CAIE P2 (Pure Mathematics 2) 2024 March

1 Use logarithms to solve the equation \(3 ^ { 4 x + 3 } = 5 ^ { 2 x + 7 }\). Give your answer correct to 3 significant figures. [4]

2

1. Sketch the graph of \(y = | 3 x - 7 |\), stating the coordinates of the points where the graph meets the axes.
2. Hence find the set of values of the constant \(k\) for which the equation \(| 3 \mathrm { x } - 7 | = \mathrm { k } ( \mathrm { x } - 4 )\) has exactly two real roots.

3 The polynomial \(\mathrm { p } ( x )\) is defined by $$p ( x ) = 6 x ^ { 3 } + a x ^ { 2 } + 3 x - 10$$ where \(a\) is a constant. It is given that \(( 2 x - 1 )\) is a factor of \(\mathrm { p } ( x )\).

1. Find the value of \(a\) and hence factorise \(\mathrm { p } ( x )\) completely.
2. Solve the equation \(\mathrm { p } ( \operatorname { cosec } \theta ) = 0\) for \(- 90 ^ { \circ } < \theta < 90 ^ { \circ }\).
   \includegraphics[max width=\textwidth, alt={}, center]{7b39a2ab-305d-43c5-a1e7-9442d6c13886-06_442_706_278_667} The diagram shows the curve with equation \(\mathrm { y } = \sqrt { 1 + \mathrm { e } ^ { 0.5 \mathrm { x } } }\). The shaded region is bounded by the curve and the straight lines \(x = 0 , x = 6\) and \(y = 0\).
3. Use the trapezium rule with three intervals to find an approximation to the area of the shaded region. Give your answer correct to 3 significant figures.
4. The shaded region is rotated completely about the \(x\)-axis. Find the exact volume of the solid produced.

5
\includegraphics[max width=\textwidth, alt={}, center]{7b39a2ab-305d-43c5-a1e7-9442d6c13886-08_615_469_260_799} The diagram shows part of the curve with equation \(\mathrm { y } = \frac { \mathrm { x } ^ { 3 } } { \mathrm { x } + 2 }\). At the point \(P\), the gradient of the curve is 6 .

1. Show that the \(x\)-coordinate of \(P\) satisfies the equation \(x = \sqrt [ 3 ] { 12 x + 12 }\).
2. Show by calculation that the \(x\)-coordinate of \(P\) lies between 3.8 and 4.0 .
3. Use an iterative formula, based on the equation in part (a), to find the \(x\)-coordinate of \(P\) correct to 3 significant figures. Show the result of each iteration to 5 significant figures.

6
\includegraphics[max width=\textwidth, alt={}, center]{7b39a2ab-305d-43c5-a1e7-9442d6c13886-10_629_620_278_717} The diagram shows the curve with parametric equations $$x = 1 + \sqrt { t } , \quad y = ( \ln t + 2 ) ( \ln t - 3 ) ,$$ for \(0 < t < 25\). The curve crosses the \(x\)-axis at the points \(A\) and \(B\) and has a minimum point \(M\).

1. Show that \(\frac { \mathrm { dy } } { \mathrm { dx } } = \frac { 4 \mathrm { Int } - 2 } { \sqrt { \mathrm { t } } }\).
2. Find the exact gradient of the curve at \(B\).
3. Find the exact coordinates of \(M\).

7

1. Prove that $$\sin 2 \theta ( a \cot \theta + b \tan \theta ) \equiv a + b + ( a - b ) \cos 2 \theta$$ where \(a\) and \(b\) are constants.
2. Find the exact value of \(\int _ { \frac { 1 } { 12 } \pi } ^ { \frac { 1 } { 6 } \pi } \sin 2 \theta ( 5 \cot \theta + 3 \tan \theta ) d \theta\).
3. Solve the equation \(\sin \frac { 2 } { 3 } \alpha \left( 2 \cot \frac { 1 } { 3 } \alpha + 7 \tan \frac { 1 } { 3 } \alpha \right) = 11\) for \(- \pi < \alpha < \pi\).
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