AQA C3 (Core Mathematics 3) 2005 June

1

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(y = x \sin 2 x\).
2. 1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(y = \left( x ^ { 2 } - 6 \right) ^ { 4 }\).
   2. Hence, or otherwise, find \(\int x \left( x ^ { 2 } - 6 \right) ^ { 3 } \mathrm {~d} x\).

2 The functions \(f\) and \(g\) are defined with their respective domains by $$\begin{array} { l l } \mathrm { f } ( x ) = x - 2 & \text { for all real values of } x
\mathrm {~g} ( x ) = \frac { 6 } { x + 3 } & \text { for real values of } x , \quad x \neq - 3 \end{array}$$ The composite function fg is denoted by h .

1. Find \(\mathrm { h } ( x )\).
2. 1. Find \(\mathrm { h } ^ { - 1 } ( x )\), where \(\mathrm { h } ^ { - 1 }\) is the inverse of h .
   2. Find the range of \(\mathrm { h } ^ { - 1 }\).

3

1. Find \(\int \mathrm { e } ^ { 4 x } \mathrm {~d} x\).
2. Use integration by parts to find \(\int \mathrm { e } ^ { 4 x } ( 2 x + 1 ) \mathrm { d } x\).
3. By using the substitution \(u = 1 + \ln x\), or otherwise, find \(\int \frac { 1 + \ln x } { x } \mathrm {~d} x\).

4 It is given that \(\tan ^ { 2 } x = \sec x + 11\).

1. Show that the equation \(\tan ^ { 2 } x = \sec x + 11\) can be written in the form $$\sec ^ { 2 } x - \sec x - 12 = 0$$
2. Hence show that \(\cos x = \frac { 1 } { 4 }\) or \(\cos x = - \frac { 1 } { 3 }\).
3. Hence, or otherwise, solve the equation \(\tan ^ { 2 } x = \sec x + 11\), giving all values of \(x\) to the nearest degree in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\).

5

1. Solve the equation \(2 \mathrm { e } ^ { x } = 5\), giving your answer as an exact natural logarithm.
2. 1. By substituting \(y = \mathrm { e } ^ { x }\), show that the equation \(2 \mathrm { e } ^ { x } + 5 \mathrm { e } ^ { - x } = 7\) can be written as $$2 y ^ { 2 } - 7 y + 5 = 0$$
   2. Hence solve the equation \(2 \mathrm { e } ^ { x } + 5 \mathrm { e } ^ { - x } = 7\), giving your answers as exact values of \(x\).

6

1. 1. Sketch the graph of \(y = 4 - x ^ { 2 }\), indicating the coordinates of the points where the graph crosses the coordinate axes.
   2. The region between the graph and the \(x\)-axis from \(x = 0\) to \(x = 2\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Find the exact value of the volume of the solid generated.
2. 1. Sketch the graph of \(y = \left| 4 - x ^ { 2 } \right|\).
   2. Solve \(\left| 4 - x ^ { 2 } \right| = 3\).
   3. Hence, or otherwise, solve the inequality \(\left| 4 - x ^ { 2 } \right| < 3\).

7

1. Sketch the graph of \(y = \tan ^ { - 1 } x\).
2. 1. By drawing a suitable straight line on your sketch, show that the equation \(\tan ^ { - 1 } x = 2 x - 1\) has only one root.
   2. Given that the root of this equation is \(\alpha\), show that \(0.8 < \alpha < 0.9\).
3. Use the iteration \(x _ { n + 1 } = \frac { 1 } { 2 } \left( \tan ^ { - 1 } x _ { n } + 1 \right)\) with \(x _ { 1 } = 0.8\) to find the value of \(x _ { 3 }\), giving your answer to two significant figures.

8 The diagram shows part of the graph of \(y = \mathrm { e } ^ { 2 x } + 3\).
\includegraphics[max width=\textwidth, alt={}, center]{d5b78fa6-ea3c-497b-94d8-1d5f61288aa5-4_833_1034_1027_513}

1. Describe a sequence of two geometrical transformations that maps the graph of \(y = \mathrm { e } ^ { x }\) onto the graph of \(y = \mathrm { e } ^ { 2 x } + 3\).
2. Use the mid-ordinate rule with four strips of equal width to find an estimate for the area of the shaded region \(A\), giving your answer to three significant figures.
3. Find the exact value of the area of the shaded region \(A\).
4. The region \(B\) is indicated on the diagram. Find the area of the region \(B\), giving your answer in the form \(p \mathrm { e } ^ { 8 } + q \mathrm { e } ^ { 4 }\), where \(p\) and \(q\) are numbers to be determined.