AQA C3 (Core Mathematics 3) 2012 January

1

1. Use Simpson's rule with 7 ordinates (6 strips) to find an estimate for \(\int _ { 0 } ^ { 3 } 4 ^ { x } \mathrm {~d} x\).
2. A curve is defined by the equation \(y = 4 ^ { x }\). The curve intersects the line \(y = 8 - 2 x\) at a single point where \(x = \alpha\).
   1. Show that \(\alpha\) lies between 1.2 and 1.3.
   2. The equation \(4 ^ { x } = 8 - 2 x\) can be rearranged into the form \(x = \frac { \ln ( 8 - 2 x ) } { \ln 4 }\). Use the iterative formula \(x _ { n + 1 } = \frac { \ln \left( 8 - 2 x _ { n } \right) } { \ln 4 }\) with \(x _ { 1 } = 1.2\) to find the values of \(x _ { 2 }\) and \(x _ { 3 }\), giving your answers to three decimal places.
      (2 marks)

2 The curve with equation \(y = \frac { 63 } { 4 x - 1 }\) is sketched below for \(1 \leqslant x \leqslant 16\).
\includegraphics[max width=\textwidth, alt={}, center]{7aa76d26-e3c4-4374-ae4f-8bb61e61b135-2_568_698_1308_669} The function f is defined by \(\mathrm { f } ( x ) = \frac { 63 } { 4 x - 1 }\) for \(1 \leqslant x \leqslant 16\).

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 \(\mathrm { f } ^ { - 1 } ( x ) = 1\).
3. The function g is defined by \(\mathrm { g } ( x ) = x ^ { 2 }\) for \(- 4 \leqslant x \leqslant - 1\).
   1. Write down an expression for \(\mathrm { fg } ( x )\).
   2. Solve the equation \(\operatorname { fg } ( x ) = 1\).

3

1. Given that \(y = 4 x ^ { 3 } - 6 x + 1\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
   (l mark)
2. Hence find \(\int _ { 2 } ^ { 3 } \frac { 2 x ^ { 2 } - 1 } { 4 x ^ { 3 } - 6 x + 1 } \mathrm {~d} x\), giving your answer in the form \(p \ln q\), where \(p\) and \(q\) are rational numbers.

4

1. By using a suitable trigonometrical identity, solve the equation $$\tan ^ { 2 } \theta = 3 ( 3 - \sec \theta )$$ giving all solutions to the nearest \(0.1 ^ { \circ }\) in the interval \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
2. Hence solve the equation $$\tan ^ { 2 } \left( 4 x - 10 ^ { \circ } \right) = 3 \left[ 3 - \sec \left( 4 x - 10 ^ { \circ } \right) \right]$$ giving all solutions to the nearest \(0.1 ^ { \circ }\) in the interval \(0 ^ { \circ } < x < 90 ^ { \circ }\).

5

1. Describe a sequence of two geometrical transformations that maps the graph of \(y = \ln x\) onto the graph of \(y = 4 \ln ( x - \mathrm { e } )\).
2. Sketch, on the axes given below, the graph of \(y = | 4 \ln ( x - \mathrm { e } ) |\), indicating the exact value of the \(x\)-coordinate where the curve meets the \(x\)-axis.
3. 1. Solve the equation \(| 4 \ln ( x - e ) | = 4\).
   2. Hence, or otherwise, solve the inequality \(| 4 \ln ( x - e ) | \geqslant 4\).
      \includegraphics[max width=\textwidth, alt={}, center]{7aa76d26-e3c4-4374-ae4f-8bb61e61b135-3_655_1428_2023_315}

6

1. Given that \(x = \frac { 1 } { \sin \theta }\), use the quotient rule to show that \(\frac { \mathrm { d } x } { \mathrm {~d} \theta } = - \operatorname { cosec } \theta \cot \theta\).
   (3 marks)
2. Use the substitution \(x = \operatorname { cosec } \theta\) to find \(\int _ { \sqrt { 2 } } ^ { 2 } \frac { 1 } { x ^ { 2 } \sqrt { x ^ { 2 } - 1 } } \mathrm {~d} x\), giving your answer to three significant figures.
   (9 marks)

7

1. A curve has equation \(y = x ^ { 2 } \mathrm { e } ^ { - \frac { x } { 4 } }\).
   Show that the curve has exactly two stationary points and find the exact values of their coordinates.
   (7 marks)
2. 1. Use integration by parts twice to find the exact value of \(\int _ { 0 } ^ { 4 } x ^ { 2 } \mathrm { e } ^ { - \frac { x } { 4 } } \mathrm {~d} x\).
   2. The region bounded by the curve \(y = 3 x \mathrm { e } ^ { - \frac { x } { 8 } }\), the \(x\)-axis from 0 to 4 and the line \(x = 4\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis to form a solid. Use your answer to part (b)(i) to find the exact value of the volume of the solid generated.