AQA C3 (Core Mathematics 3) 2010 January

1 A curve has equation \(y = \mathrm { e } ^ { - 4 x } \left( x ^ { 2 } + 2 x - 2 \right)\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 \mathrm { e } ^ { - 4 x } \left( 5 - 3 x - 2 x ^ { 2 } \right)\).
2. Find the exact values of the coordinates of the stationary points of the curve.

2 [Figure 1, printed on the insert, is provided for use in this question.]

1. 1. Sketch the graph of \(y = \sin ^ { - 1 } x\), where \(y\) is in radians. State the coordinates of the end points of the graph.
   2. By drawing a suitable straight line on your sketch, show that the equation $$\sin ^ { - 1 } x = \frac { 1 } { 4 } x + 1$$ has only one solution.
2. The root of the equation \(\sin ^ { - 1 } x = \frac { 1 } { 4 } x + 1\) is \(\alpha\). Show that \(0.5 < \alpha < 1\).
3. The equation \(\sin ^ { - 1 } x = \frac { 1 } { 4 } x + 1\) can be rewritten as \(x = \sin \left( \frac { 1 } { 4 } x + 1 \right)\).
   1. Use the iteration \(x _ { n + 1 } = \sin \left( \frac { 1 } { 4 } x _ { n } + 1 \right)\) with \(x _ { 1 } = 0.5\) to find the values of \(x _ { 2 }\) and \(x _ { 3 }\), giving your answers to three decimal places.
   2. The sketch on Figure 1 shows parts of the graphs of \(y = \sin \left( \frac { 1 } { 4 } x + 1 \right)\) and \(y = x\), and the position of \(x _ { 1 }\). On Figure 1, draw a cobweb or staircase diagram to show how convergence takes place, indicating the positions of \(x _ { 2 }\) and \(x _ { 3 }\) on the \(x\)-axis.

3

1. Solve the equation $$\operatorname { cosec } x = 3$$ giving all values of \(x\) in radians to two decimal places, in the interval \(0 \leqslant x \leqslant 2 \pi\).
   (2 marks)
2. By using a suitable trigonometric identity, solve the equation $$\cot ^ { 2 } x = 11 - \operatorname { cosec } x$$ giving all values of \(x\) in radians to two decimal places, in the interval \(0 \leqslant x \leqslant 2 \pi\).
   (6 marks)

4

1. Sketch the graph of \(y = | 8 - 2 x |\).
2. Solve the equation \(| 8 - 2 x | = 4\).
3. Solve the inequality \(| 8 - 2 x | > 4\).

5

1. Use the mid-ordinate rule with four strips to find an estimate for \(\int _ { 0 } ^ { 12 } \ln \left( x ^ { 2 } + 5 \right) \mathrm { d } x\), giving your answer to three significant figures.
2. A curve has equation \(y = \ln \left( x ^ { 2 } + 5 \right)\).
   1. Show that this equation can be rewritten as \(x ^ { 2 } = \mathrm { e } ^ { y } - 5\).
   2. The region bounded by the curve, the lines \(y = 5\) and \(y = 10\) and the \(y\)-axis is rotated through \(360 ^ { \circ }\) about the \(y\)-axis. Find the exact value of the volume of the solid generated.
3. The graph with equation \(y = \ln \left( x ^ { 2 } + 5 \right)\) is stretched with scale factor 4 parallel to the \(x\)-axis, and then translated through \(\left[ \begin{array} { l } 0
   3 \end{array} \right]\) to give the graph with equation \(y = \mathrm { f } ( x )\). Write down an expression for \(\mathrm { f } ( x )\).

6 The functions \(f\) and \(g\) are defined with their respective domains by $$\begin{array} { l l } \mathrm { f } ( x ) = \mathrm { e } ^ { 2 x } - 3 , & \text { for all real values of } x
\mathrm {~g} ( x ) = \frac { 1 } { 3 x + 4 } , & \text { for real values of } x , x \neq - \frac { 4 } { 3 } \end{array}$$

1. Find the range of \(f\).
2. The inverse of f is \(\mathrm { f } ^ { - 1 }\).
   1. Find \(\mathrm { f } ^ { - 1 } ( x )\).
   2. Solve the equation \(\mathrm { f } ^ { - 1 } ( x ) = 0\).
3. 1. Find an expression for \(\operatorname { gf } ( x )\).
   2. Solve the equation \(\mathrm { gf } ( x ) = 1\), giving your answer in an exact form.

7 It is given that \(y = \tan 4 x\).

1. By writing \(\tan 4 x\) as \(\frac { \sin 4 x } { \cos 4 x }\), use the quotient rule to show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = p \left( 1 + \tan ^ { 2 } 4 x \right)\), where \(p\) is a number to be determined.
2. Show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = q y \left( 1 + y ^ { 2 } \right)\), where \(q\) is a number to be determined.

8

1. Using integration by parts, find \(\int x \sin ( 2 x - 1 ) \mathrm { d } x\).
2. Use the substitution \(u = 2 x - 1\) to find \(\int \frac { x ^ { 2 } } { 2 x - 1 } \mathrm {~d} x\), giving your answer in terms of \(x\).
   (6 marks)