CAIE P2 (Pure Mathematics 2) 2017 March

1 Solve the equation \(2 \ln ( 2 x ) - \ln ( x + 3 ) = \ln ( 3 x + 5 )\).

2

1. Given that \(\tan 2 \theta \cot \theta = 8\), show that \(\tan ^ { 2 } \theta = \frac { 3 } { 4 }\).
2. Hence solve the equation \(\tan 2 \theta \cot \theta = 8\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).

3

1. Solve the inequality \(| 2 x - 5 | < | x + 3 |\).
2. Hence find the largest integer \(y\) satisfying the inequality \(| 2 \ln y - 5 | < | \ln y + 3 |\).

4 Find the gradient of the curve $$x ^ { 2 } \sin y + \cos 3 y = 4$$ at the point \(\left( 2 , \frac { 1 } { 2 } \pi \right)\).

5 It is given that \(a\) is a positive constant such that $$\int _ { 0 } ^ { a } \left( 1 + 2 x + 3 \mathrm { e } ^ { 3 x } \right) \mathrm { d } x = 250$$

1. Show that \(a = \frac { 1 } { 3 } \ln \left( 251 - a - a ^ { 2 } \right)\).
2. Use an iterative formula based on the equation in part (i) to find the value of \(a\) correct to 4 significant figures. Give the result of each iteration to 6 significant figures.

6 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = a x ^ { 3 } + b x ^ { 2 } - 17 x - a$$ where \(a\) and \(b\) are constants. It is given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\) and that the remainder is 28 when \(\mathrm { p } ( x )\) is divided by \(( x - 2 )\).

1. Find the values of \(a\) and \(b\).
2. Hence factorise \(\mathrm { p } ( x )\) completely.
3. State the number of roots of the equation \(\mathrm { p } \left( 2 ^ { y } \right) = 0\), justifying your answer.
   \includegraphics[max width=\textwidth, alt={}, center]{17025451-6f07-4f35-9dfc-869e084b5ed0-10_508_538_310_799} The diagram shows part of the curve $$y = 2 \cos 2 x \cos \left( 2 x + \frac { 1 } { 6 } \pi \right)$$ The shaded region is bounded by the curve and the two axes.
4. Show that \(2 \cos 2 x \cos \left( 2 x + \frac { 1 } { 6 } \pi \right)\) can be expressed in the form $$k _ { 1 } ( 1 + \cos 4 x ) + k _ { 2 } \sin 4 x ,$$ where the values of the constants \(k _ { 1 }\) and \(k _ { 2 }\) are to be determined.
5. Find the exact area of the shaded region.