AQA C3 (Core Mathematics 3) 2007 January

1 Use the mid-ordinate rule with four strips of equal width to find an estimate for \(\int _ { 1 } ^ { 5 } \frac { 1 } { 1 + \ln x } \mathrm {~d} x\), giving your answer to three significant figures.
(4 marks)

2 Describe a sequence of two geometrical transformations that maps the graph of \(y = \sec x\) onto the graph of \(y = 1 + \sec 3 x\).

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

1. Find the range of f.
2. The inverse of g is \(\mathrm { g } ^ { - 1 }\).
   1. Find \(\mathrm { g } ^ { - 1 } ( x )\).
   2. State the range of \(\mathrm { g } ^ { - 1 }\).
3. The composite function gf is denoted by h .
   1. Find \(\mathrm { h } ( x )\), simplifying your answer.
   2. State the greatest possible domain of h .

4

1. Use integration by parts to find \(\int x \sin x \mathrm {~d} x\).
2. Using the substitution \(u = x ^ { 2 } + 5\), or otherwise, find \(\int x \sqrt { x ^ { 2 } + 5 } \mathrm {~d} x\).
3. The diagram shows the curve \(y = x ^ { 2 } - 9\) for \(x \geqslant 0\).
   \includegraphics[max width=\textwidth, alt={}, center]{6890a681-2b7f-4853-a5f0-f88b7b435367-3_844_663_685_694} The shaded region \(R\) is bounded by the curve, the lines \(y = 1\) and \(y = 2\), and the \(y\)-axis. Find the exact value of the volume of the solid generated when the region \(R\) is rotated through \(360 ^ { \circ }\) about the \(\boldsymbol { y }\)-axis.

5

1. 1. Show that the equation $$2 \cot ^ { 2 } x + 5 \operatorname { cosec } x = 10$$ can be written in the form \(2 \operatorname { cosec } ^ { 2 } x + 5 \operatorname { cosec } x - 12 = 0\).
   2. Hence show that \(\sin x = - \frac { 1 } { 4 }\) or \(\sin x = \frac { 2 } { 3 }\).
2. Hence, or otherwise, solve the equation $$2 \cot ^ { 2 } ( \theta - 0.1 ) + 5 \operatorname { cosec } ( \theta - 0.1 ) = 10$$ giving all values of \(\theta\) in radians to two decimal places in the interval \(- \pi < \theta < \pi\).
   (3 marks)

6

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when:
   1. \(y = \left( 4 x ^ { 2 } + 3 x + 2 \right) ^ { 10 }\);
   2. \(y = x ^ { 2 } \tan x\).
2. 1. Find \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) when \(x = 2 y ^ { 3 } + \ln y\).
   2. Hence find an equation of the tangent to the curve \(x = 2 y ^ { 3 } + \ln y\) at the point \(( 2,1 )\).

7

1. Sketch the graph of \(y = | 2 x |\).
2. On a separate diagram, sketch the graph of \(y = 4 - | 2 x |\), indicating the coordinates of the points where the graph crosses the coordinate axes.
3. Solve \(4 - | 2 x | = x\).
4. Hence, or otherwise, solve the inequality \(4 - | 2 x | > x\).

8 The diagram shows the curve \(y = \cos ^ { - 1 } x\) for \(- 1 \leqslant x \leqslant 1\).
\includegraphics[max width=\textwidth, alt={}, center]{6890a681-2b7f-4853-a5f0-f88b7b435367-4_492_698_1640_671}

1. Write down the exact coordinates of the points \(A\) and \(B\).
2. The equation \(\cos ^ { - 1 } x = 3 x + 1\) has only one root. Given that the root of this equation is \(\alpha\), show that \(0.1 \leqslant \alpha \leqslant 0.2\).
3. Use the iteration \(x _ { n + 1 } = \frac { 1 } { 3 } \left( \cos ^ { - 1 } x _ { n } - 1 \right)\) with \(x _ { 1 } = 0.1\) to find the values of \(x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\), giving your answers to three decimal places.

9 The sketch shows the graph of \(y = 4 - \mathrm { e } ^ { 2 x }\). The curve crosses the \(y\)-axis at the point \(A\) and the \(x\)-axis at the point \(B\).
\includegraphics[max width=\textwidth, alt={}, center]{6890a681-2b7f-4853-a5f0-f88b7b435367-5_711_921_466_557}

1. 1. Find \(\int \left( 4 - \mathrm { e } ^ { 2 x } \right) \mathrm { d } x\).
      (2 marks)
   2. Hence show that \(\int _ { 0 } ^ { \ln 2 } \left( 4 - \mathrm { e } ^ { 2 x } \right) \mathrm { d } x = 4 \ln 2 - \frac { 3 } { 2 }\).
2. 1. Write down the \(y\)-coordinate of \(A\).
   2. Show that \(x = \ln 2\) at \(B\).
3. Find the equation of the normal to the curve \(y = 4 - \mathrm { e } ^ { 2 x }\) at the point \(B\).
4. Find the area of the region enclosed by the curve \(y = 4 - \mathrm { e } ^ { 2 x }\), the normal to the curve at \(B\) and the \(y\)-axis.