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</div><h2>SL Paper 1</h2><div class="specification">
<p>The vertices A, B, C of an acute angled triangle have position vectors <strong><em>a</em></strong>, <strong><em>b</em></strong>, <strong><em>c </em></strong>with respect to an origin O.</p>
</div>
<div class="specification">
<p>The mid-point of [BC] is denoted by D. The point E lies on [AD] such that \({\text{AE}} = 2{\text{DE}}\).</p>
</div>
<div class="specification">
<p>The perpendiculars from B to [AC] and C to [AB] meet at the point F.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the position vector of E is</p>
<p style="text-align: center;">\(\frac{1}{3}\) (<em><strong>a</strong></em> + <em><strong>b</strong></em> + <em><strong>c</strong></em>).</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Explain briefly why this result shows that the three medians of a triangle are concurrent.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the position vector <strong><em>f </em></strong>of F satisfies the equations</p>
<p style="text-align: center;">(<strong><em>b </em></strong>– <strong><em>f </em></strong>) \( \bullet \) (<strong><em>c </em></strong>– <strong><em>a</em></strong>) = 0</p>
<p style="text-align: center;">(<strong><em>c </em></strong>– <strong><em>f </em></strong>) \( \bullet \) (<strong><em>a </em></strong>– <strong><em>b</em></strong>) = 0.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show, by expanding these equations, that</p>
<p style="text-align: center;">(<strong><em>a</em></strong> – <strong><em>f </em></strong>) \( \bullet \) (<strong><em>c</em></strong> – <strong><em>b</em></strong>) = 0.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Explain briefly why this result shows that the three altitudes of a triangle are concurrent.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.iii.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The point \({\rm{T}}(a{t^2},2at)\) lies on the parabola \({y^2} = 4ax\) . Show that the tangent to </span><span style="font-family: times new roman,times; font-size: medium;">the parabola at T has equation \(y = \frac{x}{t} + at\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The distinct points \({\rm{P}}(a{p^2}, 2ap)\) and \(Q(a{q^2}, 2aq)\) , where \(p\), \(q \ne 0\) , also lie on </span><span style="font-family: times new roman,times; font-size: medium;">the parabola \({y^2} = 4ax\) . Given that the line (PQ) passes through the focus, show </span><span style="font-family: times new roman,times; font-size: medium;">that</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (i) \(pq = - 1\) ;</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (ii) the tangents to the parabola at P and Q, intersect on the directrix.</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The normal at the point \({\text{T}}(a{t^2},{\text{ }}2at),{\text{ }}t \ne 0\), on the parabola \({y^2} = 4ax\) meets the parabola again at the point \({\text{S}}(a{s^2},{\text{ }}2as)\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \({t^2} + st + 2 = 0\).</p>
<div class="marks">[7]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that \({\rm{S\hat OT}}\) is a right-angle, where O is the origin, determine the possible values of \(t\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The triangle ABC is isosceles and AB = BC = 5. D is the midpoint of AC and BD = 4.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Find the lengths of the tangents from A, B and D to the circle inscribed in </span><span style="font-family: times new roman,times; font-size: medium;">the triangle ABD.</span></p>
</div>
<br><hr><br><div class="question">
<p>Given that the tangents at the points P and Q on the parabola \({y^2} = 4ax\) are perpendicular, find the locus of the midpoint of PQ.</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;">\({\text{ABCDEF}}\) is a hexagon. A circle lies inside the hexagon and touches each of the six sides.</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;">Show that \({\text{AB}} + {\text{CD}} + {\text{EF}} = {\text{BC}} + {\text{DE}} + {\text{FA}}\).</span></p>
</div>
<br><hr><br><div class="question">
<p class="p1">Consider the curve C given by \(y = {x^3}\).</p>
<p class="p1">The tangent at a point <span class="s1">P </span>on \(C\) meets the curve again at <span class="s1">Q</span>. The tangent at <span class="s1">Q </span>meets the curve again at <span class="s1">R</span>. Denote the \(x\)-coordinates of \({\text{P, Q}}\) and <span class="s1">R</span>, by \({x_1},{\text{ }}{x_2}\) and \({x_3}\) respectively where \({x_1} \ne 0\). Show that, \({x_1},{\text{ }}{x_2},{\text{ }}{x_3}\) form the first three elements of a divergent geometric sequence.</p>
</div>
<br><hr><br><div class="question">
<p class="p1">The points <span class="s1">P, Q </span>and <span class="s1">R</span>, lie on the sides <span class="s1">[AB], [AC] </span>and <span class="s1">[BC]</span>, respectively, of the triangle <span class="s1">ABC</span>. The lines <span class="s1">(AR), (BQ) </span>and <span class="s1">(CP) </span>are concurrent.</p>
<p class="p1">Use Ceva’s theorem to prove that <span class="s1">[PQ] </span>is parallel to <span class="s1">[BC] </span>if and only if <span class="s1">R </span>is the midpoint of <span class="s1">[BC]</span>.</p>
</div>
<br><hr><br><div class="specification">
<p class="p1">The points <span class="s1">A, B </span>have coordinates \(( - 3,{\text{ }}0)\), \((5,{\text{ }}0)\) respectively. Consider the Apollonius circle \(C\) which is the locus of point <span class="s1">P </span><span class="s2">where</span></p>
<p class="p2">\[\frac{{{\text{AP}}}}{{{\text{BP}}}} = k{\text{ for }}k \ne 1.\]</p>
<p class="p1">Given that the centre of \(C\) has coordinates \((13,{\text{ }}0)\), find</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>the value of \(k\)<span class="s1">;</span></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>the radius of \(C\)<span class="s1">;</span></p>
<p class="p1">(iii) <span class="Apple-converted-space"> </span>the \(x\)-intercepts of \(C\).</p>
<div class="marks">[11]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">Let </span><span class="s2">M </span>be any point on \(C\) and <span class="s2">N </span>be the \(x\)-intercept of \(C\) between <span class="s2">A </span>and <span class="s2">B</span>.</p>
<p class="p1">Prove that \({\rm{A\hat MN}} = {\rm{N\hat MB}}\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The point \({\rm{P}}(x,y)\) moves in such a way that its distance from the point (\(1\) , \(0\)) is three </span><span style="font-family: times new roman,times; font-size: medium;">times its distance from the point (\( -1\) , \(0\)) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the equation of the locus of P.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Show that this equation represents a circle and state its radius and the coordinates </span><span style="font-family: times new roman,times; font-size: medium;">of its centre.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The parabola \(P\) has equation \({y^2} = 4ax\). The distinct points \({\text{U}}\left( {a{u^2},{\text{ }}2au} \right)\) and \({\text{V}}\left( {a{v^2},{\text{ }}2av} \right)\) lie on \(P\), where \(u,{\text{ }}v \ne 0\). Given that \({\rm{U\hat OV}}\) is </span><span style="font-family: 'times new roman', times; font-size: medium;">a right angle, where \({\text{O}}\) denotes the origin,</span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">(a) show that \(v = - \frac{4}{\mu }\);</span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">(b) find expressions for the coordinates of \({\text{W}}\), the midpoint of \([{\text{UV}}]\), in terms of \(a\) and \(u\);</span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">(c) show that the locus of \({\text{W}}\), as \(u\) varies, is the parabola \({P'}\) with equation \({y^2} = 2ax - 8{a^2}\);</span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">(d) determine the coordinates of the vertex of \({P'}\).</span></p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Prove the internal angle bisector theorem, namely that the internal bisector of an </span><span style="font-family: times new roman,times; font-size: medium;">angle of a triangle divides the side opposite the angle into segments proportional </span><span style="font-family: times new roman,times; font-size: medium;">to the sides adjacent to the angle.</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The bisector of the exterior angle \(\widehat A\) of the triangle ABC meets (BC) at P. </span><span style="font-family: times new roman,times; font-size: medium;">The bisector of the interior angle \(\widehat B\) meets [AC] at Q. Given that (PQ) meets </span><span style="font-family: times new roman,times; font-size: medium;">[AB] at R, use Menelaus’ theorem to prove that (CR) bisects the angle \({\rm{A}}\widehat {\rm{C}}{\rm{B}}\) .</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Triangle ABC has points D, E and F on sides [BC], [CA] and [AB] respectively; </span><span style="font-family: times new roman,times; font-size: medium;">[AD], [BE] and [CF] intersect at the point P. If 3BD = 2DC and CE = 4EA , </span><span style="font-family: times new roman,times; font-size: medium;">calculate the ratios</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">AF : FB .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">AP : PD</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;"><img style="display: block; margin-left: auto; margin-right: auto;" src="images/energy.png" alt></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The figure shows a circle C<sub>1</sub> with centre O and diameter [PQ] and a circle C<sub>2</sub> which </span><span style="font-family: times new roman,times; font-size: medium;">intersects (PQ) at the points R and S. T is one point of intersection of the two circles </span><span style="font-family: times new roman,times; font-size: medium;">and (OT) is a tangent to C<sub>2</sub> .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Show that \(\frac{{{\rm{OR}}}}{{{\rm{OT}}}} = \frac{{{\rm{OT}}}}{{{\rm{OS}}}}\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) Show that \({\rm{PR}} - {\rm{RQ}} = 2{\rm{OR}}\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Show that \(\frac{{{\rm{PR}} - {\rm{RQ}}}}{{{\rm{PR}} + {\rm{RQ}}}} = \frac{{{\rm{PS}} - {\rm{SQ}}}}{{{\rm{PS}} + {\rm{SQ}}}}\) .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A triangle \(T\) has sides of length \(3\), \(4\) and \(5\).</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (i) Find the radius of the circumscribed circle of \(T\) .</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (ii) Find the radius of the inscribed circle of \(T\) .</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A triangle \(U\) has sides of length \(4\), \(5\) and \(7\).</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (i) Show that the orthocentre, H, of \(U\) lies outside the triangle.</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (ii) Show that the foot of the perpendicular from H to the longest side divides </span><span style="font-family: times new roman,times; font-size: medium;">it in the ratio \(29:20\).</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1"><span class="s1">A wheel of radius \(r\) </span>rolls, without slipping, along a straight path with the plane of the wheel remaining vertical. A point \({\text{A}}\) on the circumference of the wheel is initially at \({\text{O}}\)<span class="s1">. When the wheel is rolled, the radius rotates through an angle of \(\theta \) </span>and the point of contact is now at \({\text{B}}\), where the length of the arc \({\text{AB}}\) is equal to the distance \({\text{OB}}\)<span class="s1">. This is shown in the following diagram.</span></p>
<p class="p2" style="text-align: center;"><img src="images/Schermafbeelding_2015-12-15_om_09.43.46.png" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">Find the coordinates of \({\text{A}}\) </span>in terms of \(r\) and \(\theta \).</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"><span class="s1">As the wheel rolls, the point A </span>traces out a curve. Show that the gradient of this curve is \(\cot \left( {\frac{1}{2}\theta } \right)\).</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 equation of the tangent to the curve when \(\theta = \frac{\pi }{3}\).</p>
<div class="marks">[3]</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: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) The function \(g\) is defined by \(g(x,{\text{ }}y) = {x^2} + {y^2} + dx + ey + f\) and the circle \({C_1}\) has equation \(g(x, y) = 0\).</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;">(i) Show that the centre of \({C_1}\) has coordinates \(\left( { - \frac{d}{2}, - \frac{e}{2}} \right)\) and the radius of \({C_1}\) is \(\sqrt {\frac{{{d^2}}}{4} + \frac{{{e^2}}}{4} - f} \).</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;">(ii) The point \({\text{P}}(a, b)\) lies outside \({C_1}\). Show that the length of the tangents from \({\text{P}}\) to \({C_1}\) is equal to \(\sqrt {g(a,{\text{ }}b)} \).</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) The circle \({C_2}\) has equation \({x^2} + {y^2} - 6x - 2y + 6 = 0\).</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;">The line \(y = mx\) meets \({C_2}\) at the points </span><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{R}}\) and \({\text{S}}\).</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;">(i) Determine the quadratic equation whose roots are the <em>x</em>-coordinates of </span><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{R}}\) and \({\text{S}}\).</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;">(ii) <strong>Hence</strong>, given that \(L\) denotes the length of the tangents from the origin </span><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{O}}\) to \({C_2}\), show that \({\text{OR}} \times {\text{OS}} = {L^2}\).</span></p>
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<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The diagram below shows a quadrilateral ABCD and a straight line which intersects </span><span style="font-family: times new roman,times; font-size: medium;">(AB), (BC), (CD), (DA) at the points P, Q, R, S respectively.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/puck.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Using Menelaus’ theorem, show that \(\frac{{{\rm{AP}}}}{{{\rm{PB}}}} \times \frac{{{\rm{BQ}}}}{{{\rm{QC}}}} \times \frac{{{\rm{CR}}}}{{{\rm{RD}}}} \times \frac{{{\rm{DS}}}}{{{\rm{SA}}}} = 1\) .</span></p>
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<p class="p1">A circle \({x^2} + {y^2} + dx + ey + c = 0\) and a straight line \(lx + my + n = 0\) intersect. Find the general equation of a circle which passes through the points of intersection, justifying your answer.</p>
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<p class="p1">Two line segments [\(\rm{AB}\)] and [\(\rm{CD}\)] meet internally at the point \(\rm{Y}\). Given that</p>
<p class="p1">\({\text{YA}} \times {\text{YB}} = {\text{YC}} \times {\text{YD }}\) show that \(\rm{A}\), \(\rm{B}\), \(\rm{C}\) and \(\rm{D}\) <span class="s1">all lie on the circumference of a circle.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
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<p class="p1">Explain why the result also holds if the line segments meet externally at \(\rm{Y}\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
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<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The rectangle ABCD is inscribed in a circle. Sides [AD] and [AB] have lengths \(3\) cm </span><span style="font-family: times new roman,times; font-size: medium;">and (\9\) cm respectively. E is a point on side [AB] such that AE is \(3\) cm. Side [DE] is </span><span style="font-family: times new roman,times; font-size: medium;">produced to meet the circumcircle of ABCD at point P. Use Ptolemy’s theorem to </span><span style="font-family: times new roman,times; font-size: medium;">calculate the length of chord [AP].</span></p>
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