CAIE P2 (Pure Mathematics 2) 2022 June

1
\includegraphics[max width=\textwidth, alt={}, center]{ed12a4fb-e3bf-4d00-ad09-9ba5be941dd5-02_654_396_258_872} The variables \(x\) and \(y\) satisfy the equation \(y = 4 ^ { 2 x - a }\), where \(a\) is an integer. As shown in the diagram, the graph of \(\ln y\) against \(x\) is a straight line passing through the point \(( 0 , - 20.8 )\), where the second coordinate is given correct to 3 significant figures.

1. Show that the gradient of the straight line is \(\ln 16\).
2. Determine the value of \(a\).

2

1. Express the equation \(7 \tan \theta + 4 \cot \theta - 13 \sec \theta = 0\) in terms of \(\sin \theta\) only.
2. Hence solve the equation \(7 \tan \theta + 4 \cot \theta - 13 \sec \theta = 0\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).

3
\includegraphics[max width=\textwidth, alt={}, center]{ed12a4fb-e3bf-4d00-ad09-9ba5be941dd5-04_531_739_258_703} The diagram shows the curve with equation \(y = 3 \sin x - 3 \sin 2 x\) for \(0 \leqslant x \leqslant \pi\). The curve meets the \(x\)-axis at the origin and at the points with \(x\)-coordinates \(a\) and \(\pi\).

1. Find the exact value of \(a\).
2. Find the area of the shaded region.

4 A curve has equation \(x ^ { 2 } y + 2 y ^ { 3 } = 48\).
Find the equation of the normal to the curve at the point ( 4,2 ), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.

5

1. By sketching the graphs of $$y = | 5 - 2 x | \quad \text { and } \quad y = 3 \ln x$$ on the same diagram, show that the equation \(| 5 - 2 x | = 3 \ln x\) has exactly two roots.
2. Show that the value of the larger root satisfies the equation \(x = 2.5 + 1.5 \ln x\).
3. Show by calculation that the value of the larger root lies between 4.5 and 5.0.
4. Use an iterative formula, based on the equation in part (b), to find the value of the larger root correct to 3 significant figures. Give the result of each iteration to 5 significant figures.

6 A curve has equation \(y = \frac { 9 \mathrm { e } ^ { 2 x } + 16 } { \mathrm { e } ^ { x } - 1 }\).

1. Show that the \(x\)-coordinate of any stationary point on the curve satisfies the equation $$\mathrm { e } ^ { x } \left( 3 \mathrm { e } ^ { x } - 8 \right) \left( 3 \mathrm { e } ^ { x } + 2 \right) = 0$$
2. Hence show that the curve has only one stationary point and find its exact coordinates.

7 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 2 x ^ { 3 } + 5 x ^ { 2 } + a x + 2 a$$ where \(a\) is an integer.

1. Find, in terms of \(x\) and \(a\), the quotient when \(\mathrm { p } ( x )\) is divided by ( \(x + 2\) ), and show that the remainder is 4 .
2. It is given that \(\int _ { - 1 } ^ { 1 } \frac { \mathrm { p } ( x ) } { x + 2 } \mathrm {~d} x = \frac { 22 } { 3 } + \ln b\), where \(b\) is an integer. Find the values of \(a\) and \(b\).
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