OCR C3 (Core Mathematics 3) 2008 January

1 Functions f and g are defined for all real values of \(x\) by $$\mathrm { f } ( x ) = x ^ { 3 } + 4 \quad \text { and } \quad \mathrm { g } ( x ) = 2 x - 5$$ Evaluate

1. \(f g ( 1 )\),
2. \(\mathrm { f } ^ { - 1 } ( 12 )\).

2 The sequence defined by $$x _ { 1 } = 3 , \quad x _ { n + 1 } = \sqrt [ 3 ] { 31 - \frac { 5 } { 2 } x _ { n } }$$ converges to the number \(\alpha\).

1. Find the value of \(\alpha\) correct to 3 decimal places, showing the result of each iteration.
2. Find an equation of the form \(a x ^ { 3 } + b x + c = 0\), where \(a\), \(b\) and \(c\) are integers, which has \(\alpha\) as a root.

3

1. Solve, for \(0 ^ { \circ } < \alpha < 180 ^ { \circ }\), the equation \(\sec \frac { 1 } { 2 } \alpha = 4\).
2. Solve, for \(0 ^ { \circ } < \beta < 180 ^ { \circ }\), the equation \(\tan \beta = 7 \cot \beta\).

4 Earth is being added to a pile so that, when the height of the pile is \(h\) metres, its volume is \(V\) cubic metres, where $$V = \left( h ^ { 6 } + 16 \right) ^ { \frac { 1 } { 2 } } - 4$$

1. Find the value of \(\frac { \mathrm { d } V } { \mathrm {~d} h }\) when \(h = 2\).
2. The volume of the pile is increasing at a constant rate of 8 cubic metres per hour. Find the rate, in metres per hour, at which the height of the pile is increasing at the instant when \(h = 2\). Give your answer correct to 2 significant figures.

5

1. Find \(\int ( 3 x + 7 ) ^ { 9 } \mathrm {~d} x\).
2. \includegraphics[max width=\textwidth, alt={}, center]{32f90420-e1eb-47ab-b588-e3806b64813f-3_537_881_402_671} The diagram shows the curve \(y = \frac { 1 } { 2 \sqrt { x } }\). The shaded region is bounded by the curve and the lines \(x = 3 , x = 6\) and \(y = 0\). The shaded region is rotated completely about the \(x\)-axis. Find the exact volume of the solid produced, simplifying your answer.

6 \includegraphics[max width=\textwidth, alt={}, center]{32f90420-e1eb-47ab-b588-e3806b64813f-3_641_837_1306_657} The diagram shows the graph of \(y = - \sin ^ { - 1 } ( x - 1 )\).

1. Give details of the pair of geometrical transformations which transforms the graph of \(y = - \sin ^ { - 1 } ( x - 1 )\) to the graph of \(y = \sin ^ { - 1 } x\).
2. Sketch the graph of \(y = \left| - \sin ^ { - 1 } ( x - 1 ) \right|\).
3. Find the exact solutions of the equation \(\left| - \sin ^ { - 1 } ( x - 1 ) \right| = \frac { 1 } { 3 } \pi\).

7 A curve has equation \(y = \frac { x \mathrm { e } ^ { 2 x } } { x + k }\), where \(k\) is a non-zero constant.

1. Differentiate \(x \mathrm { e } ^ { 2 x }\), and show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { \mathrm { e } ^ { 2 x } \left( 2 x ^ { 2 } + 2 k x + k \right) } { ( x + k ) ^ { 2 } }\).
2. Given that the curve has exactly one stationary point, find the value of \(k\), and determine the exact coordinates of the stationary point.

8 The definite integral \(I\) is defined by $$I = \int _ { 0 } ^ { 6 } 2 ^ { x } \mathrm {~d} x$$

1. Use Simpson's rule with 6 strips to find an approximate value of \(I\).
2. By first writing \(2 ^ { x }\) in the form \(\mathrm { e } ^ { k x }\), where the constant \(k\) is to be determined, find the exact value of \(I\).
3. Use the answers to parts (i) and (ii) to deduce that \(\ln 2 \approx \frac { 9 } { 13 }\).

9

1. Use the identity for \(\cos ( A + B )\) to prove that $$4 \cos \left( \theta + 60 ^ { \circ } \right) \cos \left( \theta + 30 ^ { \circ } \right) \equiv \sqrt { 3 } - 2 \sin 2 \theta .$$
2. Hence find the exact value of \(4 \cos 82.5 ^ { \circ } \cos 52.5 ^ { \circ }\).
3. Solve, for \(0 ^ { \circ } < \theta < 90 ^ { \circ }\), the equation \(4 \cos \left( \theta + 60 ^ { \circ } \right) \cos \left( \theta + 30 ^ { \circ } \right) = 1\).
4. Given that there are no values of \(\theta\) which satisfy the equation $$4 \cos \left( \theta + 60 ^ { \circ } \right) \cos \left( \theta + 30 ^ { \circ } \right) = k ,$$ determine the set of values of the constant \(k\).