CAIE P2 (Pure Mathematics 2) 2024 November

1 Use logarithms to show that the equation \(5 ^ { 8 y } = 6 ^ { 7 x }\) can be expressed in the form \(y = k x\). Give the value of the constant \(k\) correct to 3 significant figures.

2 Let \(\mathrm { f } ( x ) = 4 \sin ^ { 2 } 3 x\).

1. Find the value of \(\mathrm { f } ^ { \prime } \left( \frac { 1 } { 4 } \pi \right)\).
2. Find \(\int \mathrm { f } ( x ) \mathrm { d } x\).
   \includegraphics[max width=\textwidth, alt={}, center]{18aea465-b5b0-48f0-970a-e9ede1dc9370-05_2723_35_101_20}

3 A curve has equation \(6 \mathrm { e } ^ { - x } y ^ { 2 } + \mathrm { e } ^ { 2 x } - 12 y + 7 = 0\).
Find the gradient of the curve at the point \(( \ln 3,2 )\).

4

1. Sketch the graphs of \(y = 1 + \mathrm { e } ^ { 2 x }\) and \(y = | x - 4 |\) on the same diagram.
2. The two graphs meet at the point \(P\) .
   Show that the \(x\)-coordinate of \(P\) satisfies the equation \(x = \frac { 1 } { 2 } \ln ( 3 - x )\) .
   \includegraphics[max width=\textwidth, alt={}, center]{18aea465-b5b0-48f0-970a-e9ede1dc9370-06_2716_38_109_2012}
3. Use an iterative formula, based on the equation in part (b), to find the \(x\)-coordinate of \(P\) correct to 3 significant figures. Use an initial value of 0.45 and give the result of each iteration to 5 significant figures.

5 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = a x ^ { 3 } + b x ^ { 2 } - a x + 8$$ where \(a\) and \(b\) are constants.It is given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\) ,and that the remainder is 24 when \(\mathrm { p } ( x )\) is divided by \(( x - 2 )\) .

1. Find the values of \(a\) and \(b\) .
   \includegraphics[max width=\textwidth, alt={}, center]{18aea465-b5b0-48f0-970a-e9ede1dc9370-09_2723_35_101_20}
2. Factorise \(\mathrm { p } ( x )\) and hence show that the equation \(\mathrm { p } ( x ) = 0\) has exactly one real root.
3. Solve the equation \(\mathrm { p } \left( \frac { 1 } { 2 } \operatorname { cosec } \theta \right) = 0\) for \(- 90 ^ { \circ } < \theta < 90 ^ { \circ }\).
   \includegraphics[max width=\textwidth, alt={}, center]{18aea465-b5b0-48f0-970a-e9ede1dc9370-10_499_696_264_680} The diagram shows the curves with equations \(y = \sqrt [ 3 ] { 5 x ^ { 2 } + 7 }\) and \(y = \frac { 27 } { 2 x + 5 }\) for \(x \geqslant 0\).
   The curves meet at the point \(( 2,3 )\).
   Region \(A\) is bounded by the curve \(y = \sqrt [ 3 ] { 5 x ^ { 2 } + 7 }\) and the straight lines \(x = 0 , x = 2\) and \(y = 0\).
   Region \(B\) is bounded by the two curves and the straight line \(x = 0\).

1. Use the trapezium rule with two intervals to find an approximation to the area of region \(A\). Give your answer correct to 3 significant figures.
   \includegraphics[max width=\textwidth, alt={}, center]{18aea465-b5b0-48f0-970a-e9ede1dc9370-10_2720_38_105_2010}
   \includegraphics[max width=\textwidth, alt={}, center]{18aea465-b5b0-48f0-970a-e9ede1dc9370-11_2716_29_107_22}
2. Find the exact total area of regions \(A\) and \(B\). Give your answer in the form \(k \ln m\), where \(k\) and \(m\) are constants.
3. Deduce an approximation to the area of region \(B\). Give your answer correct to 3 significant figures.
4. State, with a reason, whether your answer to part (c) is an over-estimate or an under-estimate of the area of region \(B\).

7

1. Express \(4 \sin \theta \sin \left( \theta + 60 ^ { \circ } \right)\) in the form $$a + R \sin ( 2 \theta - \alpha ) ,$$ where \(a\) and \(R\) are positive integers and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
   \includegraphics[max width=\textwidth, alt={}, center]{18aea465-b5b0-48f0-970a-e9ede1dc9370-13_2723_33_99_21}
2. Hence find the smallest positive value of \(\theta\) satisfying the equation $$\frac { 1 } { 5 } + 4 \sin \theta \sin \left( \theta + 60 ^ { \circ } \right) = 0 .$$ If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
   \includegraphics[max width=\textwidth, alt={}, center]{18aea465-b5b0-48f0-970a-e9ede1dc9370-14_2714_38_109_2010}