OCR C3 (Core Mathematics 3) 2009 June

1 \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} Each diagram above shows part of a curve, the equation of which is one of the following: $$y = \sin ^ { - 1 } x , \quad y = \cos ^ { - 1 } x , \quad y = \tan ^ { - 1 } x , \quad y = \sec x , \quad y = \operatorname { cosec } x , \quad y = \cot x .$$ State which equation corresponds to

1. Fig. 1,
2. Fig. 2,
3. Fig. 3.

2
\includegraphics[max width=\textwidth, alt={}, center]{6a690aa5-63a7-4569-afa8-0746814ebab4-2_477_833_1493_657} The diagram shows the curve with equation \(y = ( 2 x - 3 ) ^ { 2 }\). The shaded region is bounded by the curve and the lines \(x = 0\) and \(y = 0\). Find the exact volume obtained when the shaded region is rotated completely about the \(x\)-axis.

3 The angles \(\alpha\) and \(\beta\) are such that $$\tan \alpha = m + 2 \quad \text { and } \quad \tan \beta = m$$ where \(m\) is a constant.

1. Given that \(\sec ^ { 2 } \alpha - \sec ^ { 2 } \beta = 16\), find the value of \(m\).
2. Hence find the exact value of \(\tan ( \alpha + \beta )\).

4 It is given that \(\int _ { a } ^ { 3 a } \left( \mathrm { e } ^ { 3 x } + \mathrm { e } ^ { x } \right) \mathrm { d } x = 100\), where \(a\) is a positive constant.

1. Show that \(a = \frac { 1 } { 9 } \ln \left( 300 + 3 \mathrm { e } ^ { a } - 2 \mathrm { e } ^ { 3 a } \right)\).
2. Use an iterative process, based on the equation in part (i), to find the value of \(a\) correct to 4 decimal places. Use a starting value of 0.6 and show the result of each step of the process.

5 The functions f and g are defined for all real values of \(x\) by $$\mathrm { f } ( x ) = 3 x - 2 \quad \text { and } \quad \mathrm { g } ( x ) = 3 x + 7$$ Find the exact coordinates of the point at which

1. the graph of \(y = \operatorname { fg } ( x )\) meets the \(x\)-axis,
2. the graph of \(y = \mathrm { g } ( x )\) meets the graph of \(y = \mathrm { g } ^ { - 1 } ( x )\),
3. the graph of \(y = | \mathrm { f } ( x ) |\) meets the graph of \(y = | \mathrm { g } ( x ) |\).

6
\includegraphics[max width=\textwidth, alt={}, center]{6a690aa5-63a7-4569-afa8-0746814ebab4-3_590_606_1197_772} The diagram shows the curve with equation \(x = \left( 37 + 10 y - 2 y ^ { 2 } \right) ^ { \frac { 1 } { 2 } }\).

1. Find an expression for \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) in terms of \(y\).
2. Hence find the equation of the tangent to the curve at the point ( 7,3 ), giving your answer in the form \(y = m x + c\).
3. Express \(8 \sin \theta - 6 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
4. Hence
   (a) solve, for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\), the equation \(8 \sin \theta - 6 \cos \theta = 9\),
   (b) find the greatest possible value of $$32 \sin x - 24 \cos x - ( 16 \sin y - 12 \cos y )$$ as the angles \(x\) and \(y\) vary.

8
\includegraphics[max width=\textwidth, alt={}, center]{6a690aa5-63a7-4569-afa8-0746814ebab4-4_648_1132_262_504} The diagram shows the curves \(y = \ln x\) and \(y = 2 \ln ( x - 6 )\). The curves meet at the point \(P\) which has \(x\)-coordinate \(a\). The shaded region is bounded by the curve \(y = 2 \ln ( x - 6 )\) and the lines \(x = a\) and \(y = 0\).

1. Give details of the pair of transformations which transforms the curve \(y = \ln x\) to the curve \(y = 2 \ln ( x - 6 )\).
2. Solve an equation to find the value of \(a\).
3. Use Simpson's rule with two strips to find an approximation to the area of the shaded region.

9

1. Show that, for all non-zero values of the constant \(k\), the curve $$y = \frac { k x ^ { 2 } - 1 } { k x ^ { 2 } + 1 }$$ has exactly one stationary point.
2. Show that, for all non-zero values of the constant \(m\), the curve $$y = \mathrm { e } ^ { m x } \left( x ^ { 2 } + m x \right)$$ has exactly two stationary points.