CAIE P2 (Pure Mathematics 2) 2022 November

1 Solve the inequality \(| 2 x - 5 | > x\).

2 Use logarithms to solve the equation \(14 \mathrm { e } ^ { - 2 x } = 5 ^ { x + 1 }\), giving your answer correct to 3 significant figures. [4]

3 It is given that \(\sec \theta = \sqrt { 17 }\) where \(0 < \theta < \frac { 1 } { 2 } \pi\).
Find the exact value of \(\tan \left( \theta + \frac { 1 } { 4 } \pi \right)\).

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1. By sketching a suitable pair of graphs on the same diagram, show that the equation $$\mathrm { e } ^ { - \frac { 1 } { 2 } x } = x ^ { 5 }$$ has exactly one real root.
2. Use the iterative formula \(x _ { n + 1 } = \sqrt [ 5 ] { \mathrm { e } ^ { - \frac { 1 } { 2 } x _ { n } } }\) to determine the root correct to 4 significant figures. Give the result of each iteration to 6 significant figures.

5 A curve has equation \(4 \mathrm { e } ^ { 2 x } y + y ^ { 2 } = 21\).
Find the gradient of the curve at the point \(( 0 , - 7 )\).

6 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 12 x ^ { 3 } - 9 x ^ { 2 } + 8 x - 4$$

1. Find the quotient when \(\mathrm { p } ( x )\) is divided by \(( 4 x - 3 )\) and show that the remainder is 2 .
2. Hence find \(\int _ { 2 } ^ { 12 } \left( \frac { \mathrm { p } ( x ) } { 4 x - 3 } - 3 x ^ { 2 } \right) \mathrm { d } x\), giving your answer in the form \(a + \ln b\).

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\includegraphics[max width=\textwidth, alt={}, center]{389df578-e7a7-4d19-9416-5e580d107717-10_456_598_269_762} The diagram shows the curve with equation \(y = \frac { 2 \ln x } { 3 x + 1 }\). The curve crosses the \(x\)-axis at the point \(A\) and has a maximum point \(B\). The shaded region is bounded by the curve and the lines \(x = 3\) and \(y = 0\).

1. Find the gradient of the curve at \(A\).
2. Show by calculation that the \(x\)-coordinate of \(B\) lies between 3.0 and 3.1.
3. Use the trapezium rule with two intervals to find an approximation to the area of the shaded region. Give your answer correct to 2 decimal places.

8 The expression \(\mathrm { f } ( \theta )\) is defined by \(\mathrm { f } ( \theta ) = 12 \sin \theta \cos \theta + 16 \cos ^ { 2 } \theta\).

1. Express \(\mathrm { f } ( \theta )\) in the form \(R \cos ( 2 \theta - \alpha ) + k\), where \(R > 0,0 < \alpha < \frac { 1 } { 2 } \pi\) and \(k\) is a constant. State the values of \(R\) and \(k\), and give the value of \(\alpha\) correct to 4 significant figures.
2. Find the smallest positive value of \(\theta\) satisfying the equation \(\mathrm { f } ( \theta ) = 17\).
3. Find \(\int f ( \theta ) d \theta\).
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