OCR MEI C3 (Core Mathematics 3)

1 Fig. 9 shows the curve \(y = \frac { x ^ { 2 } } { 3 x - 1 }\).
P is a turning point, and the curve has a vertical asymptote \(x = a\). \begin{figure}[h] \end{figure}

1. Write down the value of \(a\).
2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x ( 3 x - 2 ) } { ( 3 x - 1 ) ^ { 2 } }\).
3. Find the exact coordinates of the turning point P . Calculate the gradient of the curve when \(x = 0.6\) and \(x = 0.8\), and hence verify that P is a minimum point.
4. Using the substitution \(u = 3 x - 1\), show that \(\int \frac { x ^ { 2 } } { 3 x - 1 } \mathrm {~d} x = \frac { 1 } { 27 } \int \left( u + 2 + \frac { 1 } { u } \right) \mathrm { d } u\). Hence find the exact area of the region enclosed by the curve, the \(x\)-axis and the lines \(x = \frac { 2 } { 3 }\) and \(x = 1\).

2 Differentiate \(\sqrt [ 3 ] { 1 + 6 x ^ { 2 } }\).

3 Show that the curve \(y = x ^ { 2 } \ln x\) has a stationary point when \(x = \frac { 1 } { \sqrt { \mathrm { e } } }\).

4 The equation of a curve is \(y = \frac { x ^ { 2 } } { 2 x + 1 }\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 x ( x + 1 ) } { ( 2 x + 1 ) ^ { 2 } }\).
2. Find the coordinates of the stationary points of the curve. You need not determine their nature.
3. Differentiate \(\sqrt { 1 + 2 x }\).
4. Show that the derivative of \(\ln \left( 1 - \mathrm { e } ^ { - x } \right)\) is \(\frac { 1 } { \mathrm { e } ^ { x } - 1 }\).

6 The function \(\mathrm { f } ( x ) = \frac { \sin x } { 2 - \cos x }\) has domain \(- \pi \leqslant x \leqslant \pi\).
Fig. 8 shows the graph of \(y = \mathrm { f } ( x )\) for \(0 \leqslant x \leqslant \pi\). \begin{figure}[h] \end{figure}

1. Find \(\mathrm { f } ( - x )\) in terms of \(\mathrm { f } ( x )\). Hence sketch the graph of \(y = \mathrm { f } ( x )\) for the complete domain \(- \pi \leqslant x \leqslant \pi\).
2. Show that \(\mathrm { f } ^ { \prime } ( x ) = \frac { 2 \cos x - 1 } { ( 2 - \cos x ) ^ { 2 } }\). Hence find the exact coordinates of the turning point P . State the range of the function \(\mathrm { f } ( x )\), giving your answer exactly.
3. Using the substitution \(u = 2 - \cos x\) or otherwise, find the exact value of \(\int _ { 0 } ^ { \pi } \frac { \sin x } { 2 \cos x } \mathrm {~d} x\).
4. Sketch the graph of \(y = \mathrm { f } ( 2 x )\).
5. Using your answers to parts (iii) and (iv), write down the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \frac { \sin 2 x } { 2 \cos 2 x } \mathrm {~d} x\).

7 Fig. 3 shows the curve defined by the equation \(y = \arcsin ( x - 1 )\), for \(0 \leqslant x \leqslant 2\). \begin{figure}[h] \end{figure}

1. Find \(x\) in terms of \(y\), and show that \(\frac { \mathrm { d } x } { \mathrm {~d} y } = \cos y\).
2. Hence find the exact gradient of the curve at the point where \(x = 1.5\).

8 A curve has equation \(y = \frac { x } { 2 + 3 \ln x }\). Find \(\frac { d y } { d x }\). Hence find the exact coordinates of the stationary point of the curve.