CAIE P2 (Pure Mathematics 2) 2011 November

1 Solve the inequality \(| 4 - 5 x | < 3\).

2 Show that \(\int _ { 2 } ^ { 6 } \frac { 2 } { 4 x + 1 } \mathrm {~d} x = \ln \frac { 5 } { 3 }\).

3
\includegraphics[max width=\textwidth, alt={}, center]{55794ceb-2d52-459c-8724-6a6a29ab159a-2_705_737_591_703} The diagram shows the part of the curve \(y = \frac { 1 } { 2 } \tan 2 x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\). Find the \(x\)-coordinates of the points on this part of the curve at which the gradient is 4 .

4 Solve the equation \(3 ^ { 2 x } - 7 \left( 3 ^ { x } \right) + 10 = 0\), giving your answers correct to 3 significant figures.

5 The polynomial \(4 x ^ { 3 } + a x ^ { 2 } + 9 x + 9\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that when \(\mathrm { p } ( x )\) is divided by \(( 2 x - 1 )\) the remainder is 10 .

1. Find the value of \(a\) and hence verify that ( \(x - 3\) ) is a factor of \(\mathrm { p } ( x )\).
2. When \(a\) has this value, solve the equation \(\mathrm { p } ( x ) = 0\).

6

1. Verify by calculation that the cubic equation $$x ^ { 3 } - 2 x ^ { 2 } + 5 x - 3 = 0$$ has a root that lies between \(x = 0.7\) and \(x = 0.8\).
2. Show that this root also satisfies an equation of the form $$x = \frac { a x ^ { 2 } + 3 } { x ^ { 2 } + b }$$ where the values of \(a\) and \(b\) are to be found.
3. With these values of \(a\) and \(b\), use the iterative formula $$x _ { n + 1 } = \frac { a x _ { n } ^ { 2 } + 3 } { x _ { n } ^ { 2 } + b }$$ to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

7 The parametric equations of a curve are $$x = \mathrm { e } ^ { 3 t } , \quad y = t ^ { 2 } \mathrm { e } ^ { t } + 3$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { t ( t + 2 ) } { 3 \mathrm { e } ^ { 2 t } }\).
2. Show that the tangent to the curve at the point \(( 1,3 )\) is parallel to the \(x\)-axis.
3. Find the exact coordinates of the other point on the curve at which the tangent is parallel to the \(x\)-axis.

8

1. By first expanding \(\cos ( 2 x + x )\), show that $$\cos 3 x \equiv 4 \cos ^ { 3 } x - 3 \cos x$$
2. Hence show that $$\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } \left( 2 \cos ^ { 3 } x - \cos x \right) d x = \frac { 5 } { 12 }$$