CAIE P2 (Pure Mathematics 2) 2012 November

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

2 Use logarithms to solve the equation \(5 ^ { x } = 3 ^ { 2 x - 1 }\), giving your answer correct to 3 significant figures.

3 Solve the equation $$2 \cos 2 \theta = 4 \cos \theta - 3$$ for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).

4 The parametric equations of a curve are $$x = \ln ( 1 - 2 t ) , \quad y = \frac { 2 } { t } , \quad \text { for } t < 0$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 - 2 t } { t ^ { 2 } }\).
2. Find the exact coordinates of the only point on the curve at which the gradient is 3 .

5
\includegraphics[max width=\textwidth, alt={}, center]{9e1bd528-e7c4-4936-a05a-dde1d1ace7c2-2_512_775_1318_683} The diagram shows the curve \(y = \cos x\), for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\). A rectangle \(O A B C\) is drawn, where \(B\) is the point on the curve with \(x\)-coordinate \(\theta\), and \(A\) and \(C\) are on the axes, as shown. The shaded region \(R\) is bounded by the curve and by the lines \(x = \theta\) and \(y = 0\).

1. Find the area of \(R\) in terms of \(\theta\).
2. The area of the rectangle \(O A B C\) is equal to the area of \(R\). Show that $$\theta = \frac { 1 - \sin \theta } { \cos \theta }$$
3. Use the iterative formula \(\theta _ { n + 1 } = \frac { 1 - \sin \theta _ { n } } { \cos \theta _ { n } }\), with initial value \(\theta _ { 1 } = 0.5\), to determine the value of \(\theta\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

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1. Use the trapezium rule with two intervals to estimate the value of $$\int _ { 0 } ^ { 1 } \frac { 1 } { 6 + 2 \mathrm { e } ^ { x } } \mathrm {~d} x$$ giving your answer correct to 2 decimal places.
2. Find \(\int \frac { \left( \mathrm { e } ^ { x } - 2 \right) ^ { 2 } } { \mathrm { e } ^ { 2 x } } \mathrm {~d} x\).

7 The polynomial \(2 x ^ { 3 } - 4 x ^ { 2 } + a x + b\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that when \(\mathrm { p } ( x )\) is divided by \(( x + 1 )\) the remainder is 4 , and that when \(\mathrm { p } ( x )\) is divided by \(( x - 3 )\) the remainder is 12 .

1. Find the values of \(a\) and \(b\).
2. When \(a\) and \(b\) have these values, find the quotient and remainder when \(\mathrm { p } ( x )\) is divided by ( \(x ^ { 2 } - 2\) ).

8

1. By differentiating \(\frac { 1 } { \cos \theta }\), show that if \(y = \sec \theta\) then \(\frac { \mathrm { d } y } { \mathrm {~d} \theta } = \tan \theta \sec \theta\).
2. Hence show that $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} \theta ^ { 2 } } = a \sec ^ { 3 } \theta + b \sec \theta$$ giving the values of \(a\) and \(b\).
3. Find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \left( 1 + \tan ^ { 2 } \theta - 3 \sec \theta \tan \theta \right) d \theta$$