Questions — AQA C3 (172 questions)

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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.
AQA C3 2008 June Q7
7
  1. Given that \(y = \frac { \sin \theta } { \cos \theta }\), use the quotient rule to show that \(\frac { \mathrm { d } y } { \mathrm {~d} \theta } = \sec ^ { 2 } \theta\).
  2. Given that \(x = \sin \theta\), show that \(\frac { x } { \sqrt { 1 - x ^ { 2 } } } = \tan \theta\).
  3. Use the substitution \(x = \sin \theta\) to find \(\int \frac { 1 } { \left( 1 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x\), giving your answer in terms of \(x\).
AQA C3 2009 June Q1
1
  1. The curve with equation $$y = \frac { \cos x } { 2 x + 1 } , \quad x > - \frac { 1 } { 2 }$$ intersects the line \(y = \frac { 1 } { 2 }\) at the point where \(x = \alpha\).
    1. Show that \(\alpha\) lies between 0 and \(\frac { \pi } { 2 }\).
    2. Show that the equation \(\frac { \cos x } { 2 x + 1 } = \frac { 1 } { 2 }\) can be rearranged into the form $$x = \cos x - \frac { 1 } { 2 }$$
    3. Use the iteration \(x _ { n + 1 } = \cos x _ { n } - \frac { 1 } { 2 }\) with \(x _ { 1 } = 0\) to find \(x _ { 3 }\), giving your answer to three decimal places.
    1. Given that \(y = \frac { \cos x } { 2 x + 1 }\), use the quotient rule to find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
    2. Hence find the gradient of the normal to the curve \(y = \frac { \cos x } { 2 x + 1 }\) at the point on the curve where \(x = 0\).
AQA C3 2009 June Q2
2 The functions \(f\) and \(g\) are defined with their respective domains by $$\begin{array} { l l } f ( x ) = \sqrt { 2 x + 5 } , & \text { for real values of } x , x \geqslant - 2.5
g ( x ) = \frac { 1 } { 4 x + 1 } , & \text { for real values of } x , x \neq - 0.25 \end{array}$$
  1. Find the range of f.
  2. The inverse of f is \(\mathrm { f } ^ { - 1 }\).
    1. Find \(\mathrm { f } ^ { - 1 } ( x )\).
    2. State the domain of \(\mathrm { f } ^ { - 1 }\).
  3. The composite function fg is denoted by h .
    1. Find an expression for \(\mathrm { h } ( x )\).
    2. Solve the equation \(\mathrm { h } ( x ) = 3\).
AQA C3 2009 June Q3
3
  1. Solve the equation \(\tan x = - \frac { 1 } { 3 }\), giving all the values of \(x\) in the interval \(0 < x < 2 \pi\) in radians to two decimal places.
  2. Show that the equation $$3 \sec ^ { 2 } x = 5 ( \tan x + 1 )$$ can be written in the form \(3 \tan ^ { 2 } x - 5 \tan x - 2 = 0\).
  3. Hence, or otherwise, solve the equation $$3 \sec ^ { 2 } x = 5 ( \tan x + 1 )$$ giving all the values of \(x\) in the interval \(0 < x < 2 \pi\) in radians to two decimal places.
    (4 marks)
AQA C3 2009 June Q4
4
  1. Sketch the graph of \(y = \left| 50 - x ^ { 2 } \right|\), indicating the coordinates of the point where the graph crosses the \(y\)-axis.
  2. Solve the equation \(\left| 50 - x ^ { 2 } \right| = 14\).
  3. Hence, or otherwise, solve the inequality \(\left| 50 - x ^ { 2 } \right| > 14\).
  4. Describe a sequence of two geometrical transformations that maps the graph of \(y = x ^ { 2 }\) onto the graph of \(y = 50 - x ^ { 2 }\).
AQA C3 2009 June Q5
5
  1. Given that \(2 \ln x = 5\), find the exact value of \(x\).
  2. Solve the equation $$2 \ln x + \frac { 15 } { \ln x } = 11$$ giving your answers as exact values of \(x\).
AQA C3 2009 June Q6
6 The diagram shows the curve with equation \(y = \sqrt { 100 - 4 x ^ { 2 } }\), where \(x \geqslant 0\).
\includegraphics[max width=\textwidth, alt={}, center]{a596af76-9680-4ccb-a512-5b2575414429-5_518_494_367_758}
  1. Calculate the volume of the solid generated when the region bounded by the curve shown above and the coordinate axes is rotated through \(360 ^ { \circ }\) about the \(\boldsymbol { y }\)-axis, giving your answer in terms of \(\pi\).
  2. Use the mid-ordinate rule with five strips of equal width to find an estimate for \(\int _ { 0 } ^ { 5 } \sqrt { 100 - 4 x ^ { 2 } } \mathrm {~d} x\), giving your answer to three significant figures.
  3. The point \(P\) on the curve has coordinates \(( 3,8 )\).
    1. Find the gradient of the curve \(y = \sqrt { 100 - 4 x ^ { 2 } }\) at the point \(P\).
    2. Hence show that the equation of the tangent to the curve at the point \(P\) can be written as \(2 y + 3 x = 25\).
  4. The shaded regions on the diagram below are bounded by the curve, the tangent at \(P\) and the coordinate axes.
    \includegraphics[max width=\textwidth, alt={}, center]{a596af76-9680-4ccb-a512-5b2575414429-5_642_546_1800_731} Use your answers to part (b) and part (c)(ii) to find an approximate value for the total area of the shaded regions. Give your answer to three significant figures.
AQA C3 2009 June Q7
7
  1. Use integration by parts to find \(\int ( t - 1 ) \ln t \mathrm {~d} t\).
  2. Use the substitution \(t = 2 x + 1\) to show that \(\int 4 x \ln ( 2 x + 1 ) \mathrm { d } x\) can be written as \(\int ( t - 1 ) \ln t \mathrm {~d} t\).
  3. Hence find the exact value of \(\int _ { 0 } ^ { 1 } 4 x \ln ( 2 x + 1 ) \mathrm { d } x\).
AQA C3 2010 June Q1
1 The curve \(y = 3 ^ { x }\) intersects the curve \(y = 10 - x ^ { 3 }\) at the point where \(x = \alpha\).
  1. Show that \(\alpha\) lies between 1 and 2 .
    1. Show that the equation \(3 ^ { x } = 10 - x ^ { 3 }\) can be rearranged into the form $$x = \sqrt [ 3 ] { 10 - 3 ^ { x } }$$
    2. Use the iteration \(x _ { n + 1 } = \sqrt [ 3 ] { 10 - 3 ^ { x _ { n } } }\) with \(x _ { 1 } = 1\) to find the values of \(x _ { 2 }\) and \(x _ { 3 }\), giving your answers to three decimal places.
AQA C3 2010 June Q2
2
  1. The diagram shows the graph of \(y = \sec x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
    \includegraphics[max width=\textwidth, alt={}, center]{33ca7e6d-b9eb-46be-b5b0-c5685212d7ff-2_816_1447_1087_287}
    1. The point \(A\) on the curve is where \(x = 0\). State the \(y\)-coordinate of \(A\).
    2. Sketch, on the axes given on page 3, the graph of \(y = | \sec 2 x |\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
  2. Solve the equation \(\sec x = 2\), giving all values of \(x\) in degrees in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
  3. Solve the equation \(\left| \sec \left( 2 x - 10 ^ { \circ } \right) \right| = 2\), giving all values of \(x\) in degrees in the interval \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
    \includegraphics[max width=\textwidth, alt={}, center]{33ca7e6d-b9eb-46be-b5b0-c5685212d7ff-3_839_1475_317_351}
AQA C3 2010 June Q3
3
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when:
    1. \(\quad y = \ln ( 5 x - 2 )\);
    2. \(y = \sin 2 x\).
  2. The functions f and g are defined with their respective domains by $$\begin{array} { l l } \mathrm { f } ( x ) = \ln ( 5 x - 2 ) , & \text { for real values of } x \text { such that } x \geqslant \frac { 1 } { 2 }
    \mathrm {~g} ( x ) = \sin 2 x , & \text { for real values of } x \text { in the interval } - \frac { \pi } { 4 } \leqslant x \leqslant \frac { \pi } { 4 } \end{array}$$
    1. Find the range of f .
    2. Find an expression for \(\operatorname { gf } ( x )\).
    3. Solve the equation \(\operatorname { gf } ( x ) = 0\).
    4. The inverse of g is \(\mathrm { g } ^ { - 1 }\). Find \(\mathrm { g } ^ { - 1 } ( x )\).
AQA C3 2010 June Q4
4
  1. Use Simpson's rule with 7 ordinates ( 6 strips) to find an approximation to \(\int _ { 0.5 } ^ { 2 } \frac { x } { 1 + x ^ { 3 } } \mathrm {~d} x\), giving your answer to three significant figures.
  2. Find the exact value of \(\int _ { 0 } ^ { 1 } \frac { x ^ { 2 } } { 1 + x ^ { 3 } } \mathrm {~d} x\).