CAIE P2 (Pure Mathematics 2) 2013 November

1

1. Find \(\int \frac { 2 } { 4 x - 1 } \mathrm {~d} x\).
2. Hence find \(\int _ { 1 } ^ { 7 } \frac { 2 } { 4 x - 1 } \mathrm {~d} x\), expressing your answer in the form \(\ln a\), where \(a\) is an integer.

2 The curve \(y = \frac { \mathrm { e } ^ { 3 x - 1 } } { 2 x }\) has one stationary point. Find the coordinates of this stationary point.

3 Solve the equation \(2 \cot ^ { 2 } \theta - 5 \operatorname { cosec } \theta = 10\), giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

4

1. The polynomial \(a x ^ { 3 } + b x ^ { 2 } - 25 x - 6\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( x - 3 )\) and \(( x + 2 )\) are factors of \(\mathrm { p } ( x )\). Find the values of \(a\) and \(b\).
2. When \(a\) and \(b\) have these values, factorise \(\mathrm { p } ( x )\) completely.

5 The parametric equations of a curve are $$x = 1 + \sqrt { } t , \quad y = 3 \ln t$$

1. Find the exact value of the gradient of the curve at the point \(P\) where \(y = 6\).
2. Show that the tangent to the curve at \(P\) passes through the point \(( 1,0 )\).

6

1. Find \(\int ( \sin x - \cos x ) ^ { 2 } \mathrm {~d} x\).
2. 1. Use the trapezium rule with 2 intervals to estimate the value of $$\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 2 } \pi } \operatorname { cosec } x d x$$ giving your answer correct to 3 decimal places.
   2. Using a sketch of the graph of \(y = \operatorname { cosec } x\) for \(0 < x \leqslant \frac { 1 } { 2 } \pi\), explain whether the trapezium rule gives an under-estimate or an over-estimate of the true value of the integral in part (i).

7
\includegraphics[max width=\textwidth, alt={}, center]{0900b607-6136-4bf7-a42e-6824d1a21e43-3_451_451_255_845} The diagram shows part of the curve \(y = 8 x + \frac { 1 } { 2 } \mathrm { e } ^ { x }\). The shaded region \(R\) is bounded by the curve and by the lines \(x = 0 , y = 0\) and \(x = a\), where \(a\) is positive. The area of \(R\) is equal to \(\frac { 1 } { 2 }\).

1. Find an equation satisfied by \(a\), and show that the equation can be written in the form $$a = \sqrt { } \left( \frac { 2 - \mathrm { e } ^ { a } } { 8 } \right)$$
2. Verify by calculation that the equation \(a = \sqrt { } \left( \frac { 2 - \mathrm { e } ^ { a } } { 8 } \right)\) has a root between 0.2 and 0.3.
3. Use the iterative formula \(a _ { n + 1 } = \sqrt { } \left( \frac { 2 - \mathrm { e } ^ { a _ { n } } } { 8 } \right)\) to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.