OCR MEI C3 (Core Mathematics 3)

2

1. Expand \(\left( \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } \right) ^ { 2 }\).
2. Hence find \(\int \left( \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } \right) ^ { 2 } \mathrm {~d} x\).

3

1. Sketch the graph of \(y = | 3 x - 6 |\).
2. Solve the equation \(| 3 x - 6 | = x + 4\) and illustrate your answer on your graph.
   \(4 \quad\) Find \(\int x \sin 3 x \mathrm {~d} x\).
   \(5 \quad\) Make \(x\) the subject of \(t = \ln \sqrt { \frac { 5 } { ( x - 3 ) } }\).

6 The function \(\mathrm { f } ( x )\) is defined as \(\mathrm { f } ( x ) = \frac { \ln x } { x }\). The graph of the function is shown in Fig. 6 . \begin{figure}[h] \end{figure}

1. Give the coordinates of the point, P , where the curve crosses the \(x\)-axis.
2. Use calculus to find the coordinates of the stationary point, Q , and show that it is a maximum.

7 An oil slick is circular with radius \(r \mathrm {~km}\) and area \(A \mathrm {~km} ^ { 2 }\). The radius increases with time at a rate given by \(\frac { \mathrm { d } r } { \mathrm {~d} t } = 0.5\), in kilometres per hour.

1. Show that \(\frac { \mathrm { dA } } { \mathrm { d } t } = \pi r\).
2. Find the rate of increase of the area of the slick at a time when the radius is 6 km .

8 Fig. 8 shows the graph of \(y = x \sqrt { 1 + x }\). The point P on the curve is on the \(x\)-axis. \begin{figure}[h] \end{figure}

1. Write down the coordinates of P .
2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 x + 2 } { 2 \sqrt { 1 + x } }\).
3. Hence find the coordinates of the turning point on the curve. What can you say about the gradient of the curve at P ?
4. By using a suitable substitution, show that \(\int _ { - 1 } ^ { 0 } x \sqrt { 1 + x } \mathrm {~d} x = \int _ { 0 } ^ { 1 } \left( u ^ { \frac { 3 } { 2 } } - u ^ { \frac { 1 } { 2 } } \right) \mathrm { d } u\). Evaluate this integral, giving your answer in an exact form.
   What does this value represent?
5. Use your answer to part (ii) to differentiate \(y = x \sqrt { 1 + x } \sin 2 x\) with respect to \(x\).
   (You need not simplify your result.)

9 The functions \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) are defined by $$\mathrm { f } ( x ) = x ^ { 2 } , \quad \mathrm {~g} ( x ) = 2 x - 1$$ for all real values of \(x\).

1. State the ranges of \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\). Explain why \(\mathrm { f } ( x )\) has no inverse.
2. Find an expression for the inverse function \(\mathrm { g } ^ { - 1 } ( x )\) in terms of \(x\). Sketch the graphs of \(y = \mathrm { g } ( x )\) and \(y = \mathrm { g } ^ { - 1 } ( x )\) on the same axes.
3. Find expressions for \(\operatorname { gf } ( x )\) and \(\operatorname { fg } ( x )\).
4. Solve the equation \(\operatorname { gf } ( x ) = \mathrm { fg } ( x )\). Sketch the graphs of \(y = \operatorname { gf } ( x )\) and \(y = \operatorname { fg } ( x )\) on the same axes to illustrate your answer.
5. Show that the equation \(\mathrm { f } ( x + a ) = \mathrm { g } ^ { 2 } ( x )\) has no solution if \(a > \frac { 1 } { 4 }\).