CAIE P2 (Pure Mathematics 2) 2014 November

1 Use the trapezium rule with four intervals to find an approximation to $$\int _ { 1 } ^ { 5 } \left| 2 ^ { x } - 8 \right| \mathrm { d } x$$

2
\includegraphics[max width=\textwidth, alt={}, center]{72d50061-ead5-466a-96fc-2203438d1407-2_654_693_532_724} The variables \(x\) and \(y\) satisfy the equation \(y = a \left( b ^ { x } \right)\), where \(a\) and \(b\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points ( \(0.75,1.70\) ) and ( \(1.53,2.18\) ), as shown in the diagram. Find the values of \(a\) and \(b\) correct to 2 decimal places.

3

1. Find \(\int 4 \cos ^ { 2 } \left( \frac { 1 } { 2 } \theta \right) \mathrm { d } \theta\).
2. Find the exact value of \(\int _ { - 1 } ^ { 6 } \frac { 1 } { 2 x + 3 } \mathrm {~d} x\).

4 For each of the following curves, find the exact gradient at the point indicated:

1. \(y = 3 \cos 2 x - 5 \sin x\) at \(\left( \frac { 1 } { 6 } \pi , - 1 \right)\),
2. \(x ^ { 3 } + 6 x y + y ^ { 3 } = 21\) at \(( 1,2 )\).

5

1. Given that ( \(x + 2\) ) and ( \(x + 3\) ) are factors of $$5 x ^ { 3 } + a x ^ { 2 } + b$$ find the values of the constants \(a\) and \(b\).
2. When \(a\) and \(b\) have these values, factorise $$5 x ^ { 3 } + a x ^ { 2 } + b$$ completely, and hence solve the equation $$5 ^ { 3 y + 1 } + a \times 5 ^ { 2 y } + b = 0$$ giving any answers correct to 3 significant figures.

6
\includegraphics[max width=\textwidth, alt={}, center]{72d50061-ead5-466a-96fc-2203438d1407-3_296_675_945_735} The diagram shows part of the curve \(y = \frac { x ^ { 2 } } { 1 + \mathrm { e } ^ { 3 x } }\) and its maximum point \(M\). The \(x\)-coordinate of \(M\) is denoted by \(m\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence show that \(m\) satisfies the equation \(x = \frac { 2 } { 3 } \left( 1 + \mathrm { e } ^ { - 3 x } \right)\).
2. Show by calculation that \(m\) lies between 0.7 and 0.8 .
3. Use an iterative formula based on the equation in part (i) to find \(m\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

7 The angle \(\alpha\) lies between \(0 ^ { \circ }\) and \(90 ^ { \circ }\) and is such that $$2 \tan ^ { 2 } \alpha + \sec ^ { 2 } \alpha = 5 - 4 \tan \alpha$$

1. Show that $$3 \tan ^ { 2 } \alpha + 4 \tan \alpha - 4 = 0$$ and hence find the exact value of \(\tan \alpha\).
2. It is given that the angle \(\beta\) is such that \(\cot ( \alpha + \beta ) = 6\). Without using a calculator, find the exact value of \(\cot \beta\).