OCR C3 (Core Mathematics 3) 2007 January

1 Find the equation of the tangent to the curve \(y = \frac { 2 x + 1 } { 3 x - 1 }\) at the point \(\left( 1 , \frac { 3 } { 2 } \right)\), giving your answer in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers.

2 It is given that \(\theta\) is the acute angle such that \(\sin \theta = \frac { 12 } { 13 }\). Find the exact value of

1. \(\cot \theta\),
2. \(\cos 2 \theta\).

3

1. It is given that \(a\) and \(b\) are positive constants. By sketching graphs of $$y = x ^ { 5 } \quad \text { and } \quad y = a - b x$$ on the same diagram, show that the equation $$x ^ { 5 } + b x - a = 0$$ has exactly one real root.
2. Use the iterative formula \(x _ { n + 1 } = \sqrt [ 5 ] { 53 - 2 x _ { n } }\), with a suitable starting value, to find the real root of the equation \(x ^ { 5 } + 2 x - 53 = 0\). Show the result of each iteration, and give the root correct to 3 decimal places.

4

1. Given that \(x = ( 4 t + 9 ) ^ { \frac { 1 } { 2 } }\) and \(y = 6 \mathrm { e } ^ { \frac { 1 } { 2 } x + 1 }\), find expressions for \(\frac { \mathrm { d } x } { \mathrm {~d} t }\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Hence find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} t }\) when \(t = 4\), giving your answer correct to 3 significant figures.

5

1. Express \(4 \cos \theta - \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
2. Hence solve the equation \(4 \cos \theta - \sin \theta = 2\), giving all solutions for which \(- 180 ^ { \circ } < \theta < 180 ^ { \circ }\).

6
\includegraphics[max width=\textwidth, alt={}, center]{1216a06e-7e14-48d7-a7ca-7acd8d71af5f-3_483_956_264_593} The diagram shows the curve with equation \(y = \frac { 1 } { \sqrt { 3 x + 2 } }\). The shaded region is bounded by the curve and the lines \(x = 0 , x = 2\) and \(y = 0\).

1. Find the exact area of the shaded region.
2. The shaded region is rotated completely about the \(x\)-axis. Find the exact volume of the solid formed, simplifying your answer.

7 The curve \(y = \ln x\) is transformed to the curve \(y = \ln \left( \frac { 1 } { 2 } x - a \right)\) by means of a translation followed by a stretch. It is given that \(a\) is a positive constant.

1. Give full details of the translation and stretch involved.
2. Sketch the graph of \(y = \ln \left( \frac { 1 } { 2 } x - a \right)\).
3. Sketch, on another diagram, the graph of \(y = \left| \ln \left( \frac { 1 } { 2 } x - a \right) \right|\).
4. State, in terms of \(a\), the set of values of \(x\) for which \(\left| \ln \left( \frac { 1 } { 2 } x - a \right) \right| = - \ln \left( \frac { 1 } { 2 } x - a \right)\).

8
\includegraphics[max width=\textwidth, alt={}, center]{1216a06e-7e14-48d7-a7ca-7acd8d71af5f-4_538_1443_262_351} The diagram shows the curve with equation \(y = x ^ { 8 } \mathrm { e } ^ { - x ^ { 2 } }\). The curve has maximum points at \(P\) and \(Q\). The shaded region \(A\) is bounded by the curve, the line \(y = 0\) and the line through \(Q\) parallel to the \(y\)-axis. The shaded region \(B\) is bounded by the curve and the line \(P Q\).

1. Show by differentiation that the \(x\)-coordinate of \(Q\) is 2 .
2. Use Simpson's rule with 4 strips to find an approximation to the area of region \(A\). Give your answer correct to 3 decimal places.
3. Deduce an approximation to the area of region \(B\).

9 Functions \(f\) and \(g\) are defined by $$\begin{array} { l l } \mathrm { f } ( x ) = 2 \sin x & \text { for } - \frac { 1 } { 2 } \pi \leqslant x \leqslant \frac { 1 } { 2 } \pi
\mathrm {~g} ( x ) = 4 - 2 x ^ { 2 } & \text { for } x \in \mathbb { R } . \end{array}$$

1. State the range of f and the range of g .
2. Show that \(\operatorname { gf } ( 0.5 ) = 2.16\), correct to 3 significant figures, and explain why \(\mathrm { fg } ( 0.5 )\) is not defined.
3. Find the set of values of \(x\) for which \(\mathrm { f } ^ { - 1 } \mathrm {~g} ( x )\) is not defined.