CAIE P2 (Pure Mathematics 2) 2024 June

1 A curve has equation \(\mathrm { y } = 2 \tan \mathrm { x } - 5 \sin \mathrm { x }\) for \(0 \leqslant x < \frac { 1 } { 2 } \pi\).
Find the \(x\)-coordinate of the stationary point of the curve. Give your answer correct to 3 significant figures.

2 A curve has equation \(x ^ { 2 } \ln y + y ^ { 2 } + 4 x = 9\).
Find the gradient of the curve at the point \(( 2,1 )\).

3

1. Sketch on the same diagram the graphs of \(y = | 3 x - 8 |\) and \(y = 5 - x\).
2. Solve the inequality \(| 3 x - 8 | < 5 - x\).
3. Hence determine the largest integer \(N\) satisfying the inequality \(\left| 3 e ^ { 0.1 N } - 8 \right| < 5 - e ^ { 0.1 N }\).

4

1. Show that \(3 \tan 2 \theta + \tan \left( \theta + 45 ^ { \circ } \right) \equiv \frac { \tan ^ { 2 } \theta + 8 \tan \theta + 1 } { 1 - \tan ^ { 2 } \theta }\).
2. Hence solve the equation \(3 \tan 2 \theta + \tan \left( \theta + 45 ^ { \circ } \right) = 4\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).

5 A curve has equation \(\mathrm { y } = \frac { 1 + \mathrm { e } ^ { 2 \mathrm { x } } } { 1 + 3 \mathrm { x } }\). The curve has exactly one stationary point \(P\).

1. Find \(\frac { \mathrm { dy } } { \mathrm { dx } }\) and hence show that the \(x\)-coordinate of \(P\) satisfies the equation \(x = \frac { 1 } { 6 } + \frac { 1 } { 2 } \mathrm { e } ^ { - 2 x }\).
2. Show by calculation that the \(x\)-coordinate of \(P\) lies between 0.35 and 0.45 .
3. Use an iterative formula based on the equation in part (a) to find the \(x\)-coordinate of \(P\) correct to 3 significant figures. Give the result of each iteration to 5 significant figures.
   \includegraphics[max width=\textwidth, alt={}, center]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-10_451_647_258_699} The diagram shows the curve with equation \(\mathrm { y } = \sqrt { \sin 2 \mathrm { x } + \sin ^ { 2 } 2 \mathrm { x } }\) for \(0 \leqslant x \leqslant \frac { 1 } { 6 } \pi\). The shaded region is bounded by the curve and the straight lines \(x = \frac { 1 } { 6 } \pi\) and \(y = 0\).

1. Use the trapezium rule with two intervals to find an approximation to the area of the shaded region. Give your answer correct to 2 significant figures.
2. The shaded region is rotated completely about the \(x\)-axis. Find the exact volume of the solid produced.
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7 The polynomial \(\mathrm { p } ( x )\) is defined by $$p ( x ) = 9 x ^ { 3 } + 6 x ^ { 2 } + 12 x + k$$ where \(k\) is a constant.

1. Find the quotient when \(\mathrm { p } ( x )\) is divided by \(( 3 x + 2 )\) and show that the remainder is \(( k - 8 )\).
2. It is given that \(\int _ { 1 } ^ { 6 } \frac { \mathrm { p } ( \mathrm { x } ) } { 3 \mathrm { x } + 2 } \mathrm { dx } = \mathrm { a } + \ln 64\), where \(a\) is an integer. Find the values of \(a\) and \(k\).
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