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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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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 \(\overrightarrow {{\text{OA}}} \times \overrightarrow {{\text{OB}}} \).</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>
<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>
<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>
<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>
<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>
<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>Π</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>Π</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>Π</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>Π</em>\(_2\) and <em>Π</em>\(_3\).</p>
<div class="marks">[4]</div>
<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"> \({\left| {\overrightarrow {{\text{OC}}} } \right|^2} = |\)</span><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="p1">(ii) <span class="Apple-converted-space"> \({\left| {\overrightarrow {{\text{AB}}} } \right|^2} = |\)</span><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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
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<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>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 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>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 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>
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<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>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 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>
<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>
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<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>
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<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">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>
<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>
<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>
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<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>
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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;" src="data:image/png;base64,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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>
<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>
<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>
<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>
<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>
<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;">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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<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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<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>
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<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>
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