CAIE P2 (Pure Mathematics 2) 2011 June

1 Use logarithms to solve the equation \(3 ^ { x } = 2 ^ { x + 2 }\), giving your answer correct to 3 significant figures.

2
\includegraphics[max width=\textwidth, alt={}, center]{2c27f384-5289-4c1b-9199-6b4c6ac81e38-2_645_750_429_699} The diagram shows the curve \(y = \sqrt { } \left( 1 + x ^ { 3 } \right)\). Region \(A\) is bounded by the curve and the lines \(x = 0\), \(x = 2\) and \(y = 0\). Region \(B\) is bounded by the curve and the lines \(x = 0\) and \(y = 3\).

1. Use the trapezium rule with two intervals to find an approximation to the area of region \(A\). Give your answer correct to 2 decimal places.
2. Deduce an approximation to the area of region \(B\) and explain why this approximation underestimates the true area of region \(B\).

3 The sequence \(x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots\) defined by $$x _ { 1 } = 1 , \quad x _ { n + 1 } = \frac { 1 } { 2 } \sqrt [ 3 ] { } \left( x _ { n } ^ { 2 } + 6 \right)$$ converges to the value \(\alpha\).

1. Find the value of \(\alpha\) correct to 3 decimal places. Show your working, giving each calculated value of the sequence to 5 decimal places.
2. Find, in the form \(a x ^ { 3 } + b x ^ { 2 } + c = 0\), an equation of which \(\alpha\) is a root.

4

1. Find the value of \(\int _ { 0 } ^ { \frac { 2 } { 3 } \pi } \sin \left( \frac { 1 } { 2 } x \right) \mathrm { d } x\).
2. Find \(\int \mathrm { e } ^ { - x } \left( 1 + \mathrm { e } ^ { x } \right) \mathrm { d } x\).

5 A curve has equation \(x ^ { 2 } + 2 y ^ { 2 } + 5 x + 6 y = 10\). Find the equation of the tangent to the curve at the point \(( 2 , - 1 )\). Give your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
[0pt] [6]

6 The curve \(y = 4 x ^ { 2 } \ln x\) has one stationary point.

1. Find the coordinates of this stationary point, giving your answers correct to 3 decimal places.
2. Determine whether this point is a maximum or a minimum point.

7 The cubic polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 6 x ^ { 3 } + a x ^ { 2 } + b x + 10$$ where \(a\) and \(b\) are constants. 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 24 .

1. Find the values of \(a\) and \(b\).
2. When \(a\) and \(b\) have these values, factorise \(\mathrm { p } ( x )\) completely.

8

1. Prove that \(\sin ^ { 2 } 2 \theta \left( \operatorname { cosec } ^ { 2 } \theta - \sec ^ { 2 } \theta \right) \equiv 4 \cos 2 \theta\).
2. Hence
   (a) solve for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\) the equation \(\sin ^ { 2 } 2 \theta \left( \operatorname { cosec } ^ { 2 } \theta - \sec ^ { 2 } \theta \right) = 3\),
   (b) find the exact value of \(\operatorname { cosec } ^ { 2 } 15 ^ { \circ } - \sec ^ { 2 } 15 ^ { \circ }\).