CAIE P3 (Pure Mathematics 3) 2024 June

1

1. Sketch the graph of \(\mathrm { y } = | \mathrm { x } - 2 \mathrm { a } |\), where \(a\) is a positive constant.
2. Solve the inequality \(2 \mathrm { x } - 3 \mathrm { a } < | \mathrm { x } - 2 \mathrm { a } |\).

2 Express \(\frac { 6 x ^ { 2 } - 9 x - 16 } { 2 x ^ { 2 } - 5 x - 12 }\) in partial fractions.

3 The variables \(x\) and \(y\) satisfy the equation \(\mathrm { a } ^ { 2 \mathrm { y } - 1 } = \mathrm { b } ^ { \mathrm { x } - \mathrm { y } }\), where \(a\) and \(b\) are constants.

1. Show that the graph of \(y\) against \(x\) is a straight line.
2. Given that \(\mathrm { a } = \mathrm { b } ^ { 3 }\), state the equation of the straight line in the form \(\mathrm { y } = \mathrm { px } + \mathrm { q }\), where \(p\) and \(q\) are rational numbers in their simplest form.

4 The equation of a curve is \(\mathrm { ye } ^ { 2 \mathrm { x } } + \mathrm { y } ^ { 2 } \mathrm { e } ^ { \mathrm { x } } = 6\).
Find the gradient of the curve at the point where \(y = 1\).

5

1. It is given that the equation \(\mathrm { e } ^ { 2 x } = 5 + \cos 3 x\) has only one root.
   Show by calculation that this root lies in the interval \(0.7 < x < 0.8\).
2. Show that if a sequence of values in the interval \(0.7 < x < 0.8\) given by the iterative formula $$x _ { n + 1 } = \frac { 1 } { 2 } \ln \left( 5 + \cos 3 x _ { n } \right)$$ converges then it converges to the root of the equation in part (a).
3. Use this iterative formula to determine the root correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

6 \includegraphics[max width=\textwidth, alt={}, center]{5eb2657c-ed74-4ed2-b8c4-08e9e0f657b5-08_351_1031_264_516} The diagram shows the curve \(\mathrm { y } = \mathrm { xe } ^ { - \mathrm { ax } }\), where \(a\) is a positive constant, and its maximum point \(M\).

1. Find the exact coordinates of \(M\).
2. Find the exact value of \(\int _ { 0 } ^ { \frac { 2 } { a } } x e ^ { - a x } d x\).

7

1. Show that \(\cos ^ { 4 } \theta - \sin ^ { 4 } \theta \equiv \cos 2 \theta\).
2. Hence find the exact value of \(\int _ { - \frac { 1 } { 8 } \pi } ^ { \frac { 1 } { 8 } \pi } \left( \cos ^ { 4 } \theta - \sin ^ { 4 } \theta + 4 \sin ^ { 2 } \theta \cos ^ { 2 } \theta \right) \mathrm { d } \theta\).

8 The points \(A , B\) and \(C\) have position vectors \(\overrightarrow { \mathrm { OA } } = - 2 \mathbf { i } + \mathbf { j } + 4 \mathbf { k } , \overrightarrow { \mathrm { OB } } = 5 \mathbf { i } + 2 \mathbf { j }\) and \(\overrightarrow { \mathrm { OC } } = 8 \mathbf { i } + 5 \mathbf { j } - 3 \mathbf { k }\), where \(O\) is the origin. The line \(l _ { 1 }\) passes through \(B\) and \(C\).

1. Find a vector equation for \(l _ { 1 }\).
   The line \(l _ { 2 }\) has equation \(\mathbf { r } = - 2 \mathbf { i } + \mathbf { j } + 4 \mathbf { k } + \mu ( 3 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } )\).
2. Find the coordinates of the point of intersection of \(l _ { 1 }\) and \(l _ { 2 }\).
3. The point \(D\) on \(l _ { 2 }\) is such that \(\mathrm { AB } = \mathrm { BD }\). Find the position vector of \(D\). \includegraphics[max width=\textwidth, alt={}, center]{5eb2657c-ed74-4ed2-b8c4-08e9e0f657b5-13_58_1545_388_349}

9 The complex numbers \(z\) and \(\omega\) are defined by \(z = 1 - i\) and \(\omega = - 3 + 3 \sqrt { 3 } i\).

1. Express \(z \omega\) in the form \(\mathrm { a } + \mathrm { bi }\), where \(a\) and \(b\) are real and in exact surd form.
2. Express \(z\) and \(\omega\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\). Give the exact values of \(r\) and \(\theta\) in each case.
3. On an Argand diagram, the points representing \(\omega\) and \(z \omega\) are \(A\) and \(B\) respectively. Prove that \(O A B\) is an isosceles right-angled triangle, where \(O\) is the origin.
4. Using your answers to part (b), prove that \(\tan \frac { 5 } { 12 } \pi = \frac { \sqrt { 3 } + 1 } { \sqrt { 3 } - 1 }\).

10

1. By writing \(y = \sec ^ { 3 } \theta\) as \(\frac { 1 } { \cos ^ { 3 } \theta }\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} \theta } = 3 \sin \theta \sec ^ { 4 } \theta\).
2. The variables \(x\) and \(\theta\) satisfy the differential equation $$\left( x ^ { 2 } + 9 \right) \sin \theta \frac { d \theta } { d x } = ( x + 3 ) \cos ^ { 4 } \theta$$ It is given that \(x = 3\) when \(\theta = \frac { 1 } { 3 } \pi\).
   Solve the differential equation to find the value of \(\cos \theta\) when \(x = 0\). Give your answer correct to 3 significant figures.
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