CAIE P2 (Pure Mathematics 2) 2020 June

1 Given that \(2 ^ { y } = 9 ^ { 3 x }\), use logarithms to show that \(y = k x\) and find the value of \(k\) correct to 3 significant figures.

2 Find the exact coordinates of the stationary point on the curve with equation \(y = 5 x \mathrm { e } ^ { \frac { 1 } { 2 } x }\).

3 The equation of a curve is \(\cos 3 x + 5 \sin y = 3\).
Find the gradient of the curve at the point \(\left( \frac { 1 } { 9 } \pi , \frac { 1 } { 6 } \pi \right)\).

4
\includegraphics[max width=\textwidth, alt={}, center]{01f2de2b-3482-4694-889e-7fcd016b57e3-06_659_828_262_660} The variables \(x\) and \(y\) satisfy the equation \(y = A x ^ { - 2 p }\), where \(A\) and \(p\) are constants. The graph of \(\ln y\) against \(\ln x\) is a straight line passing through the points \(( - 0.68,3.02 )\) and \(( 1.07 , - 1.53 )\), as shown in the diagram. Find the values of \(A\) and \(p\).

5

1. Sketch, on the same diagram, the graphs of \(y = | 2 x - 3 |\) and \(y = 3 x + 5\).
2. Solve the inequality \(3 x + 5 < | 2 x - 3 |\).

6 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 6 x ^ { 3 } + a x ^ { 2 } - 4 x - 3$$ where \(a\) is a constant. It is given that \(( x + 3 )\) is a factor of \(\mathrm { p } ( x )\).

1. Find the value of \(a\).
2. Using this value of \(a\), factorise \(\mathrm { p } ( x )\) completely.
3. Hence solve the equation \(\mathrm { p } ( \operatorname { cosec } \theta ) = 0\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).

7 It is given that \(\int _ { 0 } ^ { a } \left( \frac { 4 } { 2 x + 1 } + 8 x \right) \mathrm { d } x = 10\), where \(a\) is a positive constant.

1. Show that \(a = \sqrt { 2.5 - 0.5 \ln ( 2 a + 1 ) }\).
2. Using the equation in part (a), show by calculation that \(1 < a < 2\).
3. Use an iterative formula, based on the equation in part (a), to find the value of \(a\) correct to 4 significant figures. Give the result of each iteration to 6 significant figures.

8

1. Show that \(3 \sin 2 \theta \cot \theta \equiv 6 \cos ^ { 2 } \theta\).
2. Solve the equation \(3 \sin 2 \theta \cot \theta = 5\) for \(0 < \theta < \pi\).
3. Find the exact value of \(\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 2 } \pi } 3 \sin x \cot \frac { 1 } { 2 } x \mathrm {~d} x\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.