CAIE P2 (Pure Mathematics 2) 2016 November

1 The sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 4 } { x _ { n } ^ { 2 } } + \frac { 2 x _ { n } } { 3 }$$ with initial value \(x _ { 1 } = 2\), converges to \(\alpha\).

1. Use this iterative formula to find \(\alpha\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
2. State an equation that is satisfied by \(\alpha\), and hence find the exact value of \(\alpha\).

2
\includegraphics[max width=\textwidth, alt={}, center]{8b051aee-4920-42a0-8b74-cbfa9f3c1ab1-2_429_821_826_662} The variables \(x\) and \(y\) satisfy the equation \(y = K x ^ { p }\), where \(K\) and \(p\) are constants. The graph of \(\ln y\) against \(\ln x\) is a straight line passing through the points \(( 1.28,3.69 )\) and \(( 2.11,4.81 )\), as shown in the diagram. Find the values of \(K\) and \(p\) correct to 2 decimal places.

1. Find \(\int \tan ^ { 2 } 4 x \mathrm {~d} x\).
2. Without using a calculator, find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 12 } \pi } ( 4 \cos 2 x + 6 \sin 3 x ) \mathrm { d } x\).

4 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = a x ^ { 3 } + 3 x ^ { 2 } + 4 a x - 5 ,$$ where \(a\) is a constant. It is given that ( \(2 x - 1\) ) is a factor of \(\mathrm { p } ( x )\).

1. Use the factor theorem to find the value of \(a\).
2. Factorise \(\mathrm { p } ( x )\) and hence show that the equation \(\mathrm { p } ( x ) = 0\) has only one real root.
3. Use logarithms to solve the equation \(\mathrm { p } \left( 6 ^ { y } \right) = 0\) correct to 3 significant figures.

5
\includegraphics[max width=\textwidth, alt={}, center]{8b051aee-4920-42a0-8b74-cbfa9f3c1ab1-3_373_623_260_760} The diagram shows the curve \(y = \sqrt { } \left( 1 + \mathrm { e } ^ { \frac { 1 } { 3 } x } \right)\) for \(0 \leqslant x \leqslant 6\). The region bounded by the curve and the lines \(x = 0 , x = 6\) and \(y = 0\) is denoted by \(R\).

1. Use the trapezium rule with 2 strips to find an estimate of the area of \(R\), giving your answer correct to 2 decimal places.
2. With reference to the diagram, explain why this estimate is greater than the exact area of \(R\).
3. The region \(R\) is rotated completely about the \(x\)-axis. Find the exact volume of the solid produced.

6 A curve has parametric equations $$x = \ln ( t + 1 ) , \quad y = t ^ { 2 } \ln t$$

1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
2. Find the exact value of \(t\) at the stationary point.
3. Find the gradient of the curve at the point where it crosses the \(x\)-axis.
4. Express \(\sin 2 \theta ( 3 \sec \theta + 4 \operatorname { cosec } \theta )\) in the form \(a \sin \theta + b \cos \theta\), where \(a\) and \(b\) are integers.
5. Hence express \(\sin 2 \theta ( 3 \sec \theta + 4 \operatorname { cosec } \theta )\) in the form \(R \sin ( \theta + \alpha )\) where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
6. Using the result of part (ii), solve the equation \(\sin 2 \theta ( 3 \sec \theta + 4 \operatorname { cosec } \theta ) = 7\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).