CAIE P3 (Pure Mathematics 3) 2022 June

1 Find, in terms of \(a\), the set of values of \(x\) satisfying the inequality $$2 | 3 x + a | < | 2 x + 3 a |$$ where \(a\) is a positive constant.

2 Solve the equation \(\cos \left( \theta - 60 ^ { \circ } \right) = 3 \sin \theta\), for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

3

1. Show that the equation \(\log _ { 3 } ( 2 x + 1 ) = 1 + 2 \log _ { 3 } ( x - 1 )\) can be written as a quadratic equation in \(x\).
2. Hence solve the equation \(\log _ { 3 } ( 4 y + 1 ) = 1 + 2 \log _ { 3 } ( 2 y - 1 )\), giving your answer correct to 2 decimal places.

4 The curve \(y = \mathrm { e } ^ { - 4 x } \tan x\) has two stationary points in the interval \(0 \leqslant x < \frac { 1 } { 2 } \pi\).

1. Obtain an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and show it can be written in the form \(\sec ^ { 2 } x ( a + b \sin 2 x ) \mathrm { e } ^ { - 4 x }\), where \(a\) and \(b\) are constants.
2. Hence find the exact \(x\)-coordinates of the two stationary points.

5 The complex number \(3 - \mathrm { i }\) is denoted by \(u\).

1. Show, on an Argand diagram with origin \(O\), the points \(A , B\) and \(C\) representing the complex numbers \(u , u ^ { * }\) and \(u ^ { * } - u\) respectively. State the type of quadrilateral formed by the points \(O , A , B\) and \(C\).
2. Express \(\frac { u ^ { * } } { u }\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
3. By considering the argument of \(\frac { u ^ { * } } { u }\), or otherwise, prove that \(\tan ^ { - 1 } \left( \frac { 3 } { 4 } \right) = 2 \tan ^ { - 1 } \left( \frac { 1 } { 3 } \right)\).

6 The parametric equations of a curve are \(x = \frac { 1 } { \cos t } , y = \ln \tan t\), where \(0 < t < \frac { 1 } { 2 } \pi\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { \cos t } { \sin ^ { 2 } t }\).
2. Find the equation of the tangent to the curve at the point where \(y = 0\).

7 Let \(\mathrm { f } ( x ) = \frac { 5 x ^ { 2 } + 8 x - 3 } { ( x - 2 ) \left( 2 x ^ { 2 } + 3 \right) }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Hence obtain the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).

8 At time \(t\) days after the start of observations, the number of insects in a population is \(N\). The variation in the number of insects is modelled by a differential equation of the form \(\frac { \mathrm { d } N } { \mathrm {~d} t } = k N ^ { \frac { 3 } { 2 } } \cos 0.02 t\), where \(k\) is a constant and \(N\) is a continuous variable. It is given that when \(t = 0 , N = 100\).

1. Solve the differential equation, obtaining a relation between \(N , k\) and \(t\).
2. Given also that \(N = 625\) when \(t = 50\), find the value of \(k\).
3. Obtain an expression for \(N\) in terms of \(t\), and find the greatest value of \(N\) predicted by this model.

9 With respect to the origin \(O\), the point \(A\) has position vector given by \(\overrightarrow { O A } = \mathbf { i } + 5 \mathbf { j } + 6 \mathbf { k }\). The line \(l\) has vector equation \(\mathbf { r } = 4 \mathbf { i } + \mathbf { k } + \lambda ( - \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } )\).

1. Find in degrees the acute angle between the directions of \(O A\) and \(l\).
2. Find the position vector of the foot of the perpendicular from \(A\) to \(l\).
3. Hence find the position vector of the reflection of \(A\) in \(l\).

10 The constant \(a\) is such that \(\int _ { 1 } ^ { a } x ^ { 2 } \ln x \mathrm {~d} x = 4\).

1. Show that \(a = \left( \frac { 35 } { 3 \ln a - 1 } \right) ^ { \frac { 1 } { 3 } }\).
2. Verify by calculation that \(a\) lies between 2.4 and 2.8.
3. Use an iterative formula based on the equation in part (a) to determine \(a\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
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