CAIE P3 (Pure Mathematics 3) 2018 November

1 Solve the inequality \(3 | 2 x - 1 | > | x + 4 |\).

2 Showing all necessary working, solve the equation \(\sin \left( \theta - 30 ^ { \circ } \right) + \cos \theta = 2 \sin \theta\), for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\). [4]

3

1. Find \(\int \frac { \ln x } { x ^ { 3 } } \mathrm {~d} x\).
2. Hence show that \(\int _ { 1 } ^ { 2 } \frac { \ln x } { x ^ { 3 } } \mathrm {~d} x = \frac { 1 } { 16 } ( 3 - \ln 4 )\).

4 Showing all necessary working, solve the equation $$\frac { \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } } { \mathrm { e } ^ { x } + 1 } = 4$$ giving your answer correct to 3 decimal places.

5 The equation of a curve is \(y = x \ln ( 8 - x )\). The gradient of the curve is equal to 1 at only one point, when \(x = a\).

1. Show that \(a\) satisfies the equation \(x = 8 - \frac { 8 } { \ln ( 8 - x ) }\).
2. Verify by calculation that \(a\) lies between 2.9 and 3.1.
3. Use an iterative formula based on the equation in part (i) to determine \(a\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

6 A certain curve is such that its gradient at a general point with coordinates \(( x , y )\) is proportional to \(\frac { y ^ { 2 } } { x }\). The curve passes through the points with coordinates \(( 1,1 )\) and (e, 2). By setting up and solving a differential equation, find the equation of the curve, expressing \(y\) in terms of \(x\).

7 A curve has equation \(y = \frac { 3 \cos x } { 2 + \sin x }\), for \(- \frac { 1 } { 2 } \pi \leqslant x \leqslant \frac { 1 } { 2 } \pi\).

1. Find the exact coordinates of the stationary point of the curve.
2. The constant \(a\) is such that \(\int _ { 0 } ^ { a } \frac { 3 \cos x } { 2 + \sin x } \mathrm {~d} x = 1\). Find the value of \(a\), giving your answer correct to 3 significant figures.
   \(8 \quad\) Let \(\mathrm { f } ( x ) = \frac { 7 x ^ { 2 } - 15 x + 8 } { ( 1 - 2 x ) ( 2 - x ) ^ { 2 } }\).

9

1. 1. Without using a calculator, express the complex number \(\frac { 2 + 6 \mathrm { i } } { 1 - 2 \mathrm { i } }\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
   2. Hence, without using a calculator, express \(\frac { 2 + 6 \mathrm { i } } { 1 - 2 \mathrm { i } }\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\), giving the exact values of \(r\) and \(\theta\).
2. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying both the inequalities \(| z - 3 \mathrm { i } | \leqslant 1\) and \(\operatorname { Re } z \leqslant 0\), where \(\operatorname { Re } z\) denotes the real part of \(z\). Find the greatest value of \(\arg z\) for points in this region, giving your answer in radians correct to 2 decimal places.

10 The line \(l\) has equation \(\mathbf { r } = 5 \mathbf { i } - 3 \mathbf { j } - \mathbf { k } + \lambda ( \mathbf { i } - 2 \mathbf { j } + \mathbf { k } )\). The plane \(p\) has equation $$( \mathbf { r } - \mathbf { i } - 2 \mathbf { j } ) \cdot ( 3 \mathbf { i } + \mathbf { j } + \mathbf { k } ) = 0$$ The line \(l\) intersects the plane \(p\) at the point \(A\).

1. Find the position vector of \(A\).
2. Calculate the acute angle between \(l\) and \(p\).
3. Find the equation of the line which lies in \(p\) and intersects \(l\) at right angles.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.