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AQA C3 2013 January Q7
7 A curve has equation \(y = 4 x \cos 2 x\).
  1. Find an exact equation of the tangent to the curve at the point on the curve where $$x = \frac { \pi } { 4 }$$
  2. The region shaded on the diagram below is bounded by the curve \(y = 4 x \cos 2 x\) and the \(x\)-axis from \(x = 0\) to \(x = \frac { \pi } { 4 }\).
    \includegraphics[max width=\textwidth, alt={}, center]{b8614dd6-2197-40c3-a673-5bef3e3653a5-8_487_878_740_591} By using integration by parts, find the exact value of the area of the shaded region.
    (5 marks)
    \includegraphics[max width=\textwidth, alt={}]{b8614dd6-2197-40c3-a673-5bef3e3653a5-8_1275_1717_1432_150}
AQA C3 2013 January Q8
8
  1. Show that $$\int _ { 0 } ^ { \ln 2 } \mathrm { e } ^ { 1 - 2 x } \mathrm {~d} x = \frac { 3 } { 8 } \mathrm { e }$$
  2. Use the substitution \(u = \tan x\) to find the exact value of $$\int _ { 0 } ^ { \frac { \pi } { 4 } } \sec ^ { 4 } x \sqrt { \tan x } d x$$ (8 marks)
AQA C3 2005 June Q1
1
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(y = x \sin 2 x\).
    1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(y = \left( x ^ { 2 } - 6 \right) ^ { 4 }\).
    2. Hence, or otherwise, find \(\int x \left( x ^ { 2 } - 6 \right) ^ { 3 } \mathrm {~d} x\).
AQA C3 2005 June Q2
2 The functions \(f\) and \(g\) are defined with their respective domains by $$\begin{array} { l l } \mathrm { f } ( x ) = x - 2 & \text { for all real values of } x
\mathrm {~g} ( x ) = \frac { 6 } { x + 3 } & \text { for real values of } x , \quad x \neq - 3 \end{array}$$ The composite function fg is denoted by h .
  1. Find \(\mathrm { h } ( x )\).
    1. Find \(\mathrm { h } ^ { - 1 } ( x )\), where \(\mathrm { h } ^ { - 1 }\) is the inverse of h .
    2. Find the range of \(\mathrm { h } ^ { - 1 }\).
AQA C3 2005 June Q3
3
  1. Find \(\int \mathrm { e } ^ { 4 x } \mathrm {~d} x\).
  2. Use integration by parts to find \(\int \mathrm { e } ^ { 4 x } ( 2 x + 1 ) \mathrm { d } x\).
  3. By using the substitution \(u = 1 + \ln x\), or otherwise, find \(\int \frac { 1 + \ln x } { x } \mathrm {~d} x\).
AQA C3 2005 June Q4
4 It is given that \(\tan ^ { 2 } x = \sec x + 11\).
  1. Show that the equation \(\tan ^ { 2 } x = \sec x + 11\) can be written in the form $$\sec ^ { 2 } x - \sec x - 12 = 0$$
  2. Hence show that \(\cos x = \frac { 1 } { 4 }\) or \(\cos x = - \frac { 1 } { 3 }\).
  3. Hence, or otherwise, solve the equation \(\tan ^ { 2 } x = \sec x + 11\), giving all values of \(x\) to the nearest degree in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\).
AQA C3 2005 June Q5
5
  1. Solve the equation \(2 \mathrm { e } ^ { x } = 5\), giving your answer as an exact natural logarithm.
    1. By substituting \(y = \mathrm { e } ^ { x }\), show that the equation \(2 \mathrm { e } ^ { x } + 5 \mathrm { e } ^ { - x } = 7\) can be written as $$2 y ^ { 2 } - 7 y + 5 = 0$$
    2. Hence solve the equation \(2 \mathrm { e } ^ { x } + 5 \mathrm { e } ^ { - x } = 7\), giving your answers as exact values of \(x\).
AQA C3 2005 June Q6
6
    1. Sketch the graph of \(y = 4 - x ^ { 2 }\), indicating the coordinates of the points where the graph crosses the coordinate axes.
    2. The region between the graph and the \(x\)-axis from \(x = 0\) to \(x = 2\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Find the exact value of the volume of the solid generated.
    1. Sketch the graph of \(y = \left| 4 - x ^ { 2 } \right|\).
    2. Solve \(\left| 4 - x ^ { 2 } \right| = 3\).
    3. Hence, or otherwise, solve the inequality \(\left| 4 - x ^ { 2 } \right| < 3\).
AQA C3 2005 June Q7
7
  1. Sketch the graph of \(y = \tan ^ { - 1 } x\).
    1. By drawing a suitable straight line on your sketch, show that the equation \(\tan ^ { - 1 } x = 2 x - 1\) has only one root.
    2. Given that the root of this equation is \(\alpha\), show that \(0.8 < \alpha < 0.9\).
  3. Use the iteration \(x _ { n + 1 } = \frac { 1 } { 2 } \left( \tan ^ { - 1 } x _ { n } + 1 \right)\) with \(x _ { 1 } = 0.8\) to find the value of \(x _ { 3 }\), giving your answer to two significant figures.
AQA C3 2005 June Q8
8 The diagram shows part of the graph of \(y = \mathrm { e } ^ { 2 x } + 3\).
\includegraphics[max width=\textwidth, alt={}, center]{d5b78fa6-ea3c-497b-94d8-1d5f61288aa5-4_833_1034_1027_513}
  1. Describe a sequence of two geometrical transformations that maps the graph of \(y = \mathrm { e } ^ { x }\) onto the graph of \(y = \mathrm { e } ^ { 2 x } + 3\).
  2. Use the mid-ordinate rule with four strips of equal width to find an estimate for the area of the shaded region \(A\), giving your answer to three significant figures.
  3. Find the exact value of the area of the shaded region \(A\).
  4. The region \(B\) is indicated on the diagram. Find the area of the region \(B\), giving your answer in the form \(p \mathrm { e } ^ { 8 } + q \mathrm { e } ^ { 4 }\), where \(p\) and \(q\) are numbers to be determined.
AQA C3 2006 June Q1
1 The curve \(y = x ^ { 3 } - x - 7\) intersects the \(x\)-axis at the point where \(x = \alpha\).
  1. Show that \(\alpha\) lies between 2.0 and 2.1.
  2. Show that the equation \(x ^ { 3 } - x - 7 = 0\) can be rearranged in the form \(x = \sqrt [ 3 ] { x + 7 }\).
  3. Use the iteration \(x _ { n + 1 } = \sqrt [ 3 ] { x _ { n } + 7 }\) with \(x _ { 1 } = 2\) to find the values of \(x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\), giving your answers to three significant figures.
AQA C3 2006 June Q2
2
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(y = ( 3 x - 1 ) ^ { 10 }\).
  2. Use the substitution \(u = 2 x + 1\) to find \(\int x ( 2 x + 1 ) ^ { 8 } \mathrm {~d} x\), giving your answer in terms of \(x\).
AQA C3 2006 June Q3
3
  1. Solve the equation \(\sec x = 5\), giving all the values of \(x\) in the interval \(0 \leqslant x \leqslant 2 \pi\) in radians to two decimal places.
  2. Show that the equation \(\tan ^ { 2 } x = 3 \sec x + 9\) can be written as $$\sec ^ { 2 } x - 3 \sec x - 10 = 0$$
  3. Solve the equation \(\tan ^ { 2 } x = 3 \sec x + 9\), giving all the values of \(x\) in the interval \(0 \leqslant x \leqslant 2 \pi\) in radians to two decimal places.
AQA C3 2006 June Q4
4
  1. Sketch and label on the same set of axes the graphs of:
    1. \(y = | x |\);
    2. \(y = | 2 x - 4 |\).
    1. Solve the equation \(| x | = | 2 x - 4 |\).
    2. Hence, or otherwise, solve the inequality \(| x | > | 2 x - 4 |\).
AQA C3 2006 June Q5
5
  1. A curve has equation \(y = \mathrm { e } ^ { 2 x } - 10 \mathrm { e } ^ { x } + 12 x\).
    1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
      (2 marks)
    2. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
      (1 mark)
  2. The points \(P\) and \(Q\) are the stationary points of the curve.
    1. Show that the \(x\)-coordinates of \(P\) and \(Q\) are given by the solutions of the equation $$\mathrm { e } ^ { 2 x } - 5 \mathrm { e } ^ { x } + 6 = 0$$ (1 mark)
    2. By using the substitution \(z = \mathrm { e } ^ { x }\), or otherwise, show that the \(x\)-coordinates of \(P\) and \(Q\) are \(\ln 2\) and \(\ln 3\).
    3. Find the \(y\)-coordinates of \(P\) and \(Q\), giving each of your answers in the form \(m + 12 \ln n\), where \(m\) and \(n\) are integers.
    4. Using the answer to part (a)(ii), determine the nature of each stationary point.
AQA C3 2006 June Q6
6
  1. Use the mid-ordinate rule with four strips to find an estimate for \(\int _ { 1 } ^ { 5 } \ln x \mathrm {~d} x\), giving your answer to three significant figures.
    1. Given that \(y = x \ln x\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
    2. Hence, or otherwise, find \(\int \ln x \mathrm {~d} x\).
    3. Find the exact value of \(\int _ { 1 } ^ { 5 } \ln x \mathrm {~d} x\).
AQA C3 2006 June Q7
7
  1. Given that \(z = \frac { \sin x } { \cos x }\), use the quotient rule to show that \(\frac { \mathrm { d } z } { \mathrm {~d} x } = \sec ^ { 2 } x\).
  2. Sketch the curve with equation \(y = \sec x\) for \(- \frac { \pi } { 2 } < x < \frac { \pi } { 2 }\).
  3. The region \(R\) is bounded by the curve \(y = \sec x\), the \(x\)-axis and the lines \(x = 0\) and \(x = 1\). Find the volume of the solid formed when \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis, giving your answer to three significant figures.
AQA C3 2006 June Q8
8 A function f is defined by \(\mathrm { f } ( x ) = 2 \mathrm { e } ^ { 3 x } - 1\) for all real values of \(x\).
  1. Find the range of f.
  2. Show that \(\mathrm { f } ^ { - 1 } ( x ) = \frac { 1 } { 3 } \ln \left( \frac { x + 1 } { 2 } \right)\).
  3. Find the gradient of the curve \(y = \mathrm { f } ^ { - 1 } ( x )\) when \(x = 0\).
AQA C3 2006 June Q9
9 The diagram shows the curve with equation \(y = \sin ^ { - 1 } 2 x\), where \(- \frac { 1 } { 2 } \leqslant x \leqslant \frac { 1 } { 2 }\).
\includegraphics[max width=\textwidth, alt={}, center]{0ab0e757-270b-4c15-9202-9df2f02dddf3-4_790_752_906_644}
  1. Find the \(y\)-coordinate of the point \(A\), where \(x = \frac { 1 } { 2 }\).
    1. Given that \(y = \sin ^ { - 1 } 2 x\), show that \(x = \frac { 1 } { 2 } \sin y\).
    2. Given that \(x = \frac { 1 } { 2 } \sin y\), find \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) in terms of \(y\).
  3. Using the answers to part (b) and a suitable trigonometrical identity, show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 } { \sqrt { 1 - 4 x ^ { 2 } } }$$
AQA C3 2008 June Q1
1 Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when:
  1. \(y = ( 3 x + 1 ) ^ { 5 }\);
  2. \(y = \ln ( 3 x + 1 )\);
  3. \(y = ( 3 x + 1 ) ^ { 5 } \ln ( 3 x + 1 )\).
AQA C3 2008 June Q2
2
  1. Solve the equation \(\sec x = 3\), giving the values of \(x\) in radians to two decimal places in the interval \(0 \leqslant x < 2 \pi\).
  2. Show that the equation \(\tan ^ { 2 } x = 2 \sec x + 2\) can be written as \(\sec ^ { 2 } x - 2 \sec x - 3 = 0\).
  3. Solve the equation \(\tan ^ { 2 } x = 2 \sec x + 2\), giving the values of \(x\) in radians to two decimal places in the interval \(0 \leqslant x < 2 \pi\).
AQA C3 2008 June Q3
3 A curve is defined for \(0 \leqslant x \leqslant \frac { \pi } { 4 }\) by the equation \(y = x \cos 2 x\), and is sketched below.
\includegraphics[max width=\textwidth, alt={}, center]{6ce5aa0d-0a73-4bc4-aabc-314c0434e4f5-3_757_878_402_559}
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. The point \(A\), where \(x = \alpha\), on the curve is a stationary point.
    1. Show that \(1 - 2 \alpha \tan 2 \alpha = 0\).
    2. Show that \(0.4 < \alpha < 0.5\).
    3. Show that the equation \(1 - 2 x \tan 2 x = 0\) can be rearranged to become \(x = \frac { 1 } { 2 } \tan ^ { - 1 } \left( \frac { 1 } { 2 x } \right)\).
    4. Use the iteration \(x _ { n + 1 } = \frac { 1 } { 2 } \tan ^ { - 1 } \left( \frac { 1 } { 2 x _ { n } } \right)\) with \(x _ { 1 } = 0.4\) to find \(x _ { 3 }\), giving your answer to two significant figures.
  3. Use integration by parts to find \(\int _ { 0 } ^ { 0.5 } x \cos 2 x \mathrm {~d} x\), giving your answer to three significant figures.
AQA C3 2008 June Q4
4 The functions \(f\) and \(g\) are defined with their respective domains by $$\begin{array} { l l } \mathrm { f } ( x ) = x ^ { 2 } , & \text { for all real values of } x
\mathrm {~g} ( x ) = \frac { 1 } { 2 x - 3 } , & \text { for real values of } x , x \neq \frac { 3 } { 2 } \end{array}$$
  1. State the range of f.
    1. The inverse of g is \(\mathrm { g } ^ { - 1 }\). Find \(\mathrm { g } ^ { - 1 } ( x )\).
    2. State the range of \(\mathrm { g } ^ { - 1 }\).
  3. Solve the equation \(\operatorname { fg } ( x ) = 9\).
AQA C3 2008 June Q5
5
  1. The diagram shows part of the curve with equation \(y = \mathrm { f } ( x )\). The curve crosses the \(x\)-axis at the point \(( a , 0 )\) and the \(y\)-axis at the point \(( 0 , - b )\).
    \includegraphics[max width=\textwidth, alt={}, center]{6ce5aa0d-0a73-4bc4-aabc-314c0434e4f5-4_569_853_1206_589} On separate diagrams, sketch the curves with the following equations. On each diagram, indicate, in terms of \(a\) or \(b\), the coordinates of the points where the curve crosses the coordinate axes.
    1. \(y = | \mathrm { f } ( x ) |\).
    2. \(\quad y = 2 \mathrm { f } ( x )\).
    1. Describe a sequence of geometrical transformations that maps the graph of \(y = \ln x\) onto the graph of \(y = 4 \ln ( x + 1 ) - 2\).
    2. Find the exact values of the coordinates of the points where the graph of \(y = 4 \ln ( x + 1 ) - 2\) crosses the coordinate axes.
AQA C3 2008 June Q6
6 The diagram shows the curve with equation \(y = \left( \mathrm { e } ^ { 3 x } + 1 \right) ^ { \frac { 1 } { 2 } }\) for \(x \geqslant 0\).
\includegraphics[max width=\textwidth, alt={}, center]{6ce5aa0d-0a73-4bc4-aabc-314c0434e4f5-5_483_611_402_717}
  1. Find the gradient of the curve \(y = \left( \mathrm { e } ^ { 3 x } + 1 \right) ^ { \frac { 1 } { 2 } }\) at the point where \(x = \ln 2\).
  2. Use the mid-ordinate rule with four strips to find an estimate for \(\int _ { 0 } ^ { 2 } \left( \mathrm { e } ^ { 3 x } + 1 \right) ^ { \frac { 1 } { 2 } } \mathrm {~d} x\), giving your answer to three significant figures.
  3. The shaded region \(R\) is bounded by the curve, the lines \(x = 0 , x = 2\) and the \(x\)-axis. Find the exact value of the volume of the solid generated when the region \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.