CAIE P2 (Pure Mathematics 2) 2020 Specimen

1

1. The polynomial \(2 x ^ { 3 } + a x ^ { 2 } - a x - 12\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that \(( x + 1 )\) is a factor of \(\mathrm { p } ( x )\). Find the value of \(a\).
2. When \(a\) has this value, find the remainder when \(\mathrm { p } ( x )\) is divided by \(( x + 3 )\).

2 Solve the equation \(3 \sin 2 \theta \tan \theta = 2\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).

3 It is given that \(a\) is a positive constant.

1. 1. Sketch on a single diagram the graphs of \(y = | 2 x - 3 a |\) and \(y = | 2 x + 4 a |\).
   2. State the coordinates of each of the points where each graph meets an axis.
2. Solve the inequality \(| 2 x - 3 a | < | 2 x + 4 a |\).

4

1. Solve the equation \(5 ^ { 2 x } + 5 ^ { x } = 12\), giving your answer correct to 3 significant figures.
2. It is given that \(\ln ( y + 5 ) - \ln y = 2 \ln x\). Express \(y\) in terms of \(x\), in a form not involving logarithms.

5 \includegraphics[max width=\textwidth, alt={}, center]{0af2714b-d3eb-4112-a869-eda5cf266cd8-08_410_977_274_543} The diagram shows the curve \(y = \frac { \sin 2 x } { x + 2 }\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\). The \(x\)-coordinate of the maximum point \(M\) is denoted by \(\alpha\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and show that \(\alpha\) satisfies the equation \(\tan 2 x = 2 x + 4\).
2. Show by calculation that \(\alpha\) lies between 0.6 and 0.7 .
3. Use the iterative formula \(x _ { n + 1 } = \frac { 1 } { 2 } \tan ^ { - 1 } \left( 2 x _ { n } + 4 \right)\) to find the value of \(\alpha\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

6 The parametric equations of a curve are $$x = \mathrm { e } ^ { 2 t } , \quad y = 4 t \mathrm { e } ^ { t } .$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 ( t + 1 ) } { \mathrm { e } ^ { t } }\).
2. Find the equation of the normal to the curve at the point where \(t = 0\).

7

1. Show that \(\tan ^ { 2 } x + \cos ^ { 2 } x \equiv \sec ^ { 2 } x + \frac { 1 } { 2 } \cos 2 x - \frac { 1 } { 2 }\) and hence find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \left( \tan ^ { 2 } x + \cos ^ { 2 } x \right) \mathrm { d } x$$
2. \includegraphics[max width=\textwidth, alt={}, center]{0af2714b-d3eb-4112-a869-eda5cf266cd8-13_535_771_274_648} The region enclosed by the curve \(y = \tan x + \cos x\) and the lines \(x = 0 , x = \frac { 1 } { 4 } \pi\) and \(y = 0\) is shown in the diagram. Find the exact volume of the solid produced when this region is rotated completely about the \(x\)-axis.