CAIE P3 (Pure Mathematics 3) 2022 June

1 Solve the equation \(2 \left( 3 ^ { 2 x - 1 } \right) = 4 ^ { x + 1 }\), giving your answer correct to 2 decimal places.

2

1. Expand \(\left( 2 - x ^ { 2 } \right) ^ { - 2 }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 4 }\), simplifying the coefficients.
2. State the set of values of \(x\) for which the expansion is valid.

3 Solve the equation \(2 \cot 2 x + 3 \cot x = 5\), for \(0 ^ { \circ } < x < 180 ^ { \circ }\).

4 The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x y } { 1 + x ^ { 2 } }$$ and \(y = 2\) when \(x = 0\).
Solve the differential equation, obtaining a simplified expression for \(y\) in terms of \(x\).

5 The polynomial \(a x ^ { 3 } - 10 x ^ { 2 } + b x + 8\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( x - 2 )\) is a factor of both \(\mathrm { p } ( x )\) and \(\mathrm { p } ^ { \prime } ( x )\).

1. Find the values of \(a\) and \(b\).
2. When \(a\) and \(b\) have these values, factorise \(\mathrm { p } ( x )\) completely.

6 Let \(I = \int _ { 0 } ^ { 3 } \frac { 27 } { \left( 9 + x ^ { 2 } \right) ^ { 2 } } \mathrm {~d} x\).

1. Using the substitution \(x = 3 \tan \theta\), show that \(I = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \cos ^ { 2 } \theta \mathrm {~d} \theta\).
2. Hence find the exact value of \(I\).

7 The complex number \(u\) is defined by \(u = \frac { \sqrt { 2 } - a \sqrt { 2 } \mathrm { i } } { 1 + 2 \mathrm { i } }\), where \(a\) is a positive integer.

1. Express \(u\) in terms of \(a\), in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact.
   It is now given that \(a = 3\).
2. Express \(u\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\), giving the exact values of \(r\) and \(\theta\).
3. Using your answer to part (b), find the two square roots of \(u\). Give your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\), giving the exact values of \(r\) and \(\theta\).

8 The equation of a curve is \(x ^ { 3 } + y ^ { 3 } + 2 x y + 8 = 0\).

1. Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
   The tangent to the curve at the point where \(x = 0\) and the tangent at the point where \(y = 0\) intersect at the acute angle \(\alpha\).
2. Find the exact value of \(\tan \alpha\).

9
\includegraphics[max width=\textwidth, alt={}, center]{c1fbc9ef-2dc6-43c3-bc58-179f683c9acf-16_696_1104_264_518} In the diagram, \(O A B C D E F G\) is a cuboid in which \(O A = 2\) units, \(O C = 4\) units and \(O G = 2\) units. Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O C\) and \(O G\) respectively. The point \(M\) is the midpoint of \(D F\). The point \(N\) on \(A B\) is such that \(A N = 3 N B\).

1. Express the vectors \(\overrightarrow { O M }\) and \(\overrightarrow { M N }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
2. Find a vector equation for the line through \(M\) and \(N\).
3. Show that the length of the perpendicular from \(O\) to the line through \(M\) and \(N\) is \(\sqrt { \frac { 53 } { 6 } }\).

10
\includegraphics[max width=\textwidth, alt={}, center]{c1fbc9ef-2dc6-43c3-bc58-179f683c9acf-18_471_686_276_717} The curve \(y = x \sqrt { \sin x }\) has one stationary point in the interval \(0 < x < \pi\), where \(x = a\) (see diagram).

1. Show that \(\tan a = - \frac { 1 } { 2 } a\).
2. Verify by calculation that \(a\) lies between 2 and 2.5.
3. Show that if a sequence of values in the interval \(0 < x < \pi\) given by the iterative formula \(x _ { n + 1 } = \pi - \tan ^ { - 1 } \left( \frac { 1 } { 2 } x _ { n } \right)\) converges, then it converges to \(a\), the root of the equation in part (a). [2]
4. Use the iterative formula given in part (c) to determine \(a\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.