 3} \\ <br> 4 \\ <br> { - 1} <br>\end{array}} \right)\) <em><strong>(M</strong></em></span><em><strong><span style="font-family: times new roman,times; font-size: medium;">1)</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( \Rightarrow \) B(\(4 - 3\mu \), \( - 7 + 4\mu \), \( - 3 - \mu \)) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\overrightarrow {{\text{BA}}} = {\boldsymbol{a}} - {\boldsymbol{b}} = \left( {\begin{array}{*{20}{c}}<br> {3\lambda + 3\mu - 7} \\ <br> {2\lambda - 4\mu + 3} \\ <br> { - 2\lambda + \mu + 9} <br>\end{array}} \right)\) <em><strong>(M1)A1</strong></em></span></p>
<p><strong><span style="font-family: times new roman,times; font-size: medium;">EITHER</span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{BA}} \bot {l_1} \Rightarrow {\text{BA}} \cdot \left( {\begin{array}{*{20}{c}}<br> 3 \\ <br> 2 \\ <br> { - 2} <br>\end{array}} \right) = 0 \Rightarrow 3\left( {3\lambda + 3\mu - 7} \right) + 2\left( {2\lambda - 4\mu + 3} \right) - 2\left( { - 2\lambda + \mu + 9} \right) = 0\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">M1</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( \Rightarrow 17\lambda - \mu = 33\) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{BA}} \bot {l_2} \Rightarrow {\text{BA}} \cdot \left( {\begin{array}{*{20}{c}}<br> { - 3} \\ <br> 4 \\ <br> { - 1} <br>\end{array}} \right) = 0 \Rightarrow - 3\left( {3\lambda + 3\mu - 7} \right) + 4\left( {2\lambda - 4\mu + 3} \right) - \left( { - 2\lambda + \mu + 9} \right) = 0\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;"> M1</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( \Rightarrow \lambda - 26\mu = - 24\) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">solving both equations above simultaneously gives</span><br><span style="font-family: times new roman,times; font-size: medium;">\(\lambda = 2\); \(\mu = 1 \Rightarrow \) A(3, 0, 2), B(1, –3, –4) <em><strong>A1A1A1A1</strong></em></span></p>
<p><strong><span style="font-family: times new roman,times; font-size: medium;">OR</span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\left| {\begin{array}{*{20}{c}}<br> {\boldsymbol{i}}&{\boldsymbol{j}}&{\boldsymbol{k}} \\ <br> 3&2&{ - 2} \\ <br> { - 3}&4&{ - 1} <br>\end{array}} \right| = 6{\boldsymbol{i}} + 9{\boldsymbol{j}} + 18{\boldsymbol{k}}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">M1A1</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">so \(\overrightarrow {{\text{AB}}} = p\left( {\begin{array}{*{20}{c}}<br> 2 \\ <br> 3 \\ <br> 6 <br>\end{array}} \right) = \left( {\begin{array}{*{20}{c}}<br> {3\lambda + 3\mu - 7} \\ <br> {2\lambda - 4\mu + 3} \\ <br> { - 2\lambda + \mu + 9} <br>\end{array}} \right)\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">M1A1</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({3\lambda + 3\mu - 2p = 7}\)</span><br><span style="font-family: times new roman,times; font-size: medium;">\({2\lambda - 4\mu - 3p = - 3}\)</span><br><span style="font-family: times new roman,times; font-size: medium;">\({ - 2\lambda + \mu - 6p = - 9}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\lambda = 2\), \(\mu = 1\), \(p = 1\) <em><strong>A1A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">A(\( - 3 + 6\), \( - 4 + 4\), \(6 - 4\)) \(=\) (\(3\), \(0\), \(2\)) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">B(\(4 - 3\), \( - 7 + 4\), \( - 3 - 1\)) \(=\) (\(1\), \( - 3\), \( - 4\)) <em><strong>A1</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[13 marks]</span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> </span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(b) </span><span style="font-family: times new roman,times; font-size: medium;">\({\text{AB}} = \left( {\begin{array}{*{20}{c}}<br> 1 \\ <br> { - 3} \\ <br> { - 4} <br>\end{array}} \right) - \left( {\begin{array}{*{20}{c}}<br> 3 \\ <br> 0 \\ <br> 2 <br>\end{array}} \right) = \left( {\begin{array}{*{20}{c}}<br> { - 2} \\ <br> { - 3} \\ <br> { - 6} <br>\end{array}} \right)\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">(A1)</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\left| {{\text{AB}}} \right| = \sqrt {{{\left( { - 2} \right)}^2} + {{\left( { - 3} \right)}^2} + {{\left( { - 6} \right)}^2}} = \sqrt {49} = 7\) <em><strong>M1A1</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> </span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(c) from (b) \(2{\boldsymbol{i}} + 3{\boldsymbol{j}} + 6{\boldsymbol{k}}\) is normal to both lines</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({l_1}\) goes through (–</span><span style="font-family: times new roman,times; font-size: medium;">3, –</span><span style="font-family: times new roman,times; font-size: medium;">4, 6) \( \Rightarrow \left( {\begin{array}{*{20}{c}}<br> { - 3} \\ <br> { - 4} \\ <br> 6 <br>\end{array}} \right) \cdot \left( {\begin{array}{*{20}{c}}<br> 2 \\ <br> 3 \\ <br> 6 <br>\end{array}} \right) = 18\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">M1A1</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">hence, the Cartesian equation of the plane through \({l_1}\) , but not \({l_2}\) , </span><span style="font-family: times new roman,times; font-size: medium;">is \(2x + 3y + 6z = 18\) <em><strong>A1</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">Total [19 marks]</span></strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p><span style="font-size: medium; font-family: times new roman,times;">There were a lot of arithmetic errors in the treatment of this question, even though it was </span><span style="font-size: medium; font-family: times new roman,times;">apparent that many students did understand the methods involved. In (a) many students </span><span style="font-size: medium; font-family: times new roman,times;">failed to realise that \(\overrightarrow {{\text{AB}}} \) should be a multiple of the cross product of the two direction </span><span style="font-size: medium; font-family: times new roman,times;">vectors, rather than the cross product itself, and many students failed to give the final answer </span><span style="font-size: medium; font-family: times new roman,times;">as coordinates.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The angle between the vector <strong><em>a</em></strong> = <strong><em>i</em></strong> − 2<strong><em>j</em></strong> + 3<strong><em>k</em></strong> and the vector <strong><em>b</em></strong> = 3<strong><em>i</em></strong> − 2<strong><em>j</em></strong> + <em>m</em><strong><em>k</em></strong> is 30° .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the values of <em>m</em>.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\boldsymbol{a}} \cdot {\boldsymbol{b}} = \left| {\boldsymbol{a}} \right|\left| {\boldsymbol{b}} \right|\cos \theta \) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\boldsymbol{a}} \cdot {\boldsymbol{b}} = \left( {\begin{array}{*{20}{c}}<br> 1 \\ <br> { - 2} \\ <br> 3 <br>\end{array}} \right) \cdot \left( {\begin{array}{*{20}{c}}<br> 3 \\ <br> { - 2} \\ <br> m <br>\end{array}} \right) = 7 + 3m\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left| {\boldsymbol{a}} \right| = \sqrt {14} \) \(\left| {\boldsymbol{b}} \right| = \sqrt {13 + {m^2}} \) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left| {\boldsymbol{a}} \right|\left| {\boldsymbol{b}} \right|\cos \theta = \sqrt {14} \sqrt {13 + {m^2}} \cos 30^\circ \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\(7 + 3m = \sqrt {14} \sqrt {13 + {m^2}} \cos 30^\circ \) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>m</em> = 2.27, <em>m</em> = 25.7 <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Many candidates gained the first 4 marks by obtaining the equation, in unsimplified form, satisfied by <em>m</em> but then made mistakes in simplifying and solving this equation.</span></p>
</div>
<br><hr><br><div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the angle between the lines \(\frac{{x - 1}}{2} = 1 - y = 2z\)</span><span style="font-family: times new roman,times; font-size: medium;"> and \(x = y = 3z\) .</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">consider a vector parallel to each line,</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\boldsymbol{u}} = \left( {\begin{array}{*{20}{c}}<br> 4 \\ <br> { - 2} \\ <br> 1 <br>\end{array}} \right)\)</span><span style="font-family: times new roman,times; font-size: medium;"> and \({\boldsymbol{v}} = \left( {\begin{array}{*{20}{c}}<br> 3 \\ <br> 3 \\ <br> 1 <br>\end{array}} \right)\) <em><strong>A1A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">let \(\theta \) be the angle between the lines</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\cos \theta = \frac{{\left| {{\boldsymbol{u \times v}}} \right|}}{{\left| {\boldsymbol{u}} \right|\left| {\boldsymbol{v}} \right|}} = \frac{{\left| {12 - 6 + 1} \right|}}{{\sqrt {21} \sqrt {19} }}\) <em><strong>M1A1</strong></em></span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;">\( = \frac{7}{{\sqrt {21} \sqrt {19} }} = 0.350...\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">(A1)</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">so \(\theta = 69.5\)</span><span style="font-family: times new roman,times; font-size: medium;"> \(\left( {{\text{or }}1.21{\text{ rad or }}\arccos \left( {\frac{7}{{\sqrt {21} \sqrt {19} }}} \right)} \right)\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1 N4</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note:</strong> Allow FT from incorrect reasonable vectors.</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[6 marks]</span></strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">Most students knew how to find the angle between two vectors, although many could not find the correct two direction vectors.</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">A curve is defined \({x^2} - 5xy + {y^2} = 7\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{5y - 2x}}{{2y - 5x}}\).</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 the equation of the normal to the curve at the point \((6,{\text{ }}1)\).</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">Find the distance between the two points on the curve where each tangent is parallel to the line \(y = x\).</p>
<div class="marks">[8]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">attempt at implicit differentiation <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p2">\(2x - 5x\frac{{{\text{d}}y}}{{{\text{d}}x}} - 5y + 2y\frac{{{\text{d}}y}}{{{\text{d}}x}} = 0\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1A1</em></strong></span></p>
<p class="p3"> </p>
<p class="p2"><strong>Note:</strong> <span class="Apple-converted-space"> </span><strong><em>A1</em></strong> for differentiation of \({x^2} - 5xy\), <strong><em>A1</em></strong> for differentiation of \({y^2}\) and \(7\).</p>
<p class="p3"> </p>
<p class="p2">\(2x - 5y + \frac{{{\text{d}}y}}{{{\text{d}}x}}(2y - 5x) = 0\)</p>
<p class="p2">\(\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{5y - 2x}}{{2y - 5x}}\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>AG</em></strong></span></p>
<p class="p2"><span class="s1"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{5 \times 1 - 2 \times 6}}{{2 \times 1 - 5 \times 6}} = \frac{1}{4}\) <strong><em>A1</em></strong></p>
<p>gradient of normal \( = - 4\) <strong><em>A1</em></strong></p>
<p>equation of normal \(y = - 4x + c\) <strong><em>M1</em></strong></p>
<p>substitution of \((6,{\text{ }}1)\)</p>
<p>\(y = - 4x + 25\) <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Accept \(y - 1 = - 4(x - 6)\)</p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>setting \(\frac{{5y - 2x}}{{2y - 5x}} = 1\) <strong><em>M1</em></strong></p>
<p>\(y = - x\) <strong><em>A1</em></strong></p>
<p>substituting into original equation <strong><em>M1</em></strong></p>
<p>\({x^2} + 5{x^2} + {x^2} = 7\) <strong><em>(A1)</em></strong></p>
<p>\(7{x^2} = 7\)</p>
<p>\(x = \pm 1\) <strong><em>A1</em></strong></p>
<p>points \((1,{\text{ }} - 1)\) and \(( - 1,{\text{ }}1)\) <strong><em>(A1)</em></strong></p>
<p>distance \( = \sqrt 8 \;\;\;\left( { = 2\sqrt 2 } \right)\) <strong><em>(M1)A1</em></strong></p>
<p><strong><em>[8 marks]</em></strong></p>
<p><strong><em>Total [15 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The diagram shows a cube OABCDEFG.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="font: normal normal normal 29px/normal Helvetica; text-align: center; margin: 0px;"><img 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" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Let O be the origin, (OA) the <em>x</em>-axis, (OC) the <em>y</em>-axis and (OD) the <em>z</em>-axis.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Let M, N and P be the midpoints of [FG], [DG] and [CG], respectively.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The coordinates of F are (2, 2, 2).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Find the position vectors \(\overrightarrow {{\text{OM}}} \), \(\overrightarrow {{\text{ON}}} \) and \(\overrightarrow {{\text{OP}}} \) in component form.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Find \(\overrightarrow {{\text{MP}}} \times \overrightarrow {{\text{MN}}} \).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) <strong>Hence</strong>,</span></p>
<p style="margin: 0px 0px 0px 30px; font: 29px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (i) calculate the area of the triangle MNP;</span></p>
<p style="margin: 0px 0px 0px 30px; font: 29px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) show that the line (AG) is perpendicular to the plane MNP;</span></p>
<p style="margin: 0px 0px 0px 30px; font: 29px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (iii) find the equation of the plane MNP.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(d) Determine the coordinates of the point where the line (AG) meets the plane MNP.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) \(\overrightarrow {{\text{OM}}} = \left( {\begin{array}{*{20}{c}}<br> 1 \\ <br> 2 \\ <br> 2 <br>\end{array}} \right)\), \(\overrightarrow {{\text{ON}}} = \left( {\begin{array}{*{20}{c}}<br> 0 \\ <br> 1 \\ <br> 2 <br>\end{array}} \right)\) and \(\overrightarrow {{\text{OP}}} = \left( {\begin{array}{*{20}{c}}<br> 0 \\ <br> 2 \\ <br> 1 <br>\end{array}} \right)\) <strong><em>A1A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) \(\overrightarrow {{\text{MP}}} = \left( {\begin{array}{*{20}{c}}<br> { - 1} \\ <br> 0 \\ <br> { - 1} <br>\end{array}} \right)\) and \(\overrightarrow {{\text{MN}}} = \left( {\begin{array}{*{20}{c}}<br> { - 1} \\ <br> { - 1} \\ <br> 0 <br>\end{array}} \right)\) <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\overrightarrow {{\text{MP}}} \times \overrightarrow {{\text{MN}}} = \left( {\begin{array}{*{20}{c}}<br> i&j&k \\ <br> { - 1}&0&{ - 1} \\ <br> { - 1}&{ - 1}&0 <br>\end{array}} \right) = \left( {\begin{array}{*{20}{c}}<br> { - 1} \\ <br> 1 \\ <br> 1 <br>\end{array}} \right)\) <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) (i) area of MNP \( = \frac{1}{2}\left| {\overrightarrow {{\text{MP}}} \times \overrightarrow {{\text{MN}}} } \right|\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{1}{2}\left| {\left( {\begin{array}{*{20}{c}}<br> { - 1} \\ <br> 1 \\ <br> 1 <br>\end{array}} \right)} \right|\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{\sqrt 3 }}{2}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \(\overrightarrow {{\text{OA}}} = \left( {\begin{array}{*{20}{c}}<br> 2 \\ <br> 0 \\ <br> 0 <br>\end{array}} \right)\), \(\overrightarrow {{\text{OG}}} = \left( {\begin{array}{*{20}{c}}<br> 0 \\ <br> 2 \\ <br> 2 <br>\end{array}} \right)\)<br></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\overrightarrow {{\text{AG}}} = \left( {\begin{array}{*{20}{c}}<br> { - 2} \\ <br> 2 \\ <br> 2 <br>\end{array}} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">since \(\overrightarrow {{\text{AG}}} = 2(\overrightarrow {{\text{MP}}} \times \overrightarrow {{\text{MN}}} )\) AG is perpendicular to MNP <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) \(r \cdot \left( {\begin{array}{*{20}{c}}<br> { - 1} \\ <br> 1 \\ <br> 1 <br>\end{array}} \right) = \left( {\begin{array}{*{20}{c}}<br> 1 \\ <br> 2 \\ <br> 2 <br>\end{array}} \right) \cdot \left( {\begin{array}{*{20}{c}}<br> { - 1} \\ <br> 1 \\ <br> 1 <br>\end{array}} \right)\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(r \cdot \left( {\begin{array}{*{20}{c}}<br> { - 1} \\ <br> 1 \\ <br> 1 <br>\end{array}} \right) = 3\) (accept \( - x + y + z = 3\)) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[7 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(d) \(r = \left( {\begin{array}{*{20}{c}}<br> 2 \\ <br> 0 \\ <br> 0 <br>\end{array}} \right) + \lambda \left( {\begin{array}{*{20}{c}}<br> { - 2} \\ <br> 2 \\ <br> 2 <br>\end{array}} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left( {\begin{array}{*{20}{c}}<br> {2 - 2\lambda } \\ <br> {2\lambda } \\ <br> {2\lambda } <br>\end{array}} \right) \cdot \left( {\begin{array}{*{20}{c}}<br> { - 1} \\ <br> 1 \\ <br> 1 <br>\end{array}} \right) = 3\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( - 2 + 2\lambda + 2\lambda + 2\lambda = 3\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\lambda = \frac{5}{6}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(r = \left( {\begin{array}{*{20}{c}}<br> 2 \\ <br> 0 \\ <br> 0 <br>\end{array}} \right) + \frac{5}{6}\left( {\begin{array}{*{20}{c}}<br> { - 2} \\ <br> 2 \\ <br> 2 <br>\end{array}} \right)\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">coordinates of point \(\left( {\frac{1}{3},\frac{5}{3},\frac{5}{3}} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [20 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">This was the most successfully answered question in part B, with many candidates achieving full marks. There were a few candidates who misread the question and treated the cube as a unit cube. The most common errors were either algebraic or arithmetic mistakes. A variety of notation forms were seen but in general were used consistently. In a few cases, candidates failed to show all the work or set it properly.</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">OABCDE is a regular hexagon and <strong><em>a</em></strong> , <strong><em>b</em></strong> denote respectively the position vectors of A, B with respect to O.</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: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that OC = 2AB .</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: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the position vectors of C, D and E in terms of <strong><em>a</em></strong> and <strong><em>b</em></strong> .</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0px; font: 25px Helvetica; text-align: justify;"><img src="data:image/png;base64,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" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{OC}} = {\text{AB}} + {\text{OA}}\cos 60 + {\text{BC}}\cos 60\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = {\text{AB}} + {\text{AB}} \times \frac{1}{2} + {\text{AB}} \times \frac{1}{2}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 2{\text{AB}}\) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\overrightarrow {{\text{OC}}} = 2\overrightarrow {{\text{AB}}} = \)2(<strong><em>b</em></strong> – <strong><em>a</em></strong>) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\overrightarrow {{\text{OD}}} = \overrightarrow {{\text{OC}}} + \overrightarrow {{\text{CD}}} \) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \overrightarrow {{\text{OC}}} + \overrightarrow {{\text{AO}}} \) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">= 2<strong><em>b</em></strong> – 2<strong><em>a</em></strong> – <strong><em>a</em></strong> = 2<strong><em>b</em></strong> – 3<strong><em>a</em></strong> <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\overrightarrow {{\text{OE}}} = \overrightarrow {{\text{BC}}} \) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">= 2<strong><em>b</em></strong> – 2<strong><em>a</em></strong> – <strong><em>b</em></strong> = <strong><em>b</em></strong> – 2<strong><em>a</em></strong> <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[7 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Two planes \({\Pi _1}\) and \({\Pi _2}\) have equations \(2x + y + z = 1\) and \(3x + y - z = 2\) respectively.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the vector equation of <em>L</em>, the line of intersection of \({\Pi _1}\) and \({\Pi _2}\).</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that the plane \({\Pi _3}\) which is perpendicular to \({\Pi _1}\) and contains <em>L</em>, has equation \(x - 2z = 1\).</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 34.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The point P has coordinates (−2, 4, 1) , the point Q lies on \({\Pi _3}\) and PQ is perpendicular to \({\Pi _2}\). Find the coordinates of Q.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) <strong>METHOD 1</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">solving simultaneously (gdc) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(x = 1 + 2z;{\text{ }}y = - 1 - 5z\) <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(L:\boldsymbol{r} = \left( {\begin{array}{*{20}{c}}<br> 1 \\ <br> { - 1} \\ <br> 0 <br>\end{array}} \right) + \lambda \left( {\begin{array}{*{20}{c}}<br> 2 \\ <br> { - 5} \\ <br> 1 <br>\end{array}} \right)\) <strong><em>A1A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> \({1^{{\text{st}}}}\) <strong><em>A1</em></strong> is for <strong><em>r</em></strong> =.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">METHOD 2</strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">direction of line \( = \left| {\begin{array}{*{20}{c}}<br> \boldsymbol{i}&\boldsymbol{j}&\boldsymbol{k} \\ <br> 3&1&{ - 1} \\ <br> 2&1&1 <br>\end{array}} \right|\) (last two rows swapped) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">= 2<strong><em>i</em></strong> − 5<strong><em>j</em></strong> + <strong><em>k</em></strong> <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">putting <em>z</em> = 0, a point on the line satisfies \(2x + y = 1,{\text{ }}3x + y = 2\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>i.e.</em> (1, −1, 0) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">the equation of the line is</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left( {\begin{array}{*{20}{c}}<br> x \\ <br> y \\ <br> z <br>\end{array}} \right) = \left( {\begin{array}{*{20}{c}}<br> 1 \\ <br> { - 1} \\ <br> 0 <br>\end{array}} \right) + \lambda \left( {\begin{array}{*{20}{c}}<br> 2 \\ <br> { - 5} \\ <br> 1 <br>\end{array}} \right)\) <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>A0A1</em></strong> if \(\left( {\begin{array}{*{20}{c}}<br> x \\ <br> y \\ <br> z <br>\end{array}} \right)\) is missing.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left( {\begin{array}{*{20}{c}}<br> 2 \\ <br> 1 \\ <br> 1 <br>\end{array}} \right) \times \left( {\begin{array}{*{20}{c}}<br> 2 \\ <br> { - 5} \\ <br> 1 <br>\end{array}} \right)\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">= 6<strong><em>i</em></strong> − 12<strong><em>k</em></strong> <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">hence, <strong><em>n</em></strong> = <strong><em>i</em></strong> − 2<strong><em>k</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\boldsymbol{n}} \cdot {\boldsymbol{a}} = \left( {\begin{array}{*{20}{c}}<br> 1 \\ <br> 0 \\ <br> { - 2} <br>\end{array}} \right) \cdot \left( {\begin{array}{*{20}{c}}<br> 1 \\ <br> { - 1} \\ <br> 0 <br>\end{array}} \right) = 1\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">therefore <strong><em>r </em></strong>\( \cdot \) <strong><em>n</em></strong> = <strong><em>a </em></strong>\( \cdot \) <strong><em>n</em></strong> \( \Rightarrow x - 2z = 1\) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 1</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">P = (−2, 4, 1), Q = \((x,{\text{ }}y,{\text{ }}z)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\overrightarrow {{\text{PQ}}} = \left( {\begin{array}{*{20}{c}}<br> {x + 2} \\ <br> {y - 4} \\ <br> {z - 1} <br>\end{array}} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\overrightarrow {{\text{PQ}}} \) is perpendicular to \(3x + y - z = 2\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow \overrightarrow {{\text{PQ}}} \) is parallel to 3<strong><em>i</em></strong> + <strong><em>j</em></strong> − <strong><em>k</em></strong> <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow x + 2 = 3t;{\text{ }}y - 4 = t;{\text{ }}z - 1 = - t\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(1 - z = t \Rightarrow x + 2 = 3 - 3z \Rightarrow x + 3z = 1\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">solving simultaneously \(x + 3z = 1;{\text{ }}x - 2z = 1\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(5z = 0 \Rightarrow z = 0;{\text{ }}x = 1,{\text{ }}y = 5\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">hence, Q = (1, 5, 0)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong> </strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 2</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Line passing through PQ has equation</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\mathbf{r}} = \begin{array}{*{20}{c}}<br> { - 2} \\ <br> 4 \\ <br> 1 <br>\end{array} + t\begin{array}{*{20}{c}}<br> 3 \\ <br> 1 \\ <br> { - 1} <br>\end{array}\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Meets \({\pi _3}\) when:</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( - 2 + 3t - 2(1 - t) = 1\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>t</em> = 1 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Q has coordinates (1, 5, 0) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Candidates generally attempted this question but with varying degrees of success. Although (a) was answered best of all the parts, quite a few did not use correct notation to designate the vector equation of a line, i.e., <strong><em>r</em></strong> =, or its equivalent. In (b) some candidates incorrectly assumed the result and worked the question from there. In (c) some candidates did not understand the necessary relationships to make a meaningful attempt.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Candidates generally attempted this question but with varying degrees of success. Although (a) was answered best of all the parts, quite a few did not use correct notation to designate the vector equation of a line, i.e., <strong><em>r</em></strong> =, or its equivalent. In (b) some candidates incorrectly assumed the result and worked the question from there. In (c) some candidates did not understand the necessary relationships to make a meaningful attempt.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Candidates generally attempted this question but with varying degrees of success. Although (a) was answered best of all the parts, quite a few did not use correct notation to designate the vector equation of a line, i.e., <strong><em>r</em></strong> =, or its equivalent. In (b) some candidates incorrectly assumed the result and worked the question from there. In (c) some candidates did not understand the necessary relationships to make a meaningful attempt.</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p class="p1">Consider the two planes</p>
<p class="p1"> \({\pi _1}:4x + 2y - z = 8\)</p>
<p class="p1"> \({\pi _2}:x + 3y + 3z = 3\).</p>
<p class="p1">Find the angle between \({\pi _1}\) and \({\pi _2}\), giving your answer correct to the nearest degree.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p class="p1">\({{{n}}_1} = \left( {\begin{array}{*{20}{c}} 4 \\ 2 \\ { - 1} \end{array}} \right)\;\;\;{\text{and}}\;\;\;{{{n}}_2} = \left( {\begin{array}{*{20}{c}} 1 \\ 3 \\ 3 \end{array}} \right)\) <strong><em>(A1)(A1)</em></strong></p>
<p class="p1">use of \(\cos \theta = \frac{{{{{n}}_1} \bullet {{{n}}_2}}}{{\left| {{{{n}}_1}} \right|\left| {{{{n}}_2}} \right|}}\) <strong><em>(M1)</em></strong></p>
<p class="p1">\(\cos \theta = \frac{7}{{\sqrt {21} \sqrt {19} }}\;\;\;\left( { = \frac{7}{{\sqrt {399} }}} \right)\) (<strong><em>A1)(A1)</em></strong></p>
<p class="p1"> </p>
<p class="p1"><strong>Note: </strong>Award <strong><em>A1</em></strong> for a correct numerator and <strong><em>A1</em></strong> for a correct denominator.</p>
<p class="p1"> </p>
<p class="p1">\(\theta = 69^\circ \) <strong><em>A1</em></strong></p>
<p class="p1"> </p>
<p class="p1"><strong>Note: </strong>Award <strong><em>A1</em></strong> for 111°.</p>
<p class="p1"> </p>
<p class="p1"><strong><em>[6 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p class="p1">Reasonably well answered. A large number of candidates did not express their final answer correct to the nearest degree.</p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A ray of light coming from the point (−1, 3, 2) is travelling in the direction of vector \(\left( {\begin{array}{*{20}{c}}<br> 4 \\ <br> 1 \\ <br> { - 2} <br>\end{array}} \right)\) and meets the plane \(\pi :x + 3y + 2z - 24 = 0\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the angle that the ray of light makes with the plane.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The normal vector to the plane is </span><span style="font-family: 'times new roman', times; font-size: medium;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left( {\begin{array}{*{20}{c}}<br> 1 \\ <br> 3 \\ <br> 2 <br>\end{array}} \right)\)</span> . <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>EITHER</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\theta \) is the angle between the line and the normal to the plane.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\cos \theta = \frac{{\left( {\begin{array}{*{20}{c}}<br> 4 \\ <br> 1 \\ <br> { - 2} <br>\end{array}} \right) \cdot \left( {\begin{array}{*{20}{c}}<br> 1 \\ <br> 3 \\ <br> 2 <br>\end{array}} \right)}}{{\sqrt {14} \sqrt {21} }} = \frac{3}{{\sqrt {14} \sqrt {21} }} = \left( {\frac{3}{{7\sqrt 6 }}} \right)\) <strong><em>(M1)A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow \theta = 79.9^\circ {\text{ }}( = 1.394…)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The required angle is 10.1° (= 0.176) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\phi \) is the angle between the line and the plane.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\sin \phi = \frac{{\left( {\begin{array}{*{20}{c}}<br> 4 \\ <br> 1 \\ <br> { - 2} <br>\end{array}} \right) \cdot \left( {\begin{array}{*{20}{c}}<br> 1 \\ <br> 3 \\ <br> 2 <br>\end{array}} \right)}}{{\sqrt {14} \sqrt {21} }} = \frac{3}{{\sqrt {14} \sqrt {21} }}\) <strong><em>(M1)A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\phi \) = 10.1° (= 0.176) <strong><em>A2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">On the whole this question was well answered. Some candidates failed to find the complementary angle when using the formula with cosine.</span></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 vectors <strong><em>a</em></strong> and <strong><em>b</em></strong> are such that <strong><em>a</em></strong> \( = (3\cos \theta + 6)\)<strong><em>i</em></strong> \( + 7\) <strong><em>j</em></strong> and <strong><em>b</em></strong> \( = (\cos \theta - 2)\)<strong><em>i</em></strong> \( + (1 + \sin \theta )\)<strong><em>j</em></strong>.</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 <strong><em>a</em></strong> and <strong><em>b</em></strong> are perpendicular,</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;">show that \(3{\sin ^2}\theta - 7\sin \theta + 2 = 0\);</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">find the smallest possible positive value of \(\theta \).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">attempting to form \((3\cos \theta + 6)(\cos \theta - 2) + 7(1 + \sin \theta ) = 0\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(3{\cos ^2}\theta - 12 + 7\sin \theta + 7 = 0\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(3\left( {1 - {{\sin }^2}\theta } \right) + 7\sin \theta - 5 = 0\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(3{\sin ^2}\theta - 7\sin \theta + 2 = 0\) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">attempting to solve algebraically (including substitution) or graphically for \(\sin \theta \) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(\sin \theta = \frac{1}{3}\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(\theta = 0.340{\text{ }}( = 19.5^\circ )\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Part (a) was very well done. Most candidates were able to use the scalar product and \({\cos ^2}\theta = 1 - {\sin ^2}\theta \) to show the required result.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Part (b) was reasonably well done. A few candidates confused ‘smallest possible positive value’ with a minimum function value. Some candidates gave \(\theta = 0.34\) as their final answer.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">Given that \({\boldsymbol{a}} = 2\sin \theta {\boldsymbol{i}} + \left( {1 - \sin \theta } \right){\boldsymbol{j}}\) , find the value of the acute angle \(\theta \) , so that \(\boldsymbol{a}\) is perpendicular to the line \(x + y = 1\).</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">direction vector for line \( = \left( {\begin{array}{*{20}{c}}<br> 1 \\ <br> { - 1} <br>\end{array}} \right)\) </span><span style="font-family: times new roman,times; font-size: medium;">or any multiple <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\left( {\begin{array}{*{20}{c}}<br> {2\sin \theta } \\ <br> {1 - \sin \theta } <br>\end{array}} \right) \cdot \left( {\begin{array}{*{20}{c}}<br> 1 \\ <br> { - 1} <br>\end{array}} \right) = 0\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">M1</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(2\sin \theta - 1 + \sin \theta = 0\) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note:</strong> Allow <em><strong>FT</strong></em> on candidate’s direction vector just for line above only.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(3\sin \theta = 1\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\sin \theta = \frac{1}{3}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\theta = 0.340\) or \(19.5\) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note:</strong> A coordinate geometry method using perpendicular gradients is acceptable.</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[5 marks]</span></strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">A variety of approaches were seen, either using a scalar product of vectors, or based on the </span><span style="font-family: times new roman,times; font-size: medium;">rule for perpendicular gradients of lines. The main problem encountered in the first approach </span><span style="font-family: times new roman,times; font-size: medium;">was in the choice of the correct vector direction for the line.</span></p>
</div>
<br><hr><br><div class="question">
<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 line \({L_1}\) has equation <strong><em>r</em></strong> = \(\left( \begin{array}{c} - 5\\ - 3\\2\end{array} \right) + \lambda \left( \begin{array}{c} - 1\\2\\2\end{array} \right)\).</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 line \({L_2}\) passing through the origin intersects \({L_1}\) and is perpendicular to \({L_1}\).</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) Find a vector equation of \({L_2}\).</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;">(b) Determine the shortest distance from the origin to \({L_1}\).</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(a) <strong>METHOD 1</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">for P on \({L_1},{\text{ }}\overrightarrow {{\text{OP}}} = \) \(\left( \begin{array}{c} - 5 - \lambda \\ - 3 + 2\lambda \\2 + 2\lambda \end{array} \right)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">require \(\left( \begin{array}{c} - 5 - \lambda \\ - 3 + 2\lambda \\2 + 2\lambda \end{array} \right) \bullet \left( \begin{array}{c} - 1\\2\\2\end{array} \right) = 0\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(5 + \lambda - 6 + 4\lambda + 4 + 4\lambda = 0\) (or equivalent) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(\lambda = - \frac{1}{3}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(\therefore \overrightarrow {{\rm{OP}}} = \left( \begin{array}{c} - \frac{{14}}{3}\\ - \frac{{11}}{3}\\\frac{4}{3}\end{array} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\({L_2}:\boldsymbol{r} = \mu \left( \begin{array}{c} - 14\\ - 11\\4\end{array} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman'; min-height: 23.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Do not award the final <strong><em>A1 </em></strong>if \(\boldsymbol{r} =\) is not seen.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman'; min-height: 23.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 2</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Calculating either \(\left| {\overrightarrow {{\text{OP}}} } \right|\) or \({\left| {\overrightarrow {{\text{OP}}} } \right|^2}\) <em>eg</em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left| {\overrightarrow {{\text{OP}}} } \right| = \sqrt {{{( - 5 - \lambda )}^2} + {{( - 3 + 2\lambda )}^2} + {{(2 + 2\lambda )}^2}} \) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \sqrt {9{\lambda ^2} + 6\lambda + 38} \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Solving either \(\frac{{\text{d}}}{{{\text{d}}\lambda }}\left( {\left| {\overrightarrow {{\text{OP}}} } \right|} \right) = 0\) or \(\frac{{\text{d}}}{{{\text{d}}\lambda }}\left( {{{\left| {\overrightarrow {{\text{OP}}} } \right|}^2}} \right) = 0\) for \(\lambda \). <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(\lambda = - \frac{1}{3}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(\overrightarrow {{\rm{OP}}} = \left( \begin{array}{c} - \frac{{14}}{3}\\ - \frac{{11}}{3}\\\frac{4}{3}\end{array} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><span style="font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium; background-color: #f7f7f7;">\({L_2}:\boldsymbol{r} = \mu \left( \begin{array}{c} - 14\\ - 11\\4\end{array} \right)\)</span><span style="font-family: 'times new roman', times; font-size: medium; background-color: #f7f7f7;"> </span> <strong><em>A1</em></strong></span></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman'; min-height: 23.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Do not award the final <strong><em>A1 </em></strong>if \(\boldsymbol{r} =\) is not seen.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman'; min-height: 23.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman'; min-height: 23.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) <strong>METHOD 1</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left| {\overrightarrow {{\text{OP}}} } \right| = \sqrt {{{\left( { - \frac{{14}}{3}} \right)}^2} + {{\left( { - \frac{{11}}{3}} \right)}^2} + {{\left( {\frac{4}{3}} \right)}^2}} \) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 6.08{\text{ }}\left( { = \sqrt {37} } \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 2</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">shortest distance = \(\frac{{\left| {\left( \begin{array}{c} - 1\\2\\2\end{array} \right) \times \left( \begin{array}{c} - 5\\ - 3\\2\end{array} \right)} \right|}}{{\left| {\left( \begin{array}{c} - 1\\2\\2\end{array} \right)} \right|}}\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{\left| {10i + 8j + 13k} \right|}}{{\sqrt {1 + 4 + 4} }}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 6.08{\text{ }}\left( { = \sqrt {37} } \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [7 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Part (a) was not well done. Most candidates recognised the need to calculate a scalar product. Some candidates made careless sign or arithmetic errors when solving for \(\lambda \). A few candidates neglected to express their final answer in the form ‘<strong><em>r</em></strong> =’.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Candidates who were successful in answering part (a) generally answered part (b) correctly. The large majority of successful candidates calculated \(\left| {\overrightarrow {{\text{OP}}} } \right|\).</span></p>
</div>
<br><hr><br><div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">Port A is defined to be the origin of a set of coordinate axes and port B is located at the point (70, 30), where distances are measured in kilometres. A ship <em>S</em><sub>1</sub> sails from port A at 10:00 in a straight line such that its position \(t\) hours after 10:00 is given by \(r = t\left( {\begin{array}{*{20}{c}}<br> {10} \\ <br> {20} <br>\end{array}} \right)\)</span><span style="font-family: times new roman,times; font-size: medium;">.</span><br><span style="font-family: times new roman,times; font-size: medium;">A speedboat <em>S</em><sub>2</sub> is capable of three times the speed of <em>S</em><sub>1</sub> and is to meet <em>S</em><sub>1</sub> by travelling the shortest possible distance. What is the latest time that <em>S</em><sub>2</sub> can leave port B?</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 1</span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">equation of journey of ship <em>S</em><sub>1</sub></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({r_1} = t\left( {\begin{array}{*{20}{c}}<br> {10} \\ <br> {20} <br>\end{array}} \right)\)<br></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">equation of journey of speedboat <em>S</em><sub>2</sub> ,setting off \(k\) minutes later</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({r_2} = \left( {\begin{array}{*{20}{c}}<br> {70} \\ <br> {30} <br>\end{array}} \right) + \left( {t - k} \right)\left( {\begin{array}{*{20}{c}}<br> { - 60} \\ <br> {30} <br>\end{array}} \right)\) <em><strong>M1A1A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note:</strong> Award <em><strong>M1</strong></em> for perpendicular direction, <em><strong>A1</strong></em> for speed, <em><strong>A1</strong></em> for change in </span><span style="font-family: times new roman,times; font-size: medium;">parameter (<em>e.g.</em> by using \(t - k\) or \(T\), \(k\) being the time difference between the </span><span style="font-family: times new roman,times; font-size: medium;">departure of the ships).</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">solve </span><span style="font-family: times new roman,times; font-size: medium;">\(t\left( {\begin{array}{*{20}{c}}<br> {10} \\ <br> {20} <br>\end{array}} \right) = \left( {\begin{array}{*{20}{c}}<br> {70} \\ <br> {30} <br>\end{array}} \right) + \left( {t - k} \right)\left( {\begin{array}{*{20}{c}}<br> { - 60} \\ <br> {30} <br>\end{array}} \right)\)</span> <em><strong> <span style="font-family: times new roman,times; font-size: medium;">(M1)</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note:</strong> <em><strong>M</strong></em> mark is for equating their two expressions.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(10t = 70 - 60t + 60k\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(20t = 30 + 30t - 30k\) <em><strong>M1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note:</strong> <em><strong>M</strong></em> mark is for obtaining two equations involving two different parameters.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(7t - 6k = 7\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( - t + 3k = 3\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(k = \frac{{28}}{{15}}\) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">latest time is 11:52 <em><strong>A1</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[7 marks]</span></strong></em></p>
<p><strong><span style="font-family: times new roman,times; font-size: medium;"> </span></strong></p>
<p><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 2<br></span></strong></p>
<p><br><img src="data:image/png;base64,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" alt></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{SB}} = 22\sqrt 5 \) <em><strong>M1A1</strong></em></span><br><span style="font-family: times new roman,times; font-size: medium;">(by perpendicular distance)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{SA}} = 26\sqrt 5 \) <em><strong>M1A1</strong></em></span><br><span style="font-family: times new roman,times; font-size: medium;">(by Pythagoras or coordinates)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(t = \frac{{26\sqrt 5 }}{{10\sqrt 5 }}\) </span><strong><span style="font-family: times new roman,times; font-size: medium;"> A1</span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(t - k = \frac{{22\sqrt 5 }}{{30\sqrt 5 }}\)</span><em><strong><span style="font-family: times new roman,times; font-size: medium;"> </span></strong><strong><span style="font-family: times new roman,times; font-size: medium;">A1</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(k = \frac{{28}}{{15}}\) leading to latest time 11:52 <em><strong>A1</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[7 marks]</span></strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">Few candidates managed to make progress on this question. Many candidates did not attempt the problem and many that did make an attempt failed to draw a diagram that would have allowed them to make further progress. There were a variety of possible solution techniques but candidates seemed unable to interpret the equation of a straight line written in vector form or find a perpendicular direction. This meant that it was very difficult for meaningful progress to be made towards a solution.</span></p>
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<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The coordinates of points A, B and C are given as \((5,\, - 2,\,5)\) , \((5,\,4,\, - 1)\) and \(( - 1,\, - 2,\, - 1)\) respectively.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: times new roman,times; font-size: medium;">Show that AB = AC and that \({\rm{B\hat AC}} = 60^\circ \).<br></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: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the Cartesian equation of \(\Pi \), the plane passing through A, B, and C.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p> <span style="font-family: 'times new roman', times; font-size: medium;">(i) Find the Cartesian equation of \({\Pi _1}\)<span style="font: normal normal normal 7px/normal Helvetica;"> </span>, the plane perpendicular to (AB) passing through the midpoint of [AB] .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Find the Cartesian equation of \({\Pi _2}\)<span style="font: 7.0px Helvetica;"> </span>, the plane perpendicular to (AC) passing through the midpoint of [AC].</span></p>
<p> </p>
<div class="marks">[4]</div>
<div class="question_part_label">c(i)(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: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the vector equation of <em>L </em>, the line of intersection of \({\Pi _1}\) and \({\Pi _2}\)<span style="font: 7.0px Helvetica;"> </span>, and show that it is perpendicular to \(\Pi \) .</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A methane molecule consists of a carbon atom with four hydrogen atoms symmetrically placed around it in three dimensions.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><br><img style="display: block; margin-left: auto; margin-right: auto;" 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" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The positions of the centres of three of the hydrogen atoms are A, B and C as given. The position of the centre of the fourth hydrogen atom is D.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Using the fact that \({\text{AB}} = {\text{AD}}\) , show that the coordinates of one of the possible positions of the fourth hydrogen atom is \(( -1,\,4,\,5)\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A methane molecule consists of a carbon atom with four hydrogen atoms symmetrically placed around it in three dimensions.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><br><img style="display: block; margin-left: auto; margin-right: auto;" 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" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The positions of the centres of three of the hydrogen atoms are A, B and C as given. The position of the centre of the fourth hydrogen atom is D.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Letting D be \(( - 1,\,4,\,5)\) , show that the coordinates of G, the position of the centre of the carbon atom, are \((2,\,1,\,2)\) . Hence calculate \({\rm{D}}\hat {\rm{G}}{\rm{A}}\) , the bonding angle of carbon.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">f.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">\(\overrightarrow {\text{AB}} = \left( {\begin{array}{*{20}{c}}<br> 0 \\ <br> 6 \\ <br> { - 6} <br>\end{array}} \right) \Rightarrow {\text{AB}} = \sqrt {72} \) <strong><em>A1</em></strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\(\overrightarrow {\text{AC}} = \left( {\begin{array}{*{20}{c}}<br> { - 6} \\ <br> 0 \\ <br> { - 6} <br>\end{array}} \right) \Rightarrow {\text{AC}} = \sqrt {72} \)</span><span style="font-family: 'times new roman', times; font-size: medium;"> <strong><em>A1</em></strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">so they are the same <strong><em>AG</em></strong></span></p>
<p> </p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\(\overrightarrow {{\text{AB}}} \cdot \overrightarrow {{\text{AC}}} = 36 = \left( {\sqrt {72} } \right)\left( {\sqrt {72} } \right)\cos \theta \) <strong><em>(M1)</em></strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\(\cos \theta = \frac{{36}}{{\left( {\sqrt {72} } \right)\left( {\sqrt {72} } \right)}} = \frac{1}{2} \Rightarrow \theta = 60^\circ \) <strong><em>A1AG</em></strong></span> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Award <strong><em>M1A1 </em></strong>if candidates find BC and claim that triangle ABC is equilateral.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font-size: 11px;"><em><span style="font-family: 'times new roman', times; font-size: medium;"> </span></em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font-size: 11px;"><em><span style="font-family: 'times new roman', times; font-size: medium;">[4 marks]</span></em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>METHOD 1</strong></span></p>
<p><span style="font-family: times new roman,times;">\(\overrightarrow {{\text{AB}}} \times \overrightarrow {{\text{AC}}} = \left| {\begin{array}{*{20}{c}}<br> \boldsymbol{i}&\boldsymbol{j}&\boldsymbol{k}\\ <br> 0&6&{ - 6} \\ <br> { - 6}&0&{ - 6} <br>\end{array}} \right| = -36\boldsymbol{i} + 36\boldsymbol{j} + 36\boldsymbol{k}\)</span><span style="font-family: times new roman,times;"><span style="font-size: medium;"> <strong><em>(M1)A1</em></strong></span></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">equation of plane is \(x - y - z = k\) <strong><em>(M1)</em></strong> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">goes through A, B or C \( \Rightarrow x - y - z = 2\) <strong><em>A1</em></strong> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>METHOD 2</strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(x + by + cz = d\) (or similar) <strong><em>M1</em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(5 - 2b + 5c = d\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(5 + 4b - c = d\) <strong><em>A1</em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( -1 - 2b - c = d\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">solving simultaneously <strong><em>M1</em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(b = -1,{\text{ }}c = -1,{\text{ }}d = 2\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">so \(x - y - z = 2\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: times new roman,times;"><strong><em><span style="font-size: medium;">[4 marks]</span><br></em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) midpoint is \((5,\,1,\,2)\), so equation of \({\Pi _1}\) is \(y - z = -1\) <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) midpoint is \((2,\, - 2,\,2)\), so equation of \({\Pi _2}\) is \(x + z = 4\) <strong><em>A1A1</em></strong></span><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>In each part, award <strong><em>A1 </em></strong>for midpoint and <strong><em>A1 </em></strong>for the equation of the plane.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[4 marks]</span><br></em></strong></p>
<div class="question_part_label">c(i)(ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: times new roman,times; font-size: medium;"><strong>EITHER</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: times new roman,times; font-size: medium;">solving the two equations above <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: times new roman,times; font-size: medium;">\(L:r = \left( {\begin{array}{*{20}{c}}<br> 4 \\ <br> { - 1} \\ <br> 0 <br>\end{array}} \right) + \lambda \left( {\begin{array}{*{20}{c}}<br> { - 1} \\ <br> 1 \\ <br> 1 <br>\end{array}} \right)\)</span><span style="font-family: 'times new roman', times; font-size: medium;"> <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: times new roman,times; font-size: medium;">L has the direction of the vector product of the normal vectors to the planes \({\Pi _1}\) and \({\Pi _2}\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: times new roman,times; font-size: medium;">\(\left| {\begin{array}{*{20}{c}}<br> \boldsymbol{i}&\boldsymbol{j}&\boldsymbol{k} \\ <br> 0&1&{ - 1} \\ <br> 1&0&1 <br>\end{array}} \right| = \boldsymbol{i} - \boldsymbol{j} - \boldsymbol{k}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: times new roman,times; font-size: medium;">(or its opposite) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: times new roman,times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>THEN</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">direction is \(\left( {\begin{array}{*{20}{c}}<br> { - 1} \\ <br> 1 \\ <br> 1 <br>\end{array}} \right)\)</span><span style="font-family: 'times new roman', times; font-size: medium;">as required <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align: left;"><span style="font-family: 'times new roman', times; font-size: medium;">D is of the form \((4 - \lambda ,\, -1 + \lambda ,\,\lambda )\) <strong><em>M1</em></strong></span></p>
<p style="font: normal normal normal 11px/normal Helvetica; text-align: left; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;">\({(1 + \lambda )^2} + {( -1 - \lambda )^2} + {(5 - \lambda )^2} = 72\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(3{\lambda ^2} - 6\lambda - 45 = 0\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\lambda = 5{\text{ or }}\lambda = -3\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{D}}( -1,\,4,\,5)\) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Note: </em></strong>Award <strong><em>M0M0A0 </em></strong>if candidates just show that \({\text{D}}( -1,\,4,\,5)\) satisfies \({\text{AB}} = {\text{AD}}\);</span><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Award <strong><em>M1M1A0 </em></strong>if candidates also show that D is of the form \((4 - \lambda ,\, -1 + \lambda ,\,\lambda )\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"> <span style="font-family: 'times new roman', times; font-size: medium;"><strong>EITHER<br></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">G is of the form \((4 - \lambda ,\, - 1 + \lambda ,\,\lambda )\) and \({\text{DG}} = {\text{AG, BG or CG}}\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>e.g.</em> \({(1 + \lambda )^2} + {( - 1 - \lambda )^2} + {(5 - \lambda )^2} = {(5 - \lambda )^2} + {(5 - \lambda )^2} + {(5 - \lambda )^2}\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\({(1 + \lambda )^2} = {(5 - \lambda )^2}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\lambda = 2\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{G}}(2,\,1,\,2)\) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">G is the centre of mass (barycentre) of the regular tetrahedron ABCD <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{G}}\left( {\frac{{5 + 5 + ( - 1) + ( - 1)}}{4},\frac{{ - 2 + 4 + ( - 2) + 4}}{4},\frac{{5 + ( - 1) + ( - 1) + 5}}{4}} \right)\) <strong><em> M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>THEN </strong></span><strong style="font-family: 'times new roman', times; font-size: medium;"> </strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>the following part is independent of previous work and candidates may use <strong><em>AG </em></strong>to answer it (here it is possible to award <strong><em>M0M0A0A1M1A1</em></strong><span style="color: #26466f;"><strong><em>)</em></strong></span></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="color: #26466f; font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\(\overrightarrow {GD} = \left( {\begin{array}{*{20}{c}}<br> { - 3} \\ <br> 3 \\ <br> 3 <br>\end{array}} \right)\) and \(\overrightarrow {GA} = \left( {\begin{array}{*{20}{c}}<br> 3 \\ <br> { - 3} \\ <br> 3 <br>\end{array}} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\cos \theta = \frac{{ - 9}}{{\left( {3\sqrt 3 } \right)\left( {3\sqrt 3 } \right)}} = - \frac{1}{3} \Rightarrow \theta = 109^\circ \) (or 1.91 radians) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]<br></em></strong></span></p>
<p> </p>
<div class="question_part_label">f.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Parts (a) and (b) were well attempted with many candidates achieving full or nearly full marks in these parts. Surprisingly, many candidates were not able to find the coordinates of the midpoints in part (c) leading to incorrect equations of the planes and, in many cases this affected the performance in part (d). Very few answered part (d) correctly. In parts (e) and (f) very few candidates made good attempts of using the ‘show that’ procedure and used a parametric approach. In most attempts, candidates simply verified the condition using the answer given. In a few cases, candidates noticed that G is the barycentre of [ABCD] and found its coordinates successfully. Some candidates were also successful in determining the bounding angle using the information given. Unfortunately, many candidates instead of finding the angle simply quoted the result.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Parts (a) and (b) were well attempted with many candidates achieving full or nearly full marks in these parts. Surprisingly, many candidates were not able to find the coordinates of the midpoints in part (c) leading to incorrect equations of the planes and, in many cases this affected the performance in part (d). Very few answered part (d) correctly. In parts (e) and (f) very few candidates made good attempts of using the ‘show that’ procedure and used a parametric approach. In most attempts, candidates simply verified the condition using the answer given. In a few cases, candidates noticed that G is the barycentre of [ABCD] and found its coordinates successfully. Some candidates were also successful in determining the bounding angle using the information given. Unfortunately, many candidates instead of finding the angle simply quoted the result.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Parts (a) and (b) were well attempted with many candidates achieving full or nearly full marks in these parts. Surprisingly, many candidates were not able to find the coordinates of the midpoints in part (c) leading to incorrect equations of the planes and, in many cases this affected the performance in part (d). Very few answered part (d) correctly. In parts (e) and (f) very few candidates made good attempts of using the ‘show that’ procedure and used a parametric approach. In most attempts, candidates simply verified the condition using the answer given. In a few cases, candidates noticed that G is the barycentre of [ABCD] and found its coordinates successfully. Some candidates were also successful in determining the bounding angle using the information given. Unfortunately, many candidates instead of finding the angle simply quoted the result.</span></p>
<div class="question_part_label">c(i)(ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Parts (a) and (b) were well attempted with many candidates achieving full or nearly full marks in these parts. Surprisingly, many candidates were not able to find the coordinates of the midpoints in part (c) leading to incorrect equations of the planes and, in many cases this affected the performance in part (d). Very few answered part (d) correctly. In parts (e) and (f) very few candidates made good attempts of using the ‘show that’ procedure and used a parametric approach. In most attempts, candidates simply verified the condition using the answer given. In a few cases, candidates noticed that G is the barycentre of [ABCD] and found its coordinates successfully. Some candidates were also successful in determining the bounding angle using the information given. Unfortunately, many candidates instead of finding the angle simply quoted the result.</span></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Parts (a) and (b) were well attempted with many candidates achieving full or nearly full marks in these parts. Surprisingly, many candidates were not able to find the coordinates of the midpoints in part (c) leading to incorrect equations of the planes and, in many cases this affected the performance in part (d). Very few answered part (d) correctly. In parts (e) and (f) very few candidates made good attempts of using the ‘show that’ procedure and used a parametric approach. In most attempts, candidates simply verified the condition using the answer given. In a few cases, candidates noticed that G is the barycentre of [ABCD] and found its coordinates successfully. Some candidates were also successful in determining the bounding angle using the information given. Unfortunately, many candidates instead of finding the angle simply quoted the result.</span></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Parts (a) and (b) were well attempted with many candidates achieving full or nearly full marks in these parts. Surprisingly, many candidates were not able to find the coordinates of the midpoints in part (c) leading to incorrect equations of the planes and, in many cases this affected the performance in part (d). Very few answered part (d) correctly. In parts (e) and (f) very few candidates made good attempts of using the ‘show that’ procedure and used a parametric approach. In most attempts, candidates simply verified the condition using the answer given. In a few cases, candidates noticed that G is the barycentre of [ABCD] and found its coordinates successfully. Some candidates were also successful in determining the bounding angle using the information given. Unfortunately, many candidates instead of finding the angle simply quoted the result.</span></p>
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The equations of the lines \({L_1}\) and \({L_2}\) are</p>
<p class="p1">\({L_1}:{r_1} = \left( \begin{array}{l}<br>1\\<br>2\\<br>2<br>\end{array} \right) + \lambda \left( \begin{array}{l}<br> - 1\\<br>1\\<br>2<br>\end{array} \right)\)</p>
<p class="p1"> </p>
<p class="p1">\({L_2}:{r_2} = \left( \begin{array}{l}<br>1\\<br>2\\<br>4<br>\end{array} \right) + \mu \left( \begin{array}{l}<br> 2\\<br>1\\<br>6<br>\end{array} \right)\).</p>
<p class="p1"> </p>
<p class="p1"> </p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that the lines \({L_1}\) and \({L_2}\) are skew.</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">Find the acute angle between the lines \({L_1}\) and \({L_2}\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Find a vector perpendicular to both lines.</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Hence determine an equation of the line \({L_3}\) that is perpendicular to both \({L_1}\) and \({L_2}\) and intersects both lines.</p>
<div class="marks">[10]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>\({L_1}\) and \({L_2}\) are not parallel, since \(\left( \begin{array}{l}<br> - 1\\<br>1\\<br>2<br>\end{array} \right) \ne k\left( \begin{array}{l}<br>2\\<br>1\\<br>6<br>\end{array} \right)\) <em><strong>R1</strong></em></p>
<p>if they meet, then \(1 - \lambda = 1 + 2\mu \) and \(2 + \lambda = 2 + \mu \) <strong><em>M1</em></strong></p>
<p>solving simultaneously \( \Rightarrow \lambda = \mu = 0\) <strong><em>A1</em></strong></p>
<p>\(2 + 2\lambda = 4 + 6\mu \Rightarrow 2 \ne 4\) contradiction, <strong><em>R1</em></strong></p>
<p>so lines are skew <strong><em>AG</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Do not award the second <strong><em>R1</em></strong> if their values of parameters are incorrect.</p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(\left( \begin{array}{l}<br> - 1\\<br>1\\<br>2<br>\end{array} \right) \bullet \left( \begin{array}{l}<br>2\\<br>1\\<br>6<br>\end{array} \right)( = 11) = \sqrt 6 \sqrt {41} \cos \theta \) <strong><em>M1A1</em></strong></p>
<p>\(\cos \theta = \frac{{11}}{{\sqrt {246} }}\) <strong><em>(A1)</em></strong></p>
<p>\(\theta = 45.5^\circ \;\;\;(0.794{\text{ radians}})\) <strong><em>A1</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>(i) \(\left( {\begin{array}{*{20}{c}}<br> { - 1} \\<br> 1 \\<br> 2<br>\end{array}} \right) \times \left( {\begin{array}{*{20}{c}}<br> 2 \\<br> 1 \\<br> 6<br>\end{array}} \right) = \left( {\begin{array}{*{20}{c}}<br> {6 - 2} \\<br> {4 + 6} \\<br> { - 1 - 2}<br>\end{array}} \right)\) <strong><em>(M1)</em></strong></p>
<p>\( = \left( {\begin{array}{*{20}{c}}<br> 4 \\<br> {10} \\<br> { - 3}<br>\end{array}} \right) = 4{\mathbf{i}} + 10{\mathbf{j}} - 3{\mathbf{k}}\) <strong><em>A1</em></strong></p>
<p>(ii) <strong>METHOD 1</strong></p>
<p>let P be the intersection of \({L_1}\) and \({L_3}\)</p>
<p>let Q be the intersection of \({L_2}\) and \({L_3}\)</p>
<p>\(\overrightarrow {{\text{OP}}} = \left( {\begin{array}{*{20}{c}}<br> {1 - \lambda } \\<br> {2 + \lambda } \\<br> {2 + 2\lambda }<br>\end{array}} \right)\overrightarrow {{\text{OQ}}} = \left( {\begin{array}{*{20}{c}}<br> {1 + 2\mu } \\<br> {2 + \mu } \\<br> {4 + 6\mu }<br>\end{array}} \right)\) <strong><em>M1</em></strong></p>
<p>therefore \(\overrightarrow {{\text{PQ}}} = \overrightarrow {{\text{OQ}}} - \overrightarrow {{\text{OP}}} = \left( {\begin{array}{*{20}{c}}<br> {2\mu + \lambda } \\<br> {\mu - \lambda } \\<br> {2 + 6\mu - 2\lambda }<br>\end{array}} \right)\) <strong><em>M1A1</em></strong></p>
<p>\(\left( {\begin{array}{*{20}{c}}<br> {2\mu + \lambda } \\<br> {\mu - \lambda } \\<br> {2 + 6\mu - 2\lambda }<br>\end{array}} \right) = t\left( {\begin{array}{*{20}{c}}<br> 4 \\<br> {10} \\<br> { - 3}<br>\end{array}} \right)\) <strong><em>M1</em></strong></p>
<p>\(2\mu + \lambda - 4t = 0\)<br>\(\mu - \lambda - 10t = 0\)<br>\(6\mu - 2\lambda + 3t = - 2\)<br>
</p><p>solving simultaneously <strong><em>(M1)</em></strong></p>
<p>\(\lambda = \frac{{32}}{{125}}\;\;\;(0.256),{\text{ }}\mu = - \frac{{28}}{{125}}\;\;\;( - 0.224)\) <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>A1</em></strong> for either correct \(\lambda \) or \(\mu \).</p>
<p> </p>
<p><strong>EITHER</strong></p>
<p>therefore \(\overrightarrow {{\text{OP}}} = \left( {\begin{array}{*{20}{c}}<br> {1 - \lambda } \\<br> {2 + \lambda } \\<br> {2 + 2\lambda }<br>\end{array}} \right) = \left( {\begin{array}{*{20}{c}}<br> {\frac{{93}}{{125}}} \\<br> {\frac{{282}}{{125}}} \\<br> {\frac{{314}}{{125}}}<br>\end{array}} \right) = \left( {\begin{array}{*{20}{c}}<br> {0.744} \\<br> {2.256} \\<br> {2.512}<br>\end{array}} \right)\)<strong><em>A1</em></strong></p>
<p>therefore \({L_3}:{r_3} = \left( {\begin{array}{*{20}{c}}<br> {0.744} \\<br> {2.256} \\<br> {2.512}<br>\end{array}} \right) + \alpha \left( {\begin{array}{*{20}{c}}<br> 4 \\<br> {10} \\<br> { - 3}<br>\end{array}} \right)\) <strong><em>A1</em></strong></p>
<p><strong>OR</strong></p>
<p>therefore \(\overrightarrow {{\text{OQ}}} = \left( {\begin{array}{*{20}{c}}<br> {1 + 2\mu } \\<br> {2 + \mu } \\<br> {4 + 6\mu }<br>\end{array}} \right) = \left( {\begin{array}{*{20}{c}}<br> {\frac{{69}}{{125}}} \\<br> {\frac{{222}}{{125}}} \\<br> {\frac{{332}}{{125}}}<br>\end{array}} \right) = \left( {\begin{array}{*{20}{c}}<br> {0.552} \\<br> {1.776} \\<br> {2.656}<br>\end{array}} \right)\) <strong><em>A1</em></strong></p>
<p>therefore \({L_3}:{r_3} = \left( {\begin{array}{*{20}{c}}<br> {0.552} \\<br> {1.776} \\<br> {2.656}<br>\end{array}} \right) + \alpha \left( {\begin{array}{*{20}{c}}<br> 4 \\<br> {10} \\<br> { - 3}<br>\end{array}} \right)\) <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Allow position vector(s) to be expressed in decimal or fractional form.</p>
<p><strong>METHOD 2</strong></p>
<p>\({L_3}:{r_3} = \left( {\begin{array}{*{20}{c}}<br> a \\<br> b \\<br> c<br>\end{array}} \right) + t\left( {\begin{array}{*{20}{c}}<br> 4 \\<br> {10} \\<br> { - 3}<br>\end{array}} \right)\)</p>
<p>forming two equations as intersections with \({L_1}\) and \({L_2}\)</p>
<p>\(\left( {\begin{array}{*{20}{c}}<br> a \\<br> b \\<br> c<br>\end{array}} \right) + {t_1}\left( {\begin{array}{*{20}{c}}<br> 4 \\<br> {10} \\<br> { - 3}<br>\end{array}} \right) = \left( {\begin{array}{*{20}{c}}<br> 1 \\<br> 2 \\<br> 2<br>\end{array}} \right) + \lambda \left( {\begin{array}{*{20}{c}}<br> { - 1} \\<br> 1 \\<br> 2<br>\end{array}} \right)\)<br>\(\left( {\begin{array}{*{20}{c}}<br> a \\<br> b \\<br> c<br>\end{array}} \right) + {t_2}\left( {\begin{array}{*{20}{c}}<br> 4 \\<br> {10} \\<br> { - 3}<br>\end{array}} \right) = \left( {\begin{array}{*{20}{c}}<br> 1 \\<br> 2 \\<br> 4<br>\end{array}} \right) + \mu \left( {\begin{array}{*{20}{c}}<br> 2 \\<br> 1 \\<br> 6<br>\end{array}} \right)\) <strong><em>M1A1A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Only award <strong><em>M1A1A1</em></strong> if two different parameters \({t_1},{\text{ }}{t_2}\) used.</p>
<p> </p>
<p>attempting to solve simultaneously <strong><em>M1</em></strong></p>
<p>\(\lambda = \frac{{32}}{{125}}\;\;\;(0.256),{\text{ }}\mu = - \frac{{28}}{{125}}\;\;\;( - 0.224)\) <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>A1</em></strong> for either correct \(\lambda \) or \(\mu \).<br> </p>
<p><strong>EITHER</strong></p>
<p>\(\left( {\begin{array}{*{20}{c}}<br> a \\<br> b \\<br> c<br>\end{array}} \right) = \left( {\begin{array}{*{20}{c}}<br> {0.552} \\<br> {1.776} \\<br> {2.656}<br>\end{array}} \right)\) <strong><em>A1</em></strong></p>
<p>therefore \({L_3}:{r_3} = \left( {\begin{array}{*{20}{c}}<br> {0.552} \\<br> {1.776} \\<br> {2.656}<br>\end{array}} \right) + t\left( {\begin{array}{*{20}{c}}<br> 4 \\<br> {10} \\<br> { - 3}<br>\end{array}} \right)\) <strong><em>A1A1</em></strong></p>
<p><strong>OR</strong></p>
<p>\(\left( {\begin{array}{*{20}{c}}<br> a \\<br> b \\<br> c<br>\end{array}} \right) = \left( {\begin{array}{*{20}{c}}<br> {0.744} \\<br> {2.256} \\<br> {2.512}<br>\end{array}} \right)\) <strong><em>A1</em></strong></p>
<p>therefore \({L_3}:{r_3} = \left( {\begin{array}{*{20}{c}}<br> {0.744} \\<br> {2.256} \\<br> {2.512}<br>\end{array}} \right) + t\left( {\begin{array}{*{20}{c}}<br> 4 \\<br> {10} \\<br> { - 3}<br>\end{array}} \right)\) <strong><em>A1A1</em></strong><br> </p>
<p><strong>Note:</strong> Allow position vector(s) to be expressed in decimal or fractional form.</p>
<p><em><strong>10 marks</strong></em></p>
<p><em><strong>Total [18 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p class="p1">Find the acute angle between the planes with equations \(x + y + z = 3\) and \(2x - z = 2\).</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p class="p1"><strong><em>n</em></strong>\(_1 = \left( {\begin{array}{*{20}{c}} 1 \\ 1 \\ 1 \end{array}} \right)\) and <strong><em>n</em></strong>\(_2 = \left( {\begin{array}{*{20}{c}} 2 \\ 0 \\ { - 1} \end{array}} \right)\)Â <span class="Apple-converted-space">Â Â </span><strong><em>(A1)(A1)</em></strong></p>
<p class="p1"><strong>EITHER </strong></p>
<p class="p1"><span class="Apple-converted-space">\(\theta  = \arccos \left( {\frac{{{n_1} \bullet {n_2}}}{{\left| {{n_1}} \right|\left| {{n_2}} \right|}}} \right)\left( {\cos \theta  = \frac{{{n_1} \bullet {n_2}}}{{\left| {{n_1}} \right|\left| {{n_2}} \right|}}} \right)\)   </span><strong><em>(M1)</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\( = \arccos \left( {\frac{{2 + 0 - 1}}{{\sqrt 3 \sqrt 5 }}} \right)\left( {\cos \theta  = \frac{{2 + 0 - 1}}{{\sqrt 3 \sqrt 5 }}} \right)\)   </span><strong><em>(A1)</em></strong></p>
<p class="p1">\( = \arccos \left( {\frac{1}{{\sqrt {15} }}} \right)\left( {\cos \theta  = \frac{1}{{\sqrt {15} }}} \right)\)</p>
<p class="p1"><strong>OR</strong></p>
<p class="p1"><span class="Apple-converted-space">\(\theta  = \arcsin \left( {\frac{{\left| {{n_1} \times {n_2}} \right|}}{{\left| {{n_1}} \right|\left| {{n_2}} \right|}}} \right)\left( {\sin \theta  = \frac{{\left| {{n_1} \times {n_2}} \right|}}{{\left| {{n_1}} \right|\left| {{n_2}} \right|}}} \right)\)   </span><strong><em>(M1)</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\( = \arcsin \left( {\frac{{\sqrt {14} }}{{\sqrt 3 \sqrt 5 }}} \right)\left( {\sin \theta  = \frac{{\sqrt {14} }}{{\sqrt 3 \sqrt 5 }}} \right)\)   </span><strong><em>(A1)</em></strong></p>
<p class="p1">\( = \arcsin \left( {\frac{{\sqrt {14} }}{{\sqrt {15} }}} \right)\left( {\sin \theta  = \frac{{\sqrt {14} }}{{\sqrt {15} }}} \right)\)</p>
<p class="p2">Â </p>
<p class="p1"><strong>THEN</strong></p>
<p class="p1"><span class="Apple-converted-space">\( = 75.0^\circ {\text{ (or 1.31)}}\) Â Â </span><strong><em>A1</em></strong></p>
<p class="p1"><strong><em>[5 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">The planes \(2x + 3y - z = 5\) and \(x - y + 2z = k\) intersect in the line \(5x + 1 = 9 - 5y = - 5z\)<em> </em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the value of <em>k </em>.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><span style="font-family: 'times new roman', times; font-size: medium;"><span style="font-family: 'times new roman', times;">point on line is</span> \(x = \frac{{ - 1 - 5\lambda }}{5}{\text{, }}y = \frac{{9 + 5\lambda }}{5}{\text{, }}z = \lambda \) or similar <strong><em>M1A1</em></strong></span></p>
<p><span style="font-size: medium;"> </span><strong style="font-family: 'times new roman', times; font-size: medium; line-height: normal;">Note: </strong><span style="font-family: 'times new roman', times; font-size: medium; line-height: normal;">Accept use of point on the line or elimination of one of the variables using the equations of the planes</span></p>
<p><span style="font-size: medium;"> </span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{ - 1 - 5\lambda }}{5} - \frac{{9 + 5\lambda }}{5} + 2\lambda = k\) <strong><em>M1A1</em></strong></span></p>
<p><span style="font-size: medium;"> </span><strong style="font-family: 'times new roman', times; font-size: medium; line-height: normal;">Note: </strong><span style="font-family: 'times new roman', times; font-size: medium; line-height: normal;">Award </span><strong style="font-family: 'times new roman', times; font-size: medium; line-height: normal;"><em>M1A1 </em></strong><span style="font-family: 'times new roman', times; font-size: medium; line-height: normal;">if coordinates of point and equation of a plane is used to obtain linear equation in </span><em style="font-family: 'times new roman', times; font-size: medium; line-height: normal;">k </em><span style="font-family: 'times new roman', times; font-size: medium; line-height: normal;">or equations of the line are used in combination with equation obtained by elimination to get linear equation in </span><em style="font-family: 'times new roman', times; font-size: medium; line-height: normal;">k.</em></p>
<p><span style="font-size: medium;"> </span></p>
<p><span style="font-size: medium;"> </span><span style="font-family: 'times new roman', times; font-size: medium;">\(k = - 2\) </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A1</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[5 marks]</span><br></em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Many different attempts were seen, sometimes with success. Unfortunately many candidates wasted time with substitutions showing little understanding of the problem.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Write the vector equations of the following lines in parametric form.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[{r_1} = \left( {\begin{array}{*{20}{c}}<br> 3 \\ <br> 2 \\ <br> 7 <br>\end{array}} \right) + m\left( {\begin{array}{*{20}{c}}<br> 2 \\ <br> { - 1} \\ <br> 2 <br>\end{array}} \right)\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[{r_2} = \left( {\begin{array}{*{20}{c}}<br> 1 \\ <br> 4 \\ <br> 2 <br>\end{array}} \right) + n\left( {\begin{array}{*{20}{c}}<br> 4 \\ <br> { - 1} \\ <br> 1 <br>\end{array}} \right)\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Hence show that these two lines intersect and find the point of intersection, A.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Find the Cartesian equation of the plane \(\prod \) that contains these two lines.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(d) Let B be the point of intersection of the plane \(\prod \) and the line\({r} = \left( {\begin{array}{*{20}{c}}<br> { - 8} \\ <br> { - 3} \\ <br> 0 <br>\end{array}} \right) + \lambda \left( {\begin{array}{*{20}{c}}<br> 3 \\ <br> 8 \\ <br> 2 <br>\end{array}} \right)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the coordinates of B.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(e) If C is the mid-point of AB, find the vector equation of the line perpendicular to the plane \(\prod \) and passing through C.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) \(x = 3 + 2m\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(y = 2 - m\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(z = 7 + 2m\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(x = 1 + 4n\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(y = 4 - n\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(z = 2 + n\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) \(3 + 2m = 1 + 4n \Rightarrow 2m - 4n = - 2{\text{ }}({\text{i}})\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(2 - m = 4 - n \Rightarrow m - n = - 2{\text{ }}({\text{ii}})\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(7 + 2m = 2 + n \Rightarrow 2m - n = - 5{\text{ }}({\text{iii}})\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(({\text{iii}}) - ({\text{ii}}) \Rightarrow m = - 3\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow n = - 1\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Substitute in (i), –6 + 4 = –2 . Hence lines intersect. <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Point of intersection A is (–3, 5,1) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) \(\left| {\begin{array}{*{20}{c}}<br> i&j&k \\ <br> 2&{ - 1}&2 \\ <br> 4&{ - 1}&1 <br>\end{array}} \right| = \left( {\begin{array}{*{20}{c}}<br> 1 \\ <br> 6 \\ <br> 2 <br>\end{array}} \right)\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(r \cdot \left( {\begin{array}{*{20}{c}}<br> 1 \\ <br> 6 \\ <br> 2 <br>\end{array}} \right) = \left( {\begin{array}{*{20}{c}}<br> 3 \\ <br> 2 \\ <br> 7 <br>\end{array}} \right) \cdot \left( {\begin{array}{*{20}{c}}<br> 1 \\ <br> 6 \\ <br> 2 <br>\end{array}} \right)\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(r \cdot \left( {\begin{array}{*{20}{c}}<br> 1 \\ <br> 6 \\ <br> 2 <br>\end{array}} \right) = 29\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>x</em> + 6<em>y</em> + 2<em>z</em> = 29 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>M1A0</em></strong> if answer is not in Cartesian form.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(d) \(x = - 8 + 3\lambda \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(y = - 3 + 8\lambda \) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(z = 2\lambda \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Substitute in equation of plane.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( - 8 + 3\lambda - 18 + 48\lambda + 4\lambda = 29\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(55\lambda = 55\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\lambda = 1\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Coordinates of B are (–5, 5, 2) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(e) Coordinates of C are \(\left( { - 4,{\text{ 5, }}\frac{3}{2}} \right)\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(r = \left( {\begin{array}{*{20}{c}}<br> { - 4} \\ <br> 5 \\ <br> {\frac{3}{2}} <br>\end{array}} \right) + \mu \left( {\begin{array}{*{20}{c}}<br> 1 \\ <br> 6 \\ <br> 2 <br>\end{array}} \right)\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>M1A0</em></strong> unless candidate writes <strong><em>r</em></strong> = or \(\left( {\begin{array}{*{20}{c}}<br> x \\ <br> y \\ <br> z <br>\end{array}} \right)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [18 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Most candidates found this question to their liking and many correct solutions were seen. In (b), some candidates solved two equations for <em>m</em> and <em>n</em> but then failed to show that these values satisfied the third equation. In (e), some candidates used an incorrect formula to determine the coordinates of the mid-point of <em>AB</em> .</span></p>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A plane \(\pi \) has vector equation <strong><em>r</em></strong> = (−2<strong><em>i</em></strong> + 3<strong><em>j</em></strong> − 2<strong><em>k</em></strong>) + \(\lambda \)(2<strong><em>i</em></strong> + 3<strong><em>j</em></strong> + 2<strong><em>k</em></strong>) + \(\mu \)(6<strong><em>i</em></strong> − 3<strong><em>j</em></strong> + 2<strong><em>k</em></strong>).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Show that the Cartesian equation of the plane \(\pi \) is 3<em>x</em> + 2<em>y</em> − 6<em>z</em> = 12.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) The plane \(\pi \) meets the <em>x</em>, <em>y</em> and <em>z</em> axes at A, B and C respectively. Find the coordinates of A, B and C.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Find the volume of the pyramid OABC.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(d) Find the angle between the plane \(\pi \) and the <em>x</em>-axis.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(e) <strong>Hence</strong>, or otherwise, find the distance from the origin to the plane \(\pi \).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(f) Using your answers from (c) and (e), find the area of the triangle ABC.</span></p>
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<h2 style="margin-top: 1em">Markscheme</h2>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) <strong>EITHER</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">normal to plane given by</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left| {\begin{array}{*{20}{c}}<br> i&j&k \\ <br> 2&3&2 \\ <br> 6&{ - 3}&2 <br>\end{array}} \right|\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">= 12<strong><em>i</em></strong> + 8<strong><em>j</em></strong> – 24<strong><em>k</em></strong> <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">equation of \(\pi \) is \(3x + 2y - 6z = d\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">as goes through (–2, 3, –2) so <em>d</em> = 12 <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\pi :3x + 2y - 6z = 12\) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(x = - 2 + 2\lambda + 6\mu \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(y = 3 + 3\lambda - 3\mu \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(z = - 2 + 2\lambda + 2\mu \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">eliminating \(\mu \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(x + 2y = 4 + 8\lambda \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(2y + 3z = 12\lambda \) <strong><em>M1A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">eliminating \(\lambda \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(3(x + 2y) - 2(2y + 3z) = 12\) <strong><em>M1A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\pi :3x + 2y - 6z = 12\) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) therefore A(4, 0, 0), B(0, 6, 0) and C(0, 0, 2) <strong><em>A1A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>A1A1A0</em></strong> if position vectors given instead of coordinates.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) area of base \({\text{OAB}} = \frac{1}{2} \times 4 \times 6 = 12\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(V = \frac{1}{3} \times 12 \times 2 = 8\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(d) \(\left( {\begin{array}{*{20}{c}}<br> 3 \\ <br> 2 \\ <br> { - 6} <br>\end{array}} \right) \cdot \left( {\begin{array}{*{20}{c}}<br> 1 \\ <br> 0 \\ <br> 0 <br>\end{array}} \right) = 3 = 7 \times 1 \times \cos \phi \) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\phi = \arccos \frac{3}{7}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">so \(\theta = 90 - \arccos \frac{3}{7} = 25.4^\circ \,\,\,\,\,\)(accept 0.443 radians) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(e) \(d = 4\sin \theta = \frac{{12}}{7}\,\,\,\,\,( = 1.71)\) <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(f) \(8 = \frac{1}{3} \times \frac{{12}}{7} \times {\text{area}} \Rightarrow {\text{area}} = 14\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> If answer to part (f) is found in an earlier part, award <strong><em>M1A1</em></strong>, regardless of the fact that it has not come from their answers to part (c) and part (e).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>[2 marks]</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [20 marks]</em></strong></span></p>
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<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The question was generally well answered, although there were many students who failed to recognise that the volume was most logically found using a base as one of the coordinate planes.</span></p>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The function <em>f</em> is defined on the domain [0, 2] by \(f(x) = \ln (x + 1)\sin (\pi x)\) .</span></p>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Obtain an expression for \(f'(x)\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Sketch the graphs of <em>f</em> and \(f'\) on the same axes, showing clearly all <em>x</em>-intercepts.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the <em>x</em>-coordinates of the two points of inflexion on the graph of <em>f</em> .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the equation of the normal to the graph of <em>f</em> where <em>x</em> = 0.75 , giving your answer in the form <em>y</em> = <em>mx</em> + <em>c</em> .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the points \({\text{A}}\left( {a{\text{ }},{\text{ }}f(a)} \right)\) , \({\text{B}}\left( {b{\text{ }},{\text{ }}f(b)} \right)\) and \({\text{C}}\left( {c{\text{ }},{\text{ }}f(c)} \right)\) where <em>a</em> , <em>b</em> and <em>c</em> \((a < b < c)\) are the solutions of the equation \(f(x) = f'(x)\) . Find the area of the triangle ABC.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">e.</div>
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<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(f'(x) = \frac{1}{{x + 1}}\sin (\pi x) + \pi \ln (x + 1)\cos (\pi x)\) <strong><em>M1A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><img 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" alt><span style="font-family: 'times new roman', times; font-size: medium;"><span style="font-family: 'times new roman', times; font-size: medium;"> <em><strong>A4</strong></em></span></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>A1A1</em></strong> for graphs, <strong><em>A1A1</em></strong> for intercepts.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">0.310, 1.12 <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(f'(0.75) = - 0.839092\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">so equation of normal is \(y - 0.39570812 = \frac{1}{{0.839092}}(x - 0.75)\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(y = 1.19x - 0.498\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{A}}(0,{\text{ }}0)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{B(}}\overbrace {0.548 \ldots }^c,\overbrace {0.432 \ldots }^d)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{C(}}\overbrace {1.44 \ldots }^e,\overbrace { - 0.881 \ldots }^f)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Accept coordinates for B and C rounded to 3 significant figures.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">area \(\Delta {\text{ABC}} = \frac{1}{2}|\)(</span><em style="font-family: 'times new roman', times; font-size: medium;">c</em><strong style="font-family: 'times new roman', times; font-size: medium;"><em>i</em></strong><span style="font-family: 'times new roman', times; font-size: medium;"> + </span><em style="font-family: 'times new roman', times; font-size: medium;">d</em><strong style="font-family: 'times new roman', times; font-size: medium;"><em>j</em></strong><span style="font-family: 'times new roman', times; font-size: medium;">) \( \times \) (</span><em style="font-family: 'times new roman', times; font-size: medium;">e</em><strong style="font-family: 'times new roman', times; font-size: medium;"><em>i</em></strong><span style="font-family: 'times new roman', times; font-size: medium;"> + </span><em style="font-family: 'times new roman', times; font-size: medium;">f</em><strong style="font-family: 'times new roman', times; font-size: medium;"><em>j</em></strong><span style="font-family: 'times new roman', times; font-size: medium;">)\(|\) </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>M1A1</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{1}{2}(de - cf)\) <em><strong>A1</strong></em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 0.554\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The position vector at time \(t\) of a point \(P\) is given by\[\overrightarrow {{\text{OP}}} = \left( {1 + t} \right){\boldsymbol{i}} + \left( {2 - 2t} \right){\boldsymbol{j}} + \left( {3t - 1} \right){\boldsymbol{k}},{\text{ }}t \geqslant 0.\]</span></p>
</div>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">(a) Find the coordinates of P when \(t = 0\) .<br></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(b) Show that P moves along the line \(L\) with Cartesian equations\[x - 1 = \frac{{y - 2}}{{ - 2}} = \frac{{z + 1}}{3}\]</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(c) (i) Find the value of t when P lies on the plane with equation \(2x + y + z = 6\) .</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (ii) State the coordinates of P at this time.</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (iii) Hence find the total distance travelled by P before it meets the plane.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The position vector at time \(t\) of another point, Q, is given by\[\overrightarrow {{\text{OQ}}} = \left( {\begin{array}{*{20}{c}}<br> {{t^2}} \\ <br> {1 - t} \\ <br> {1 - {t^2}} <br>\end{array}} \right),{\text{ }}t \geqslant 0.\]</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(d) (i) Find the value of t for which the distance from Q to the origin is minimum.</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (ii) Find the coordinates of Q at this time.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(e) Let \(\boldsymbol{a}\) , </span><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: times new roman,times; font-size: medium;">\(\boldsymbol{b}\)</span> and </span><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: times new roman,times; font-size: medium;">\(\boldsymbol{c}\)</span> be the position vectors of Q at times \(t = 0\), \(t =1\) and \(t = 2\) </span><span style="font-family: times new roman,times; font-size: medium;">respectively.</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (i) Show that the equation \({\boldsymbol{a}} - {\boldsymbol{b}} = k\left( {{\boldsymbol{b}} - {\boldsymbol{c}}} \right)\) has no solution for \(k\) .</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (ii) Hence show that the path of Q is not a straight line.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">(a) \(\overrightarrow {{\text{OP}}} = {\boldsymbol{i}} + 2{\boldsymbol{j}} - {\boldsymbol{k}}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">(M1)</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">the coordinates of P are (1, 2, –1) <em><strong>A1</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(b) <strong>EITHER</strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(x = 1 + t\) , \(y = 2 - 2t\) , \(z = 3t - 1\) <em><strong>M1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(x - 1 = t\)</span><span style="font-family: times new roman,times; font-size: medium;"> , \(\frac{{y - 2}}{{ - 2}} = t\) , \(\frac{{z + 1}}{3} = t\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(x - 1 = \frac{{y - 2}}{{ - 2}} = \frac{{z + 1}}{3}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">AG N0</span></strong></em></p>
<p><strong><span style="font-family: times new roman,times; font-size: medium;">OR</span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\left( {\begin{array}{*{20}{c}}<br> x \\ <br> y \\ <br> z <br>\end{array}} \right) = \left( {\begin{array}{*{20}{c}}<br> 1 \\ <br> 2 \\ <br> { - 1} <br>\end{array}} \right) + t\left( {\begin{array}{*{20}{c}}<br> 1 \\ <br> { - 2} \\ <br> 3 <br>\end{array}} \right)\) <em><strong>M1A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(x - 1 = \frac{{y - 2}}{{ - 2}} = \frac{{z + 1}}{3}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">AG</span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> </span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(c) (i) \(2\left( {1 + t} \right) + \left( {2 - 2t} \right) + \left( {3t - 1} \right) = 6 \Rightarrow t = 1\) <em><strong>M1A1 N1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) coordinates are (2, 0, 2) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note:</strong> Award <em><strong>A0</strong></em> for position vector.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iii) distance travelled is the distance between the two points <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\sqrt {{{\left( {2 - 1} \right)}^2} + {{\left( {0 - 2} \right)}^2} + {{\left( {2 + 1} \right)}^2}} = \sqrt {14} \) (\( = 3.74\)) <em><strong>(M1)A1</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[6 marks]</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(d) (i) distance from Q to the origin is given by</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(d(t) = \sqrt {{t^4} + {{\left( {1 - t} \right)}^2} + {{\left( {1 - {t^2}} \right)}^2}} \) (or equivalent) <em><strong>M1A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>e.g.</em> for labelled sketch of graph of \(d\) or \({d^2}\) <em><strong>(M1)(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em><strong><img 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" alt> </strong></em></span><span style="font-family: times new roman,times; font-size: medium;"><em>or </em> </span><img 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" alt></p>
<p><span style="font-family: times new roman,times; font-size: medium;">the minimum value is obtained for \(t = 0.761\) <strong><em>A1 N3</em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) the coordinates are (0.579, 0.239, 0.421) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note:</strong> Accept answers given as a position vector.</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[6 marks]</span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> </span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(e) (i)</span> <span style="font-family: times new roman,times; font-size: medium;">\({\boldsymbol{a}} = \left( {\begin{array}{*{20}{c}}<br> 0 \\ <br> 1 \\ <br> 1 <br>\end{array}} \right)\), \({\boldsymbol{b}} = \left( {\begin{array}{*{20}{c}}<br> 1 \\ <br> 0 \\ <br> 0 <br>\end{array}} \right)\) and \({\boldsymbol{c}} = \left( {\begin{array}{*{20}{c}}<br> 4 \\ <br> { - 1} \\ <br> { - 3} <br>\end{array}} \right)\)</span><span style="font-family: times new roman,times; font-size: medium;"> <em><strong>(M1)A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">substituting in the equation \({\boldsymbol{a}} - {\boldsymbol{b}} = k\left( {{\boldsymbol{b}} - {\boldsymbol{c}}} \right)\) , we have <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\left( {\begin{array}{*{20}{c}}<br> 0 \\ <br> 1 \\ <br> 1 <br>\end{array}} \right) - \left( {\begin{array}{*{20}{c}}<br> 1 \\ <br> 0 \\ <br> 0 <br>\end{array}} \right) = k\left( {\left( {\begin{array}{*{20}{c}}<br> 1 \\ <br> 0 \\ <br> 0 <br>\end{array}} \right) - \left( {\begin{array}{*{20}{c}}<br> 4 \\ <br> { - 1} \\ <br> { - 3} <br>\end{array}} \right)} \right) \Leftrightarrow \left( {\begin{array}{*{20}{c}}<br> { - 1} \\ <br> 1 \\ <br> 1 <br>\end{array}} \right) = k\left( {\begin{array}{*{20}{c}}<br> { - 3} \\ <br> 1 \\ <br> 3 <br>\end{array}} \right)\)</span><span style="font-family: times new roman,times; font-size: medium;"> </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( \Rightarrow k = 1\) and</span> <span style="font-family: times new roman,times; font-size: medium;">\(k = \frac{1}{3}\)</span><span style="font-family: times new roman,times; font-size: medium;"> which is impossible</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">so there is no solution for \(k\) <em><strong>R1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \({{\text{BA}}}\)</span><span style="font-family: times new roman,times; font-size: medium;">and \(\overrightarrow {{\text{CB}}} \)</span><span style="font-family: times new roman,times; font-size: medium;">are not parallel <em><strong>R2</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(hence A, B, and C cannot be collinear)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note:</strong> Only accept answers that follow from part (i).</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[7 marks]</span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> </span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">Total [23 marks]</span></strong></em></p>
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<h2 style="margin-top: 1em">Examiners report</h2>
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<p><span style="font-family: times new roman,times; font-size: medium;">Generally this question was answered well by those students who attempted it. It was </span><span style="font-family: times new roman,times; font-size: medium;">common to see confusion between coordinates and position vectors. Part (d) was most easily </span><span style="font-family: times new roman,times; font-size: medium;">answered with the use of a GDC, but fewer candidates took advantage of this. In part (e) </span><span style="font-family: times new roman,times; font-size: medium;">many students had difficulties expressing their reasoning well to obtain the marks.</span></p>
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