CAIE P2 (Pure Mathematics 2) 2015 November

1 Find the exact value of \(\int _ { - 1 } ^ { 35 } \frac { 3 } { 2 x + 5 } \mathrm {~d} x\), giving the answer in the form \(\ln k\).

2

1. Solve the equation \(| 2 x + 3 | = | x + 8 |\).
2. Hence, using logarithms, solve the equation \(\left| 2 ^ { y + 1 } + 3 \right| = \left| 2 ^ { y } + 8 \right|\). Give the answer correct to 3 significant figures.

3 The parametric equations of a curve are $$x = ( t + 1 ) \mathrm { e } ^ { t } , \quad y = 6 ( t + 4 ) ^ { \frac { 1 } { 2 } }$$ Find the equation of the tangent to the curve when \(t = 0\), giving the answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.

4

1. Find the quotient when \(3 x ^ { 3 } + 5 x ^ { 2 } - 2 x - 1\) is divided by ( \(x - 2\) ), and show that the remainder is 39 .
2. Hence show that the equation \(3 x ^ { 3 } + 5 x ^ { 2 } - 2 x - 40 = 0\) has exactly one real root.

5 It is given that \(\int _ { 0 } ^ { a } \left( 3 \mathrm { e } ^ { 3 x } + 5 \mathrm { e } ^ { x } \right) \mathrm { d } x = 100\), where \(a\) is a positive constant.

1. Show that \(a = \frac { 1 } { 3 } \ln \left( 106 - 5 \mathrm { e } ^ { a } \right)\).
2. Use an iterative formula based on the equation in part (i) to find the value of \(a\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

6

1. Express \(( \sqrt { } 5 ) \cos \theta - 2 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the value of \(\alpha\) correct to 2 decimal places.
2. Hence
   (a) solve the equation \(( \sqrt { } 5 ) \cos \theta - 2 \sin \theta = 0.9\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\),
   (b) state the greatest and least values of $$10 + ( \sqrt { } 5 ) \cos \theta - 2 \sin \theta$$ as \(\theta\) varies.

7 The equation of a curve is \(y = \frac { \sin 2 x } { \cos x + 1 }\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 \left( \cos ^ { 2 } x + \cos x - 1 \right) } { \cos x + 1 }\).
2. Find the \(x\)-coordinate of each stationary point of the curve in the interval \(- \pi < x < \pi\). Give each answer correct to 3 significant figures.