CAIE P3 (Pure Mathematics 3) 2016 November

1 It is given that \(z = \ln ( y + 2 ) - \ln ( y + 1 )\). Express \(y\) in terms of \(z\).

2 The equation of a curve is \(y = \frac { \sin x } { 1 + \cos x }\), for \(- \pi < x < \pi\). Show that the gradient of the curve is positive for all \(x\) in the given interval.

3 Express the equation \(\cot 2 \theta = 1 + \tan \theta\) as a quadratic equation in \(\tan \theta\). Hence solve this equation for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).

4 The polynomial \(4 x ^ { 4 } + a x ^ { 2 } + 11 x + b\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(\mathrm { p } ( x )\) is divisible by \(x ^ { 2 } - x + 2\).

1. Find the values of \(a\) and \(b\).
2. When \(a\) and \(b\) have these values, find the real roots of the equation \(\mathrm { p } ( x ) = 0\).

5
\includegraphics[max width=\textwidth, alt={}, center]{ccadf73b-16f5-463a-8f69-1394839d5325-2_346_437_1155_854} The diagram shows a variable point \(P\) with coordinates \(( x , y )\) and the point \(N\) which is the foot of the perpendicular from \(P\) to the \(x\)-axis. \(P\) moves on a curve such that, for all \(x \geqslant 0\), the gradient of the curve is equal in value to the area of the triangle \(O P N\), where \(O\) is the origin.

1. State a differential equation satisfied by \(x\) and \(y\). The point with coordinates \(( 0,2 )\) lies on the curve.
2. Solve the differential equation to obtain the equation of the curve, expressing \(y\) in terms of \(x\).
3. Sketch the curve.

6 Let \(I = \int _ { 1 } ^ { 4 } \frac { ( \sqrt { } x ) - 1 } { 2 ( x + \sqrt { } x ) } \mathrm { d } x\).

1. Using the substitution \(u = \sqrt { } x\), show that \(I = \int _ { 1 } ^ { 2 } \frac { u - 1 } { u + 1 } \mathrm {~d} u\).
2. Hence show that \(I = 1 + \ln \frac { 4 } { 9 }\).

1. Find the modulus and argument of \(z\).
2. Express each of the following in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact:
   (a) \(z + 2 z ^ { * }\);
   (b) \(\frac { z ^ { * } } { \mathrm { i } z }\).
3. On a sketch of an Argand diagram with origin \(O\), show the points \(A\) and \(B\) representing the complex numbers \(z ^ { * }\) and \(\mathrm { i } z\) respectively. Prove that angle \(A O B\) is equal to \(\frac { 1 } { 6 } \pi\).

8 Let \(\mathrm { f } ( x ) = \frac { 3 x ^ { 2 } + x + 6 } { ( x + 2 ) \left( x ^ { 2 } + 4 \right) }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Hence obtain the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).

9
\includegraphics[max width=\textwidth, alt={}, center]{ccadf73b-16f5-463a-8f69-1394839d5325-3_481_483_1434_831} The diagram shows the curves \(y = x \cos x\) and \(y = \frac { k } { x }\), where \(k\) is a constant, for \(0 < x \leqslant \frac { 1 } { 2 } \pi\). The curves touch at the point where \(x = a\).

1. Show that \(a\) satisfies the equation \(\tan a = \frac { 2 } { a }\).
2. Use the iterative formula \(a _ { n + 1 } = \tan ^ { - 1 } \left( \frac { 2 } { a _ { n } } \right)\) to determine \(a\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
3. Hence find the value of \(k\) correct to 2 decimal places.

10 The line \(l\) has vector equation \(\mathbf { r } = \mathbf { i } + 2 \mathbf { j } + \mathbf { k } + \lambda ( 2 \mathbf { i } - \mathbf { j } + \mathbf { k } )\).

1. Find the position vectors of the two points on the line whose distance from the origin is \(\sqrt { } ( 10 )\).
2. The plane \(p\) has equation \(a x + y + z = 5\), where \(a\) is a constant. The acute angle between the line \(l\) and the plane \(p\) is equal to \(\sin ^ { - 1 } \left( \frac { 2 } { 3 } \right)\). Find the possible values of \(a\).