CAIE P3 (Pure Mathematics 3) 2016 June

1 Use logarithms to solve the equation \(4 ^ { 3 x - 1 } = 3 \left( 5 ^ { x } \right)\), giving your answer correct to 3 decimal places.

2 Expand \(\frac { 1 } { \sqrt { ( 1 - 2 x ) } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), simplifying the coefficients.

3 Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } x ^ { 2 } \sin 2 x \mathrm {~d} x\).

4 The curve with equation \(y = \frac { ( \ln x ) ^ { 2 } } { x }\) has two stationary points. Find the exact values of the coordinates of these points.

5

1. Prove the identity \(\cos 4 \theta - 4 \cos 2 \theta \equiv 8 \sin ^ { 4 } \theta - 3\).
2. Hence solve the equation $$\cos 4 \theta = 4 \cos 2 \theta + 3$$ for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

6 The variables \(x\) and \(\theta\) satisfy the differential equation $$( 3 + \cos 2 \theta ) \frac { \mathrm { d } x } { \mathrm {~d} \theta } = x \sin 2 \theta$$ and it is given that \(x = 3\) when \(\theta = \frac { 1 } { 4 } \pi\).

1. Solve the differential equation and obtain an expression for \(x\) in terms of \(\theta\).
2. State the least value taken by \(x\).

7 Let \(\mathrm { f } ( x ) = \frac { 4 x ^ { 2 } + 7 x + 4 } { ( 2 x + 1 ) ( x + 2 ) }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Show that \(\int _ { 0 } ^ { 4 } \mathrm { f } ( x ) \mathrm { d } x = 8 - \ln 3\).

8
\includegraphics[max width=\textwidth, alt={}, center]{9d3af34a-670f-425e-8156-0ad4d08fbdc0-3_499_552_258_792} The diagram shows the curve \(y = \operatorname { cosec } x\) for \(0 < x < \pi\) and part of the curve \(y = \mathrm { e } ^ { - x }\). When \(x = a\), the tangents to the curves are parallel.

1. By differentiating \(\frac { 1 } { \sin x }\), show that if \(y = \operatorname { cosec } x\) then \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \operatorname { cosec } x \cot x\).
2. By equating the gradients of the curves at \(x = a\), show that $$a = \tan ^ { - 1 } \left( \frac { \mathrm { e } ^ { a } } { \sin a } \right)$$
3. Verify by calculation that \(a\) lies between 1 and 1.5.
4. Use an iterative formula based on the equation in part (ii) to determine \(a\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

9 The points \(A , B\) and \(C\) have position vectors, relative to the origin \(O\), given by \(\overrightarrow { O A } = \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k }\), \(\overrightarrow { O B } = 4 \mathbf { j } + \mathbf { k }\) and \(\overrightarrow { O C } = 2 \mathbf { i } + 5 \mathbf { j } - \mathbf { k }\). A fourth point \(D\) is such that the quadrilateral \(A B C D\) is a parallelogram.

1. Find the position vector of \(D\) and verify that the parallelogram is a rhombus.
2. The plane \(p\) is parallel to \(O A\) and the line \(B C\) lies in \(p\). Find the equation of \(p\), giving your answer in the form \(a x + b y + c z = d\).

10

1. Showing all necessary working, solve the equation \(\mathrm { i } z ^ { 2 } + 2 z - 3 \mathrm { i } = 0\), giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact.
2. 1. On a sketch of an Argand diagram, show the locus representing complex numbers satisfying the equation \(| z | = | z - 4 - 3 \mathrm { i } |\).
   2. Find the complex number represented by the point on the locus where \(| z |\) is least. Find the modulus and argument of this complex number, giving the argument correct to 2 decimal places.