CAIE P3 (Pure Mathematics 3) 2021 November

1 Find the quotient and remainder when \(2 x ^ { 4 } + 1\) is divided by \(x ^ { 2 } - x + 2\).

2

1. Sketch the graph of \(y = | 2 x - 3 |\).
2. Solve the inequality \(| 2 x - 3 | < 3 x + 2\).

3 Solve the equation \(4 ^ { x - 2 } = 4 ^ { x } - 4 ^ { 2 }\), giving your answer correct to 3 decimal places.

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

5 Solve the equation \(\sin \theta = 3 \cos 2 \theta + 2\), for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

6

1. By first expanding \(\cos \left( x - 60 ^ { \circ } \right)\), show that the expression $$2 \cos \left( x - 60 ^ { \circ } \right) + \cos x$$ can be written in the form \(R \cos ( x - \alpha )\), 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.
2. Hence find the value of \(x\) in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\) for which \(2 \cos \left( x - 60 ^ { \circ } \right) + \cos x\) takes its least possible value.

7 The equation of a curve is \(\ln ( x + y ) = x - 2 y\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x + y - 1 } { 2 ( x + y ) + 1 }\).
2. Find the coordinates of the point on the curve where the tangent is parallel to the \(x\)-axis. \(\quad\) [3]

8
\includegraphics[max width=\textwidth, alt={}, center]{bbe57fc0-a8a5-4fe5-a637-4f9db00bdc13-10_588_789_260_678} In the diagram, \(O A B C D\) is a pyramid with vertex \(D\). The horizontal base \(O A B C\) is a square of side 4 units. The edge \(O D\) is vertical and \(O D = 4\) units. The unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O C\) and \(O D\) respectively. The midpoint of \(A B\) is \(M\) and the point \(N\) on \(C D\) is such that \(D N = 3 N C\).

1. Find a vector equation for the line through \(M\) and \(N\).
2. Show that the length of the perpendicular from \(O\) to \(M N\) is \(\frac { 1 } { 3 } \sqrt { 82 }\).
   \(9 \quad\) Let \(\mathrm { f } ( x ) = \frac { 1 } { ( 9 - x ) \sqrt { x } }\).

1. Find the \(x\)-coordinate of the stationary point of the curve with equation \(y = \mathrm { f } ( x )\).
2. Using the substitution \(u = \sqrt { x }\), show that \(\int _ { 0 } ^ { 4 } \mathrm { f } ( x ) \mathrm { d } x = \frac { 1 } { 3 } \ln 5\).

10 A large plantation of area \(20 \mathrm {~km} ^ { 2 }\) is becoming infected with a plant disease. At time \(t\) years the area infected is \(x \mathrm {~km} ^ { 2 }\) and the rate of increase of \(x\) is proportional to the ratio of the area infected to the area not yet infected. When \(t = 0 , x = 1\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 1\).

1. Show that \(x\) and \(t\) satisfy the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { 19 x } { 20 - x }$$
2. Solve the differential equation and show that when \(t = 1\) the value of \(x\) satisfies the equation \(x = \mathrm { e } ^ { 0.9 + 0.05 x }\).
3. Use an iterative formula based on the equation in part (b), with an initial value of 2 , to determine \(x\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
4. Calculate the value of \(t\) at which the entire plantation becomes infected.

11 The complex number \(- \sqrt { 3 } + \mathrm { i }\) is denoted by \(u\).

1. Express \(u\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\), giving the exact values of \(r\) and \(\theta\).
2. Hence show that \(u ^ { 6 }\) is real and state its value.
3. 1. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(0 \leqslant \arg ( z - u ) \leqslant \frac { 1 } { 4 } \pi\) and \(\operatorname { Re } z \leqslant 2\).
   2. Find the greatest value of \(| z |\) for points in the shaded region. Give your answer correct to 3 significant figures.
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