AQA C3 (Core Mathematics 3) 2009 January

1 Use Simpson's rule with 5 ordinates (4 strips) to find an approximation to \(\int _ { 1 } ^ { 9 } \frac { 1 } { 1 + \sqrt { x } } \mathrm {~d} x\), giving your answer to three significant figures.

2 The diagram shows the curve with equation \(y = \sqrt { ( x - 2 ) ^ { 5 } }\) for \(x \geqslant 2\).
\includegraphics[max width=\textwidth, alt={}, center]{59b896ae-60ce-49ea-9c70-0f76fc5fffae-2_885_1125_854_461} The shaded region \(R\) is bounded by the curve \(y = \sqrt { ( x - 2 ) ^ { 5 } }\), the \(x\)-axis and the lines \(x = 3\) and \(x = 4\). Find the exact value of the volume of the solid formed when the region \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.

3 [Figure 1, printed on the insert, is provided for use in this question.]
The curve with equation \(y = x ^ { 3 } + 5 x - 4\) intersects the \(x\)-axis at the point \(A\), where \(x = \alpha\).

1. Show that \(\alpha\) lies between 0.5 and 1 .
2. Show that the equation \(x ^ { 3 } + 5 x - 4 = 0\) can be rearranged into the form $$x = \frac { 1 } { 5 } \left( 4 - x ^ { 3 } \right)$$
3. Use the iteration \(x _ { n + 1 } = \frac { 1 } { 5 } \left( 4 - x _ { n } { } ^ { 3 } \right)\) with \(x _ { 1 } = 0.5\) to find \(x _ { 3 }\), giving your answer to three decimal places.
4. The sketch on Figure 1 shows parts of the graphs of \(y = \frac { 1 } { 5 } \left( 4 - x ^ { 3 } \right)\) 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.

4

1. Solve the equation \(\sec x = \frac { 3 } { 2 }\), giving all values of \(x\) to the nearest degree in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\).
2. By using a suitable trigonometrical identity, solve the equation $$2 \tan ^ { 2 } x = 10 - 5 \sec x$$ giving all values of \(x\) to the nearest degree in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\).

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

1. State the range of f .
2. Explain why the function f does not have an inverse.
3. 1. Write down an expression for fg(x).
   2. Solve the equation \(\operatorname { fg } ( x ) = - 14\).

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

1. Find the value of the \(x\)-coordinate of each of the stationary points of the curve.
2. 1. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
   2. Determine the nature of each of the stationary points of the curve.

7

1. Given that \(3 \mathrm { e } ^ { x } = 4\), find the exact value of \(x\).
2. 1. By substituting \(y = \mathrm { e } ^ { x }\), show that the equation \(3 \mathrm { e } ^ { x } + 20 \mathrm { e } ^ { - x } = 19\) can be written as \(3 y ^ { 2 } - 19 y + 20 = 0\).
   2. Hence solve the equation \(3 \mathrm { e } ^ { x } + 20 \mathrm { e } ^ { - x } = 19\), giving your answers as exact values. (3 marks)

8 The sketch shows the graph of \(y = \cos ^ { - 1 } x\).
\includegraphics[max width=\textwidth, alt={}, center]{59b896ae-60ce-49ea-9c70-0f76fc5fffae-5_593_686_383_683}

1. Write down the coordinates of \(P\) and \(Q\), the end points of the graph.
2. Describe a sequence of two geometrical transformations that maps the graph of \(y = \cos ^ { - 1 } x\) onto the graph of \(y = 2 \cos ^ { - 1 } ( x - 1 )\).
3. Sketch the graph of \(y = 2 \cos ^ { - 1 } ( x - 1 )\).
4. 1. Write the equation \(y = 2 \cos ^ { - 1 } ( x - 1 )\) in the form \(x = \mathrm { f } ( y )\).
   2. Hence find the value of \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) when \(y = 2\).

9

1. Given that \(y = \frac { 4 x } { 4 x - 3 }\), use the quotient rule to show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { k } { ( 4 x - 3 ) ^ { 2 } }\), where \(k\) is an integer.
2. 1. Given that \(y = x \ln ( 4 x - 3 )\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
   2. Find an equation of the tangent to the curve \(y = x \ln ( 4 x - 3 )\) at the point where \(x = 1\).
3. 1. Use the substitution \(u = 4 x - 3\) to find \(\int \frac { 4 x } { 4 x - 3 } \mathrm {~d} x\), giving your answer in terms of \(x\).
   2. By using integration by parts, or otherwise, find \(\int \ln ( 4 x - 3 ) \mathrm { d } x\).