CAIE P2 (Pure Mathematics 2) 2024 November

1 The variables \(x\) and \(y\) satisfy the equation \(a ^ { 2 y } = \mathrm { e } ^ { 3 x + k }\), where \(a\) and \(k\) are constants.
The graph of \(y\) against \(x\) is a straight line.

1. Use logarithms to show that the gradient of the straight line is \(\frac { 3 } { 2 \ln a }\).
2. Given that the straight line passes through the points \(( 0.4,0.95 )\) and \(( 3.3,3.80 )\), find the values of \(a\) and \(k\).
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2 Solve the inequality \(| x - 7 | > 4 x + 3\).

3 The function f is defined by \(\mathrm { f } ( x ) = \tan ^ { 2 } \left( \frac { 1 } { 2 } x \right)\) for \(0 \leqslant x < \pi\).

1. Find the exact value of \(\mathrm { f } ^ { \prime } \left( \frac { 2 } { 3 } \pi \right)\).
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2. Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } ( \mathrm { f } ( x ) + \sin x ) \mathrm { d } x\).

4 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = a x ^ { 3 } - a x ^ { 2 } - 15 x + 18$$ where \(a\) is a constant. It is given that ( \(x + 2\) ) is a factor of \(\mathrm { p } ( x )\).

1. Find the value of \(a\).
2. Hence factorise \(\mathrm { p } ( x )\) completely.
   
3. Solve the equation \(\mathrm { p } \left( \operatorname { cosec } ^ { 2 } \theta \right) = 0\) for \(- 90 ^ { \circ } < \theta < 90 ^ { \circ }\).

5 It is given that \(\int _ { a } ^ { a ^ { 3 } } \frac { 10 } { 2 x + 1 } \mathrm {~d} x = 7\), where \(a\) is a constant greater than 1 .

1. Show that \(a = \sqrt [ 3 ] { 0.5 \mathrm { e } ^ { 1.4 } ( 2 a + 1 ) - 0.5 }\).
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2. Use an iterative formula, based on the equation in part (a), to find the value of \(a\) correct to 3 significant figures. Use an initial value of 2 and give the result of each iteration to 5 significant figures.

6 A curve has parametric equations $$x = \frac { \mathrm { e } ^ { 2 t } - 2 } { \mathrm { e } ^ { 2 t } + 1 } , \quad y = \mathrm { e } ^ { 3 t } + 1$$

1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
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2. Find the exact gradient of the curve at the point where the curve crosses the \(y\)-axis.

7

1. Prove that \(\cos \left( \theta + 30 ^ { \circ } \right) \cos \left( \theta + 60 ^ { \circ } \right) \equiv \frac { 1 } { 4 } \sqrt { 3 } - \frac { 1 } { 2 } \sin 2 \theta\).
2. Solve the equation \(5 \cos \left( 2 \alpha + 30 ^ { \circ } \right) \cos \left( 2 \alpha + 60 ^ { \circ } \right) = 1\) for \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
3. Show that the exact value of \(\cos 20 ^ { \circ } \cos 50 ^ { \circ } + \cos 40 ^ { \circ } \cos 70 ^ { \circ }\) is \(\frac { 1 } { 2 } \sqrt { 3 }\).
   If you use the following page to complete the answer to any question, the question number must be clearly shown.
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