CAIE P3 (Pure Mathematics 3) 2021 June

1 Solve the inequality \(2 | 3 x - 1 | < | x + 1 |\).

2 Find the real root of the equation \(\frac { 2 \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } } { 2 + \mathrm { e } ^ { x } } = 3\), giving your answer correct to 3 decimal places. Your working should show clearly that the equation has only one real root.

3

1. Given that \(\cos \left( x - 30 ^ { \circ } \right) = 2 \sin \left( x + 30 ^ { \circ } \right)\), show that \(\tan x = \frac { 2 - \sqrt { 3 } } { 1 - 2 \sqrt { 3 } }\).
2. Hence solve the equation $$\cos \left( x - 30 ^ { \circ } \right) = 2 \sin \left( x + 30 ^ { \circ } \right)$$ for \(0 ^ { \circ } < x < 360 ^ { \circ }\).

4

1. Prove that \(\frac { 1 - \cos 2 \theta } { 1 + \cos 2 \theta } \equiv \tan ^ { 2 } \theta\).
2. Hence find the exact value of \(\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 3 } \pi } \frac { 1 - \cos 2 \theta } { 1 + \cos 2 \theta } \mathrm {~d} \theta\).

5

1. Solve the equation \(z ^ { 2 } - 2 p \mathrm { i } z - q = 0\), where \(p\) and \(q\) are real constants.
   In an Argand diagram with origin \(O\), the roots of this equation are represented by the distinct points \(A\) and \(B\).
2. Given that \(A\) and \(B\) lie on the imaginary axis, find a relation between \(p\) and \(q\).
3. Given instead that triangle \(O A B\) is equilateral, express \(q\) in terms of \(p\).

6 The parametric equations of a curve are $$x = \ln ( 2 + 3 t ) , \quad y = \frac { t } { 2 + 3 t }$$

1. Show that the gradient of the curve is always positive.
2. Find the equation of the tangent to the curve at the point where it intersects the \(y\)-axis.

7 \includegraphics[max width=\textwidth, alt={}, center]{a257f49d-5c8f-4c23-be78-46619b746fde-10_353_689_262_726} The diagram shows the curve \(y = \frac { \tan ^ { - 1 } x } { \sqrt { x } }\) and its maximum point \(M\) where \(x = a\).

1. Show that \(a\) satisfies the equation $$a = \tan \left( \frac { 2 a } { 1 + a ^ { 2 } } \right)$$
2. Verify by calculation that \(a\) lies between 1.3 and 1.5.
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.

8 With respect to the origin \(O\), the points \(A\) and \(B\) have position vectors given by \(\overrightarrow { O A } = \left( \begin{array} { l } 1 \\ 2 \\ 1 \end{array} \right)\) and \(\overrightarrow { O B } = \left( \begin{array} { r } 3 \\ 1 \\ - 2 \end{array} \right)\). The line \(l\) has equation \(\mathbf { r } = \left( \begin{array} { l } 2 \\ 3 \\ 1 \end{array} \right) + \lambda \left( \begin{array} { r } 1 \\ - 2 \\ 1 \end{array} \right)\).

1. Find the acute angle between the directions of \(A B\) and \(l\).
2. Find the position vector of the point \(P\) on \(l\) such that \(A P = B P\).

9 The equation of a curve is \(y = x ^ { - \frac { 2 } { 3 } } \ln x\) for \(x > 0\). The curve has one stationary point.

1. Find the exact coordinates of the stationary point.
2. Show that \(\int _ { 1 } ^ { 8 } y \mathrm {~d} x = 18 \ln 2 - 9\).

10 The variables \(x\) and \(t\) satisfy the differential equation \(\frac { \mathrm { d } x } { \mathrm {~d} t } = x ^ { 2 } ( 1 + 2 x )\), and \(x = 1\) when \(t = 0\).
Using partial fractions, solve the differential equation, obtaining an expression for \(t\) in terms of \(x\).
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.