CAIE P3 (Pure Mathematics 3) 2021 November

1 Find the value of \(x\) for which \(3 \left( 2 ^ { 1 - x } \right) = 7 ^ { x }\). Give your answer in the form \(\frac { \ln a } { \ln b }\), where \(a\) and \(b\) are integers.

2 Solve the inequality \(| 3 x - a | > 2 | x + 2 a |\), where \(a\) is a positive constant.

3

1. Given the complex numbers \(u = a + \mathrm { i } b\) and \(w = c + \mathrm { i } d\), where \(a , b , c\) and \(d\) are real, prove that \(( u + w ) ^ { * } = u ^ { * } + w ^ { * }\).
2. Solve the equation \(( z + 2 + \mathrm { i } ) ^ { * } + ( 2 + \mathrm { i } ) z = 0\), giving your answer in the form \(x + \mathrm { i } y\) where \(x\) and \(y\) are real.

4 Express \(\frac { 4 x ^ { 2 } - 13 x + 13 } { ( 2 x - 1 ) ( x - 3 ) }\) in partial fractions.

5

1. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z - 3 - 2 \mathbf { i } | \leqslant 1\) and \(\operatorname { Im } z \geqslant 2\).
2. Find the greatest value of \(\arg z\) for points in the shaded region, giving your answer in degrees.

6

1. Using the expansions of \(\sin ( 3 x + 2 x )\) and \(\sin ( 3 x - 2 x )\), show that $$\frac { 1 } { 2 } ( \sin 5 x + \sin x ) \equiv \sin 3 x \cos 2 x$$
2. Hence show that \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sin 3 x \cos 2 x \mathrm {~d} x = \frac { 1 } { 5 } ( 3 - \sqrt { 2 } )\).

7 The variables \(x\) and \(y\) satisfy the differential equation $$\mathrm { e } ^ { 2 x } \frac { \mathrm {~d} y } { \mathrm {~d} x } = 4 x y ^ { 2 }$$ and it is given that \(y = 1\) when \(x = 0\).
Solve the differential equation, obtaining an expression for \(y\) in terms of \(x\).

8

1. By first expanding \(\left( \cos ^ { 2 } \theta + \sin ^ { 2 } \theta \right) ^ { 2 }\), show that $$\cos ^ { 4 } \theta + \sin ^ { 4 } \theta \equiv 1 - \frac { 1 } { 2 } \sin ^ { 2 } 2 \theta .$$
2. Hence solve the equation $$\cos ^ { 4 } \theta + \sin ^ { 4 } \theta = \frac { 5 } { 9 } ,$$ for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).

9 The equation of a curve is \(y \mathrm { e } ^ { 2 x } - y ^ { 2 } \mathrm { e } ^ { x } = 2\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 y \mathrm { e } ^ { x } - y ^ { 2 } } { 2 y - \mathrm { e } ^ { x } }\).
2. Find the exact coordinates of the point on the curve where the tangent is parallel to the \(y\)-axis.

10 With respect to the origin \(O\), the position vectors of the points \(A\) and \(B\) are given by \(\overrightarrow { O A } = \left( \begin{array} { r } 1
2
- 1 \end{array} \right)\) and \(\overrightarrow { O B } = \left( \begin{array} { l } 0
3
1 \end{array} \right)\).

1. Find a vector equation for the line \(l\) through \(A\) and \(B\).
2. The point \(C\) lies on \(l\) and is such that \(\overrightarrow { A C } = 3 \overrightarrow { A B }\). Find the position vector of \(C\).
3. Find the possible position vectors of the point \(P\) on \(l\) such that \(O P = \sqrt { 14 }\).

11 The equation of a curve is \(y = \sqrt { \tan x }\), for \(0 \leqslant x < \frac { 1 } { 2 } \pi\).

1. Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\tan x\), and verify that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1\) when \(x = \frac { 1 } { 4 } \pi\).
   The value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) is also 1 at another point on the curve where \(x = a\), as shown in the diagram.
   \includegraphics[max width=\textwidth, alt={}, center]{87be326f-f638-43e9-a654-b7b53d5141ef-18_605_492_1493_822}
2. Show that \(t ^ { 3 } + t ^ { 2 } + 3 t - 1 = 0\), where \(t = \tan a\).
3. Use the iterative formula $$a _ { n + 1 } = \tan ^ { - 1 } \left( \frac { 1 } { 3 } \left( 1 - \tan ^ { 2 } a _ { n } - \tan ^ { 3 } a _ { n } \right) \right)$$ to determine \(a\) correct to 2 decimal places, giving 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.