CAIE P2 (Pure Mathematics 2) 2018 June

1 Solve the inequality \(| 3 x - 2 | < | x + 5 |\).

2 A curve has equation \(y = 3 \ln ( 2 x + 9 ) - 2 \ln x\).

1. Find the \(x\)-coordinate of the stationary point.
2. Determine whether the stationary point is a maximum or minimum point.

3

1. Find the quotient when $$x ^ { 4 } - 2 x ^ { 3 } + 8 x ^ { 2 } - 12 x + 13$$ is divided by \(x ^ { 2 } + 6\) and show that the remainder is 1 .
2. Show that the equation $$x ^ { 4 } - 2 x ^ { 3 } + 8 x ^ { 2 } - 12 x + 12 = 0$$ has no real roots.

4

1. Solve the equation \(2 \ln ( 2 x ) - \ln ( x + 3 ) = 4 \ln 2\).
2. Hence solve the equation $$2 \ln \left( 2 ^ { u + 1 } \right) - \ln \left( 2 ^ { u } + 3 \right) = 4 \ln 2$$ giving the value of \(u\) correct to 4 significant figures.

5 A curve has equation $$y ^ { 3 } \sin 2 x + 4 y = 8$$ Find the equation of the tangent to the curve at the point where it crosses the \(y\)-axis.

6 It is given that \(\int _ { 0 } ^ { a } \left( 1 + \mathrm { e } ^ { \frac { 1 } { 2 } x } \right) ^ { 2 } \mathrm {~d} x = 10\), where \(a\) is a positive constant.

1. Show that \(a = 2 \ln \left( \frac { 15 - a } { 4 + \mathrm { e } ^ { \frac { 1 } { 2 } a } } \right)\).
2. Use the equation in part (i) to show by calculation that \(1.5 < a < 1.6\).
3. Use an iterative formula based on the equation in part (i) to find the value of \(a\) correct to 3 significant figures. Give the result of each iteration to 5 significant figures.

7

1. Show that \(2 \operatorname { cosec } ^ { 2 } 2 x ( 1 - \cos 2 x ) \equiv \sec ^ { 2 } x\).
2. Solve the equation \(2 \operatorname { cosec } ^ { 2 } 2 x ( 1 - \cos 2 x ) = \tan x + 21\) for \(0 < x < \pi\), giving your answers correct to 3 significant figures.
3. Find \(\int \left[ 2 \operatorname { cosec } ^ { 2 } ( 4 y + 2 ) - 2 \operatorname { cosec } ^ { 2 } ( 4 y + 2 ) \cos ( 4 y + 2 ) \right] \mathrm { d } y\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.