CAIE P2 (Pure Mathematics 2) 2016 November

1

1. It is given that \(x\) satisfies the equation \(3 ^ { 2 x } = 5 \left( 3 ^ { x } \right) + 14\). Find the value of \(3 ^ { x }\) and, using logarithms, find the value of \(x\) correct to 3 significant figures.
2. Hence state the values of \(x\) satisfying the equation \(3 ^ { 2 | x | } = 5 \left( 3 ^ { | x | } \right) + 14\).

2
\includegraphics[max width=\textwidth, alt={}, center]{3edf4fb5-c1f9-4c99-8e23-fa666185e0ee-2_374_728_536_705} The variables \(x\) and \(y\) satisfy the equation \(y = A \mathrm { e } ^ { p x }\), where \(A\) and \(p\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points \(( 5,3.17 )\) and \(( 10,4.77 )\), as shown in the diagram. Find the values of \(A\) and \(p\) correct to 2 decimal places.

3 A curve has equation \(y = 2 \sin 2 x - 5 \cos 2 x + 6\) and is defined for \(0 \leqslant x \leqslant \pi\). Find the \(x\)-coordinates of the stationary points of the curve, giving your answers correct to 3 significant figures.

4 It is given that the positive constant \(a\) is such that $$\int _ { - a } ^ { a } \left( 4 \mathrm { e } ^ { 2 x } + 5 \right) \mathrm { d } x = 100$$

1. Show that \(a = \frac { 1 } { 2 } \ln \left( 50 + \mathrm { e } ^ { - 2 a } - 5 a \right)\).
2. Use the iterative formula \(a _ { n + 1 } = \frac { 1 } { 2 } \ln \left( 50 + \mathrm { e } ^ { - 2 a _ { n } } - 5 a _ { n } \right)\) to find \(a\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
3. Show that \(\frac { \cos 2 x + 9 \cos x + 5 } { \cos x + 4 } \equiv 2 \cos x + 1\).
4. Hence find the exact value of \(\int _ { - \pi } ^ { \pi } \frac { \cos 4 x + 9 \cos 2 x + 5 } { \cos 2 x + 4 } \mathrm {~d} x\).

6 The equation of a curve is \(3 x ^ { 2 } + 4 x y + y ^ { 2 } = 24\). Find the equation of the normal to the curve at the point ( 1,3 ), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.

7 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = a x ^ { 3 } + 3 x ^ { 2 } + b x + 12$$ where \(a\) and \(b\) are constants. It is given that \(( x + 3 )\) is a factor of \(\mathrm { p } ( x )\). It is also given that the remainder is 18 when \(\mathrm { p } ( x )\) is divided by \(( x + 2 )\).

1. Find the values of \(a\) and \(b\).
2. When \(a\) and \(b\) have these values,
   (a) show that the equation \(\mathrm { p } ( x ) = 0\) has exactly one real root,
   (b) solve the equation \(\mathrm { p } ( \sec y ) = 0\) for \(- 180 ^ { \circ } < y < 180 ^ { \circ }\).