AQA C3 (Core Mathematics 3) 2012 June

1 Use the mid-ordinate rule with four strips to find an estimate for \(\int _ { 0.4 } ^ { 1.2 } \cot \left( x ^ { 2 } \right) \mathrm { d } x\), giving your answer to three decimal places.

2 For \(0 < x \leqslant 2\), the curves with equations \(y = 4 \ln x\) and \(y = \sqrt { x }\) intersect at a single point where \(x = \alpha\).

1. Show that \(\alpha\) lies between 0.5 and 1.5.
2. Show that the equation \(4 \ln x = \sqrt { x }\) can be rearranged into the form $$x = \mathrm { e } ^ { \left( \frac { \sqrt { x } } { 4 } \right) }$$
3. Use the iterative formula $$x _ { n + 1 } = \mathrm { e } ^ { \left( \frac { \sqrt { x _ { n } } } { 4 } \right) }$$ with \(x _ { 1 } = 0.5\) to find the values of \(x _ { 2 }\) and \(x _ { 3 }\), giving your answers to three decimal places.
4. Figure 1, on the page 3, shows a sketch of parts of the graphs of \(y = \mathrm { e } ^ { \left( \frac { \sqrt { x } } { 4 } \right) }\) 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. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{d3c66c34-b09c-4223-8383-cf0a68419bf9-3_1285_1543_356_296}
   \end{figure}

3 A curve has equation \(y = x ^ { 3 } \ln x\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. 1. Find an equation of the tangent to the curve \(y = x ^ { 3 } \ln x\) at the point on the curve where \(x = \mathrm { e }\).
   2. This tangent intersects the \(x\)-axis at the point \(A\). Find the exact value of the \(x\)-coordinate of the point \(A\).

4

1. By using integration by parts, find \(\int x \mathrm { e } ^ { 6 x } \mathrm {~d} x\).
   (4 marks)
2. The diagram shows part of the curve with equation \(y = \sqrt { x } \mathrm { e } ^ { 3 x }\).
   \includegraphics[max width=\textwidth, alt={}, center]{d3c66c34-b09c-4223-8383-cf0a68419bf9-4_547_846_536_591} The shaded region \(R\) is bounded by the curve \(y = \sqrt { x } \mathrm { e } ^ { 3 x }\), the line \(x = 1\) and the \(x\)-axis from \(x = 0\) to \(x = 1\). Find the volume of the solid generated when the region \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis, giving your answer in the form \(\pi \left( p \mathrm { e } ^ { 6 } + q \right)\), where \(p\) and \(q\) are rational numbers.
   (3 marks)

5 The functions f and g are defined with their respective domains by $$\begin{array} { l l } \mathrm { f } ( x ) = \sqrt { 2 x - 5 } , & \text { for } x \geqslant 2.5
\mathrm {~g} ( x ) = \frac { 10 } { x } , & \text { for real values of } x , \quad x \neq 0 \end{array}$$

1. State the range of f .
2. 1. Find \(\mathrm { fg } ( x )\).
   2. Solve the equation \(\operatorname { fg } ( x ) = 5\).
3. The inverse of f is \(\mathrm { f } ^ { - 1 }\).
   1. Find \(\mathrm { f } ^ { - 1 } ( x )\).
   2. Solve the equation \(\mathrm { f } ^ { - 1 } ( x ) = 7\).

6 Use the substitution \(u = x ^ { 4 } + 2\) to find the value of \(\int _ { 0 } ^ { 1 } \frac { x ^ { 7 } } { \left( x ^ { 4 } + 2 \right) ^ { 2 } } \mathrm {~d} x\), giving your answer in the form \(p \ln q + r\), where \(p , q\) and \(r\) are rational numbers.

7 The sketch shows part of the curve with equation \(y = \mathrm { f } ( x )\).
\includegraphics[max width=\textwidth, alt={}, center]{d3c66c34-b09c-4223-8383-cf0a68419bf9-5_632_1029_712_541}

1. On Figure 2 on page 6, sketch the curve with equation \(y = | \mathrm { f } ( x ) |\).
2. On Figure 3 on page 6, sketch the curve with equation \(y = \mathrm { f } ( | x | )\).
3. Describe a sequence of two geometrical transformations that maps the graph of \(y = \mathrm { f } ( x )\) onto the graph of \(y = \frac { 1 } { 2 } \mathrm { f } ( x + 1 )\).
4. The maximum point of the curve with equation \(y = \mathrm { f } ( x )\) has coordinates \(( - 1,10 )\). Find the coordinates of the maximum point of the curve with equation \(y = \frac { 1 } { 2 } \mathrm { f } ( x + 1 )\).
   (2 marks)
5. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{d3c66c34-b09c-4223-8383-cf0a68419bf9-6_785_1022_358_548}
   \end{figure}
6. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{d3c66c34-b09c-4223-8383-cf0a68419bf9-6_776_1022_1395_548}
   \end{figure}

8

1. Show that the equation $$\frac { 1 } { 1 + \cos \theta } + \frac { 1 } { 1 - \cos \theta } = 32$$ can be written in the form $$\operatorname { cosec } ^ { 2 } \theta = 16$$
2. Hence, or otherwise, solve the equation $$\frac { 1 } { 1 + \cos ( 2 x - 0.6 ) } + \frac { 1 } { 1 - \cos ( 2 x - 0.6 ) } = 32$$ giving all values of \(x\) in radians to two decimal places in the interval \(0 < x < \pi\).
   (5 marks)

9

1. Given that \(x = \frac { \sin y } { \cos y }\), use the quotient rule to show that $$\frac { \mathrm { d } x } { \mathrm {~d} y } = \sec ^ { 2 } y$$ (3 marks)
2. Given that \(\tan y = x - 1\), use a trigonometrical identity to show that $$\sec ^ { 2 } y = x ^ { 2 } - 2 x + 2$$
3. Show that, if \(y = \tan ^ { - 1 } ( x - 1 )\), then $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { x ^ { 2 } - 2 x + 2 }$$ (l mark)
4. A curve has equation \(y = \tan ^ { - 1 } ( x - 1 ) - \ln x\).
   1. Find the value of the \(x\)-coordinate of each of the stationary points of the curve.
   2. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
   3. Hence show that the curve has a minimum point which lies on the \(x\)-axis.