OCR C3 (Core Mathematics 3) 2007 June

1 Differentiate each of the following with respect to \(x\).

1. \(x ^ { 3 } ( x + 1 ) ^ { 5 }\)
2. \(\sqrt { 3 x ^ { 4 } + 1 }\)

2 Solve the inequality \(| 4 x - 3 | < | 2 x + 1 |\).

3 The function \(f\) is defined for all non-negative values of \(x\) by $$f ( x ) = 3 + \sqrt { x }$$

1. Evaluate ff(169).
2. Find an expression for \(\mathrm { f } ^ { - 1 } ( \mathrm { x } )\) in terms of x .
3. On a single diagram sketch the graphs of \(y = f ( x )\) and \(y = f ^ { - 1 } ( x )\), indicating how the two graphs are related.

4 The integral I is defined by $$I = \int _ { 0 } ^ { 13 } ( 2 x + 1 ) ^ { \frac { 1 } { 3 } } d x$$

1. Use integration to find the exact value of I .
2. Use Simpson's rule with two strips to find an approximate value for I. Give your answer correct to 3 significant figures.

5 A substance is decaying in such a way that its mass, m kg , at a time t years from now is given by the formula $$\mathrm { m } = 240 \mathrm { e } ^ { - 0.04 \mathrm { t } }$$

1. Find the time taken for the substance to halve its mass.
2. Find the value of \(t\) for which the mass is decreasing at a rate of 2.1 kg per year.

6

1. Given that \(\int _ { 0 } ^ { \mathrm { a } } \left( 6 \mathrm { e } ^ { 2 \mathrm { x } } + \mathrm { x } \right) \mathrm { dx } = 42\), show that \(\mathrm { a } = \frac { 1 } { 2 } \ln \left( 15 - \frac { 1 } { 6 } \mathrm { a } ^ { 2 } \right)\).
2. Use an iterative formula, based on the equation in part (i), to find the value of a correct to 3 decimal places. Use a starting value of 1 and show the result of each iteration.

7

1. Sketch the graph of \(y = \sec x\) for \(0 \leqslant x \leqslant 2 \pi\).
2. Solve the equation \(\sec x = 3\) for \(0 \leqslant x \leqslant 2 \pi\), giving the roots correct to 3 significant figures.
3. Solve the equation \(\sec \theta = 5 \operatorname { cosec } \theta\) for \(0 \leqslant \theta \leqslant 2 \pi\), giving the roots correct to 3 significant figures.

8

1. Given that \(\mathrm { y } = \frac { 4 \ln \mathrm { x } - 3 } { 4 \ln \mathrm { x } + 3 }\), show that \(\frac { \mathrm { dy } } { \mathrm { dx } } = \frac { 24 } { \mathrm { x } ( 4 \ln \mathrm { x } + 3 ) ^ { 2 } }\).
2. Find the exact value of the gradient of the curve \(y = \frac { 4 \ln x - 3 } { 4 \ln x + 3 }\) at the point where it crosses the \(x\)-axis.
3. 
   \includegraphics[max width=\textwidth, alt={}, center]{133c38fb-307f-4f20-86cb-1bd57cc4f870-3_524_830_941_699} The diagram shows part of the curve with equation $$\mathrm { y } = \frac { 2 } { \mathrm { x } ^ { \frac { 1 } { 2 } } ( 4 \ln \mathrm { x } + 3 ) }$$ The region shaded in the diagram is bounded by the curve and the lines \(x = 1 , x = e\) and \(y = 0\). Find the exact volume of the solid produced when this shaded region is rotated completely about the x -axis.

9

1. Prove the identity $$\tan \left( \theta + 60 ^ { \circ } \right) \tan \left( \theta - 60 ^ { \circ } \right) \equiv \frac { \tan ^ { 2 } \theta - 3 } { 1 - 3 \tan ^ { 2 } \theta }$$
2. Solve, for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\), the equation $$\tan \left( \theta + 60 ^ { \circ } \right) \tan \left( \theta - 60 ^ { \circ } \right) = 4 \sec ^ { 2 } \theta - 3 ,$$ giving your answers correct to the nearest \(0.1 ^ { \circ }\).
3. Show that, for all values of the constant k , the equation $$\tan \left( \theta + 60 ^ { \circ } \right) \tan \left( \theta - 60 ^ { \circ } \right) = \mathrm { K } ^ { 2 }$$ has two roots in the interval \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).