CAIE P3 (Pure Mathematics 3) 2017 June

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

2 Expand \(\frac { 1 } { \sqrt [ 3 ] { } ( 1 + 6 x ) }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), simplifying the coefficients.

3 It is given that \(x = \ln ( 1 - y ) - \ln y\), where \(0 < y < 1\).

1. Show that \(y = \frac { \mathrm { e } ^ { - x } } { 1 + \mathrm { e } ^ { - x } }\).
2. Hence show that \(\int _ { 0 } ^ { 1 } y \mathrm {~d} x = \ln \left( \frac { 2 \mathrm { e } } { \mathrm { e } + 1 } \right)\).

4 The parametric equations of a curve are $$x = \ln \cos \theta , \quad y = 3 \theta - \tan \theta ,$$ where \(0 \leqslant \theta < \frac { 1 } { 2 } \pi\).

1. Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\tan \theta\).
2. Find the exact \(y\)-coordinate of the point on the curve at which the gradient of the normal is equal to 1 .
   \includegraphics[max width=\textwidth, alt={}, center]{b00cefad-7c3c-4672-b309-f19aafab8b01-08_378_689_260_726} The diagram shows a semicircle with centre \(O\), radius \(r\) and diameter \(A B\). The point \(P\) on its circumference is such that the area of the minor segment on \(A P\) is equal to half the area of the minor segment on \(B P\). The angle \(A O P\) is \(x\) radians.

1. Show that \(x\) satisfies the equation \(x = \frac { 1 } { 3 } ( \pi + \sin x )\).
2. Verify by calculation that \(x\) lies between 1 and 1.5.
3. Use an iterative formula based on the equation in part (i) to determine \(x\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

6 The plane with equation \(2 x + 2 y - z = 5\) is denoted by \(m\). Relative to the origin \(O\), the points \(A\) and \(B\) have coordinates \(( 3,4,0 )\) and \(( - 1,0,2 )\) respectively.

1. Show that the plane \(m\) bisects \(A B\) at right angles.
   A second plane \(p\) is parallel to \(m\) and nearer to \(O\). The perpendicular distance between the planes is 1 .
2. Find the equation of \(p\), giving your answer in the form \(a x + b y + c z = d\).

7 Throughout this question the use of a calculator is not permitted.
The complex numbers \(u\) and \(w\) are defined by \(u = - 1 + 7 \mathrm { i }\) and \(w = 3 + 4 \mathrm { i }\).

1. Showing all your working, find in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real, the complex numbers \(u - 2 w\) and \(\frac { u } { w }\).
   In an Argand diagram with origin \(O\), the points \(A , B\) and \(C\) represent the complex numbers \(u , w\) and \(u - 2 w\) respectively.
2. Prove that angle \(A O B = \frac { 1 } { 4 } \pi\).
3. State fully the geometrical relation between the line segments \(O B\) and \(C A\).

8

1. By first expanding \(2 \sin \left( x - 30 ^ { \circ } \right)\), express \(2 \sin \left( x - 30 ^ { \circ } \right) - \cos x\) in the form \(R \sin ( x - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the exact value of \(R\) and the value of \(\alpha\) correct to 2 decimal places. [5]
2. Hence solve the equation $$2 \sin \left( x - 30 ^ { \circ } \right) - \cos x = 1$$ for \(0 ^ { \circ } < x < 180 ^ { \circ }\).

9

1. Express \(\frac { 1 } { x ( 2 x + 3 ) }\) in partial fractions.
2. The variables \(x\) and \(y\) satisfy the differential equation $$x ( 2 x + 3 ) \frac { \mathrm { d } y } { \mathrm {~d} x } = y$$ and it is given that \(y = 1\) when \(x = 1\). Solve the differential equation and calculate the value of \(y\) when \(x = 9\), giving your answer correct to 3 significant figures.

10
\includegraphics[max width=\textwidth, alt={}, center]{b00cefad-7c3c-4672-b309-f19aafab8b01-18_324_677_259_734} The diagram shows the curve \(y = \sin x \cos ^ { 2 } 2 x\) for \(0 \leqslant x \leqslant \frac { 1 } { 4 } \pi\) and its maximum point \(M\).

1. Using the substitution \(u = \cos x\), find by integration the exact area of the shaded region bounded by the curve and the \(x\)-axis.
2. Find the \(x\)-coordinate of \(M\). Give your answer correct to 2 decimal places.