CAIE P2 (Pure Mathematics 2) 2017 June

1 Solve the equation \(| x + a | = | 2 x - 5 a |\), giving \(x\) in terms of the positive constant \(a\).

2 Use logarithms to solve the equation \(3 ^ { x + 4 } = 5 ^ { 2 x }\), giving your answer correct to 3 significant figures.

3

1. By sketching a suitable pair of graphs, show that the equation $$x ^ { 3 } = 11 - 2 x$$ has exactly one real root.
2. Use the iterative formula $$x _ { n + 1 } = \sqrt [ 3 ] { } \left( 11 - 2 x _ { n } \right)$$ to find the root correct to 4 significant figures. Give the result of each iteration to 6 significant figures.

4 Find the equation of the tangent to the curve \(y = \frac { \mathrm { e } ^ { 4 x } } { 2 x + 3 }\) at the point on the curve for which \(x = 0\). Give your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.

5
\includegraphics[max width=\textwidth, alt={}, center]{de2f8bf3-fd03-4199-9eb2-c9cbac4d4385-05_551_535_260_806} The variables \(x\) and \(y\) satisfy the equation \(y = \frac { K } { a ^ { 2 x } }\), where \(K\) and \(a\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points \(( 0.6,1.81 )\) and \(( 1.4,1.39 )\), as shown in the diagram. Find the values of \(K\) and \(a\) correct to 2 significant figures.

6

1. Use the factor theorem to show that ( \(x + 2\) ) is a factor of the expression $$6 x ^ { 3 } + 13 x ^ { 2 } - 33 x - 70$$ and hence factorise the expression completely.
2. Deduce the roots of the equation $$6 + 13 y - 33 y ^ { 2 } - 70 y ^ { 3 } = 0$$

7

1. Find \(\int ( 2 \cos \theta - 3 ) ( \cos \theta + 1 ) \mathrm { d } \theta\).
2. 1. Find \(\int \left( \frac { 4 } { 2 x + 1 } + \frac { 1 } { 2 x } \right) \mathrm { d } x\).
   2. Hence find \(\int _ { 1 } ^ { 4 } \left( \frac { 4 } { 2 x + 1 } + \frac { 1 } { 2 x } \right) \mathrm { d } x\), giving your answer in the form \(\ln k\).

8
\includegraphics[max width=\textwidth, alt={}, center]{de2f8bf3-fd03-4199-9eb2-c9cbac4d4385-10_549_495_258_824} The diagram shows the curve with parametric equations $$x = 2 - \cos 2 t , \quad y = 2 \sin ^ { 3 } t + 3 \cos ^ { 3 } t + 1$$ for \(0 \leqslant t \leqslant \frac { 1 } { 2 } \pi\). The end-points of the curve \(( 1,4 )\) and \(( 3,3 )\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 } { 2 } \sin t - \frac { 9 } { 4 } \cos t\).
2. Find the coordinates of the minimum point, giving each coordinate correct to 3 significant figures.
3. Find the exact gradient of the normal to the curve at the point for which \(x = 2\).