CAIE P2 (Pure Mathematics 2) 2009 June

1 Given that \(( 1.25 ) ^ { x } = ( 2.5 ) ^ { y }\), use logarithms to find the value of \(\frac { x } { y }\) correct to 3 significant figures.

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

3
\includegraphics[max width=\textwidth, alt={}, center]{b9556031-871d-4dd3-9523-e3438a41339f-2_451_775_559_683} The diagram shows the curve \(y = \frac { 1 } { 1 + \sqrt { } x }\) for values of \(x\) from 0 to 2 .

1. Use the trapezium rule with two intervals to estimate the value of $$\int _ { 0 } ^ { 2 } \frac { 1 } { 1 + \sqrt { } x } \mathrm {~d} x$$ 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 integral in part (i).

4 The parametric equations of a curve are $$x = 4 \sin \theta , \quad y = 3 - 2 \cos 2 \theta$$ where \(- \frac { 1 } { 2 } \pi < \theta < \frac { 1 } { 2 } \pi\). Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\theta\), simplifying your answer as far as possible.

5 Solve the equation \(\sec x = 4 - 2 \tan ^ { 2 } x\), giving all solutions in the interval \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).

6 The polynomial \(x ^ { 3 } + a x ^ { 2 } + b x + 6\), 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 4 .

1. Find the values of \(a\) and \(b\).
2. When \(a\) and \(b\) have these values, find the other two linear factors of \(\mathrm { p } ( x )\).

7
\includegraphics[max width=\textwidth, alt={}, center]{b9556031-871d-4dd3-9523-e3438a41339f-3_655_685_262_730} The diagram shows the curve \(y = x \mathrm { e } ^ { 2 x }\) and its minimum point \(M\).

1. Find the exact coordinates of \(M\).
2. Show that the curve intersects the line \(y = 20\) at the point whose \(x\)-coordinate is the root of the equation $$x = \frac { 1 } { 2 } \ln \left( \frac { 20 } { x } \right)$$
3. Use the iterative formula $$x _ { n + 1 } = \frac { 1 } { 2 } \ln \left( \frac { 20 } { x _ { n } } \right)$$ with initial value \(x _ { 1 } = 1.3\), to calculate the root correct to 2 decimal places, giving the result of each iteration to 4 decimal places.

8

1. Find the equation of the tangent to the curve \(y = \ln ( 3 x - 2 )\) at the point where \(x = 1\).
2. 1. Find the value of the constant \(A\) such that $$\frac { 6 x } { 3 x - 2 } \equiv 2 + \frac { A } { 3 x - 2 }$$
   2. Hence show that \(\int _ { 2 } ^ { 6 } \frac { 6 x } { 3 x - 2 } \mathrm {~d} x = 8 + \frac { 8 } { 3 } \ln 2\).