CAIE P2 (Pure Mathematics 2) 2011 November

1 Solve the inequality \(| x + 2 | > \left| \frac { 1 } { 2 } x - 2 \right|\).

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

3
\includegraphics[max width=\textwidth, alt={}, center]{322eb555-d40a-460c-8c71-5780f5772bcd-2_535_1041_573_552} The diagram shows the curve \(y = x - 2 \ln x\) and its minimum point \(M\).

1. Find the \(x\)-coordinate of \(M\).
2. Use the trapezium rule with three intervals to estimate the value of $$\int _ { 2 } ^ { 5 } ( x - 2 \ln x ) \mathrm { d } x$$ giving your answer correct to 2 decimal places.
3. State, with a reason, whether the trapezium rule gives an under-estimate or an over-estimate of the true value of the integral in part (ii).

4 Find the exact value of the positive constant \(k\) for which $$\int _ { 0 } ^ { k } e ^ { 4 x } d x = \int _ { 0 } ^ { 2 k } e ^ { x } d x$$

5

1. By sketching a suitable pair of graphs, show that the equation $$\frac { 1 } { x } = \sin x$$ where \(x\) is in radians, has only one root for \(0 < x \leqslant \frac { 1 } { 2 } \pi\).
2. Verify by calculation that this root lies between \(x = 1.1\) and \(x = 1.2\).
3. Use the iterative formula \(x _ { n + 1 } = \frac { 1 } { \sin x _ { n } }\) to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

6 The parametric equations of a curve are $$x = 1 + 2 \sin ^ { 2 } \theta , \quad y = 4 \tan \theta$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sin \theta \cos ^ { 3 } \theta }\).
2. Find the equation of the tangent to the curve at the point where \(\theta = \frac { 1 } { 4 } \pi\), giving your answer in the form \(y = m x + c\).

7 The polynomial \(a x ^ { 3 } - 3 x ^ { 2 } - 11 x + b\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\), and that when \(\mathrm { p } ( x )\) is divided by \(( x + 1 )\) the remainder is 12 .

1. Find the values of \(a\) and \(b\).
2. When \(a\) and \(b\) have these values, factorise \(\mathrm { p } ( x )\) completely.

8

1. Express \(5 \cos \theta - 3 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), giving the exact value of \(R\) and the value of \(\alpha\) correct to 2 decimal places.
2. Hence solve the equation $$5 \cos \theta - 3 \sin \theta = 4$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
3. Write down the least value of \(15 \cos \theta - 9 \sin \theta\) as \(\theta\) varies.