CAIE P2 (Pure Mathematics 2) 2010 June

1 Given that \(13 ^ { x } = ( 2.8 ) ^ { y }\), use logarithms to show that \(y = k x\) and find the value of \(k\) correct to 3 significant figures.

2
\includegraphics[max width=\textwidth, alt={}, center]{f55e1431-9443-40d7-bec0-6f5c1d9fa2d7-2_531_949_431_598} The diagram shows part of the curve \(y = x \mathrm { e } ^ { - x }\). The shaded region \(R\) is bounded by the curve and by the lines \(x = 2 , x = 3\) and \(y = 0\).

1. Use the trapezium rule with two intervals to estimate the area of \(R\), giving your answer correct to 2 decimal places.
2. State, with a reason, whether the trapezium rule gives an under-estimate or an over-estimate of the true value of the area of \(R\).

3 Solve the inequality \(| 2 x - 1 | < | x + 4 |\).

4

1. Show that \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \cos 2 x \mathrm {~d} x = \frac { 1 } { 2 }\).
2. By using an appropriate trigonometrical identity, find the exact value of $$\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 3 } \pi } 3 \tan ^ { 2 } x \mathrm {~d} x$$

5 The equation of a curve is \(y = x ^ { 3 } \mathrm { e } ^ { - x }\).

1. Show that the curve has a stationary point where \(x = 3\).
2. Find the equation of the tangent to the curve at the point where \(x = 1\).

1. By sketching a suitable pair of graphs, show that the equation $$\ln x = 2 - x ^ { 2 }$$ has only one root.
2. Verify by calculation that this root lies between \(x = 1.3\) and \(x = 1.4\).
3. Show that, if a sequence of values given by the iterative formula $$x _ { n + 1 } = \sqrt { } \left( 2 - \ln x _ { n } \right)$$ converges, then it converges to the root of the equation in part (i).
4. Use the iterative formula \(x _ { n + 1 } = \sqrt { } \left( 2 - \ln x _ { n } \right)\) to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

7 The polynomial \(2 x ^ { 3 } + a x ^ { 2 } + b x + 6\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that when \(\mathrm { p } ( x )\) is divided by ( \(x - 3\) ) the remainder is 30 , and that when \(\mathrm { p } ( x )\) is divided by ( \(x + 1\) ) the remainder is 18 .

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

8

1. Prove the identity $$\sin \left( x - 30 ^ { \circ } \right) + \cos \left( x - 60 ^ { \circ } \right) \equiv ( \sqrt { } 3 ) \sin x$$
2. Hence solve the equation $$\sin \left( x - 30 ^ { \circ } \right) + \cos \left( x - 60 ^ { \circ } \right) = \frac { 1 } { 2 } \sec x$$ for \(0 ^ { \circ } < x < 360 ^ { \circ }\).