CAIE P2 (Pure Mathematics 2) 2016 June

1 Given that \(5 ^ { 3 x } = 7 ^ { 4 y }\), use logarithms to find the value of \(\frac { x } { y }\) correct to 4 significant figures.

2

1. Find the quotient and remainder when \(2 x ^ { 3 } - 7 x ^ { 2 } - 9 x + 3\) is divided by \(x ^ { 2 } - 2 x + 5\).
2. Hence find the values of the constants \(p\) and \(q\) such that \(x ^ { 2 } - 2 x + 5\) is a factor of \(2 x ^ { 3 } - 7 x ^ { 2 } + p x + q\).

3

1. Solve the equation \(| 3 u + 1 | = | 2 u - 5 |\).
2. Hence solve the equation \(| 3 \cot x + 1 | = | 2 \cot x - 5 |\) for \(0 < x < \frac { 1 } { 2 } \pi\), giving your answer correct to 3 significant figures.

4

1. Show that \(\sin \left( \theta + 60 ^ { \circ } \right) + \sin \left( \theta + 120 ^ { \circ } \right) \equiv ( \sqrt { } 3 ) \cos \theta\).
2. Hence
   (a) find the exact value of \(\sin 105 ^ { \circ } + \sin 165 ^ { \circ }\),
   (b) solve the equation \(\sin \left( \theta + 60 ^ { \circ } \right) + \sin \left( \theta + 120 ^ { \circ } \right) = \sec \theta\) for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).

5 The equation of a curve is \(y = 6 x \mathrm { e } ^ { \frac { 1 } { 3 } x }\). At the point on the curve with \(x\)-coordinate \(p\), the gradient of the curve is 40 .

1. Show that \(p = 3 \ln \left( \frac { 20 } { p + 3 } \right)\).
2. Show by calculation that \(3.3 < p < 3.5\).
3. Use an iterative formula based on the equation in part (i) to find the value of \(p\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

6

1. Find \(\int \frac { 4 + \mathrm { e } ^ { x } } { 2 \mathrm { e } ^ { 2 x } } \mathrm {~d} x\).
2. Without using a calculator, find \(\int _ { 2 } ^ { 10 } \frac { 1 } { 2 x + 5 } \mathrm {~d} x\), giving your answer in the form \(\ln k\).
3. 
   \includegraphics[max width=\textwidth, alt={}, center]{a07e6d2f-ded1-4c62-957b-41fb94b46a2d-3_446_755_580_735} The diagram shows the curve \(y = \log _ { 10 } ( x + 2 )\) for \(0 \leqslant x \leqslant 6\). The region bounded by the curve and the lines \(x = 0 , x = 6\) and \(y = 0\) is denoted by \(R\). Use the trapezium rule with 2 strips to find an estimate of the area of \(R\), giving your answer correct to 1 decimal place.

7
\includegraphics[max width=\textwidth, alt={}, center]{a07e6d2f-ded1-4c62-957b-41fb94b46a2d-3_423_837_1352_651} The diagram shows the curve with parametric equations $$x = 2 - \cos t , \quad y = 1 + 3 \cos 2 t$$ for \(0 < t < \pi\). The minimum point is \(M\) and the curve crosses the \(x\)-axis at points \(P\) and \(Q\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - 12 \cos t\).
2. Find the coordinates of \(M\).
3. Find the gradient of the curve at \(P\) and at \(Q\).