Questions C3 (1301 questions)

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AQA C3 2011 January Q1
7 marks Moderate -0.8
1
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(y = \left( x ^ { 3 } - 1 \right) ^ { 6 }\).
  2. A curve has equation \(y = x \ln x\).
    1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
    2. Find an equation of the tangent to the curve \(y = x \ln x\) at the point on the curve where \(x = \mathrm { e }\).
AQA C3 2011 January Q2
6 marks Moderate -0.3
2 A curve is defined by the equation \(y = \left( x ^ { 2 } - 4 \right) \ln ( x + 2 )\) for \(x \geqslant 3\).
The curve intersects the line \(y = 15\) at a single point, where \(x = \alpha\).
  1. Show that \(\alpha\) lies between 3.5 and 3.6.
  2. Show that the equation \(\left( x ^ { 2 } - 4 \right) \ln ( x + 2 ) = 15\) can be arranged into the form $$x = \pm \sqrt { 4 + \frac { 15 } { \ln ( x + 2 ) } }$$ (2 marks)
  3. Use the iteration $$x _ { n + 1 } = \sqrt { 4 + \frac { 15 } { \ln \left( x _ { n } + 2 \right) } }$$ with \(x _ { 1 } = 3.5\) to find the values of \(x _ { 2 }\) and \(x _ { 3 }\), giving your answers to three decimal places.
    (2 marks)
AQA C3 2011 January Q3
6 marks Moderate -0.3
3
  1. Given that \(x = \tan ( 3 y + 1 )\) :
    1. find \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) in terms of \(y\);
    2. find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(y = - \frac { 1 } { 3 }\).
  2. Sketch the graph of \(y = \tan ^ { - 1 } x\).
AQA C3 2011 January Q4
14 marks Standard +0.3
4 The functions f and g are defined with their respective domains by $$\begin{array} { l l } \mathrm { f } ( x ) = 3 \cos \frac { 1 } { 2 } x , & \text { for } 0 \leqslant x \leqslant 2 \pi \\ \mathrm {~g} ( x ) = | x | , & \text { for all real values of } x \end{array}$$
  1. Find the range of f .
  2. The inverse of f is \(\mathrm { f } ^ { - 1 }\).
    1. Find \(\mathrm { f } ^ { - 1 } ( x )\).
    2. Solve the equation \(\mathrm { f } ^ { - 1 } ( x ) = 1\), giving your answer in an exact form.
    1. Write down an expression for \(\mathrm { gf } ( x )\).
    2. Sketch the graph of \(y = \operatorname { gf } ( x )\) for \(0 \leqslant x \leqslant 2 \pi\).
  4. Describe a sequence of two geometrical transformations that maps the graph of \(y = \cos x\) onto the graph of \(y = 3 \cos \frac { 1 } { 2 } x\).
AQA C3 2011 January Q5
6 marks Moderate -0.3
5
  1. Find \(\int \frac { 1 } { 3 + 2 x } \mathrm {~d} x\).
  2. By using integration by parts, find \(\int x \sin \frac { x } { 2 } \mathrm {~d} x\).
AQA C3 2011 January Q6
10 marks Standard +0.3
6
  1. Use the mid-ordinate rule with four strips to find an estimate for \(\int _ { 0 } ^ { 0.4 } \cos \sqrt { 3 x + 1 } \mathrm {~d} x\), giving your answer to three significant figures.
  2. Use the substitution \(u = 3 x + 1\) to find the exact value of \(\int _ { 0 } ^ { 1 } x \sqrt { 3 x + 1 } \mathrm {~d} x\).
    (6 marks)
AQA C3 2011 January Q7
10 marks Standard +0.3
7
  1. Solve the equation \(\sec x = - 5\), giving all values of \(x\) in radians to two decimal places in the interval \(0 < x < 2 \pi\).
  2. Show that the equation $$\frac { \operatorname { cosec } x } { 1 + \operatorname { cosec } x } - \frac { \operatorname { cosec } x } { 1 - \operatorname { cosec } x } = 50$$ can be written in the form $$\sec ^ { 2 } x = 25$$
  3. Hence, or otherwise, solve the equation $$\frac { \operatorname { cosec } x } { 1 + \operatorname { cosec } x } - \frac { \operatorname { cosec } x } { 1 - \operatorname { cosec } x } = 50$$ giving all values of \(x\) in radians to two decimal places in the interval \(0 < x < 2 \pi\).
    (3 marks)
AQA C3 2011 January Q8
16 marks Standard +0.3
8
  1. Given that \(\mathrm { e } ^ { - 2 x } = 4\), find the exact value of \(x\).
  2. The diagram shows the curve \(y = 4 \mathrm { e } ^ { - 2 x } - \mathrm { e } ^ { - 4 x }\). \includegraphics[max width=\textwidth, alt={}, center]{6761e676-48ae-47e9-9617-153342cdf5c4-9_490_1185_463_440} The curve crosses the \(y\)-axis at the point \(A\), the \(x\)-axis at the point \(B\), and has a stationary point at \(M\).
    1. State the \(y\)-coordinate of \(A\).
    2. Find the \(x\)-coordinate of \(B\), giving your answer in an exact form.
    3. Find the \(x\)-coordinate of the stationary point, \(M\), giving your answer in an exact form.
    4. The shaded region \(R\) is bounded by the curve \(y = 4 \mathrm { e } ^ { - 2 x } - \mathrm { e } ^ { - 4 x }\), the lines \(x = 0\) and \(x = \ln 2\) and the \(x\)-axis. 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 \(\frac { p } { q } \pi\), where \(p\) and \(q\) are integers.
      (7 marks)
AQA C3 2012 January Q1
8 marks Moderate -0.3
1
  1. Use Simpson's rule with 7 ordinates (6 strips) to find an estimate for \(\int _ { 0 } ^ { 3 } 4 ^ { x } \mathrm {~d} x\).
  2. A curve is defined by the equation \(y = 4 ^ { x }\). The curve intersects the line \(y = 8 - 2 x\) at a single point where \(x = \alpha\).
    1. Show that \(\alpha\) lies between 1.2 and 1.3.
    2. The equation \(4 ^ { x } = 8 - 2 x\) can be rearranged into the form \(x = \frac { \ln ( 8 - 2 x ) } { \ln 4 }\). Use the iterative formula \(x _ { n + 1 } = \frac { \ln \left( 8 - 2 x _ { n } \right) } { \ln 4 }\) with \(x _ { 1 } = 1.2\) to find the values of \(x _ { 2 }\) and \(x _ { 3 }\), giving your answers to three decimal places.
      (2 marks)
AQA C3 2012 January Q2
11 marks Moderate -0.3
2 The curve with equation \(y = \frac { 63 } { 4 x - 1 }\) is sketched below for \(1 \leqslant x \leqslant 16\). \includegraphics[max width=\textwidth, alt={}, center]{7aa76d26-e3c4-4374-ae4f-8bb61e61b135-2_568_698_1308_669} The function f is defined by \(\mathrm { f } ( x ) = \frac { 63 } { 4 x - 1 }\) for \(1 \leqslant x \leqslant 16\).
  1. Find the range of f .
  2. The inverse of f is \(\mathrm { f } ^ { - 1 }\).
    1. Find \(\mathrm { f } ^ { - 1 } ( x )\).
    2. Solve the equation \(\mathrm { f } ^ { - 1 } ( x ) = 1\).
  3. The function g is defined by \(\mathrm { g } ( x ) = x ^ { 2 }\) for \(- 4 \leqslant x \leqslant - 1\).
    1. Write down an expression for \(\mathrm { fg } ( x )\).
    2. Solve the equation \(\operatorname { fg } ( x ) = 1\).
AQA C3 2012 January Q3
6 marks Moderate -0.3
3
  1. Given that \(y = 4 x ^ { 3 } - 6 x + 1\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
    (l mark)
  2. Hence find \(\int _ { 2 } ^ { 3 } \frac { 2 x ^ { 2 } - 1 } { 4 x ^ { 3 } - 6 x + 1 } \mathrm {~d} x\), giving your answer in the form \(p \ln q\), where \(p\) and \(q\) are rational numbers.
AQA C3 2012 January Q4
9 marks Standard +0.3
4
  1. By using a suitable trigonometrical identity, solve the equation $$\tan ^ { 2 } \theta = 3 ( 3 - \sec \theta )$$ giving all solutions to the nearest \(0.1 ^ { \circ }\) in the interval \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
  2. Hence solve the equation $$\tan ^ { 2 } \left( 4 x - 10 ^ { \circ } \right) = 3 \left[ 3 - \sec \left( 4 x - 10 ^ { \circ } \right) \right]$$ giving all solutions to the nearest \(0.1 ^ { \circ }\) in the interval \(0 ^ { \circ } < x < 90 ^ { \circ }\).
AQA C3 2012 January Q5
13 marks Standard +0.3
5
  1. Describe a sequence of two geometrical transformations that maps the graph of \(y = \ln x\) onto the graph of \(y = 4 \ln ( x - \mathrm { e } )\).
  2. Sketch, on the axes given below, the graph of \(y = | 4 \ln ( x - \mathrm { e } ) |\), indicating the exact value of the \(x\)-coordinate where the curve meets the \(x\)-axis.
    1. Solve the equation \(| 4 \ln ( x - e ) | = 4\).
    2. Hence, or otherwise, solve the inequality \(| 4 \ln ( x - e ) | \geqslant 4\). \includegraphics[max width=\textwidth, alt={}, center]{7aa76d26-e3c4-4374-ae4f-8bb61e61b135-3_655_1428_2023_315}
AQA C3 2012 January Q6
12 marks Standard +0.3
6
  1. Given that \(x = \frac { 1 } { \sin \theta }\), use the quotient rule to show that \(\frac { \mathrm { d } x } { \mathrm {~d} \theta } = - \operatorname { cosec } \theta \cot \theta\).
    (3 marks)
  2. Use the substitution \(x = \operatorname { cosec } \theta\) to find \(\int _ { \sqrt { 2 } } ^ { 2 } \frac { 1 } { x ^ { 2 } \sqrt { x ^ { 2 } - 1 } } \mathrm {~d} x\), giving your answer to three significant figures.
    (9 marks)
AQA C3 2012 January Q7
16 marks Standard +0.3
7
  1. A curve has equation \(y = x ^ { 2 } \mathrm { e } ^ { - \frac { x } { 4 } }\).
    Show that the curve has exactly two stationary points and find the exact values of their coordinates.
    (7 marks)
    1. Use integration by parts twice to find the exact value of \(\int _ { 0 } ^ { 4 } x ^ { 2 } \mathrm { e } ^ { - \frac { x } { 4 } } \mathrm {~d} x\).
    2. The region bounded by the curve \(y = 3 x \mathrm { e } ^ { - \frac { x } { 8 } }\), the \(x\)-axis from 0 to 4 and the line \(x = 4\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis to form a solid. Use your answer to part (b)(i) to find the exact value of the volume of the solid generated.
AQA C3 2013 January Q1
5 marks Moderate -0.3
1
  1. Show that the equation \(x ^ { 3 } - 6 x + 1 = 0\) has a root \(\alpha\), where \(2 < \alpha < 3\).
  2. Show that the equation \(x ^ { 3 } - 6 x + 1 = 0\) can be rearranged into the form $$x ^ { 2 } = 6 - \frac { 1 } { x }$$ (1 mark)
  3. Use the recurrence relation \(x _ { n + 1 } = \sqrt { 6 - \frac { 1 } { x _ { n } } }\), with \(x _ { 1 } = 2.5\), to find the value of \(x _ { 3 }\), giving your answer to four significant figures.
    (2 marks)
    \includegraphics[max width=\textwidth, alt={}]{b8614dd6-2197-40c3-a673-5bef3e3653a5-2_142_116_2560_157}\(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)
AQA C3 2013 January Q2
9 marks Moderate -0.3
2
  1. Use Simpson's rule, with five ordinates (four strips), to calculate an estimate for $$\int _ { 0 } ^ { 4 } \frac { x } { x ^ { 2 } + 2 } \mathrm {~d} x$$ Give your answer to four significant figures.
  2. Show that the exact value of \(\int _ { 0 } ^ { 4 } \frac { x } { x ^ { 2 } + 2 } \mathrm {~d} x\) is \(\ln k\), where \(k\) is an integer. (5 marks)
AQA C3 2013 January Q3
7 marks Moderate -0.3
3
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when $$y = \mathrm { e } ^ { 3 x } + \ln x$$
    1. Given that \(u = \frac { \sin x } { 1 + \cos x }\), show that \(\frac { \mathrm { d } u } { \mathrm {~d} x } = \frac { 1 } { 1 + \cos x }\).
    2. Hence show that if \(y = \ln \left( \frac { \sin x } { 1 + \cos x } \right)\), then \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \operatorname { cosec } x\).
AQA C3 2013 January Q4
6 marks Standard +0.3
4 The diagram shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\). \includegraphics[max width=\textwidth, alt={}, center]{b8614dd6-2197-40c3-a673-5bef3e3653a5-5_629_1113_370_461}
  1. On the axes below, sketch the curve with equation \(y = | \mathrm { f } ( x ) |\).
  2. Describe a sequence of two geometrical transformations that maps the graph of \(y = \mathrm { f } ( x )\) onto the graph of \(y = \mathrm { f } ( 2 x - 1 )\).
AQA C3 2013 January Q5
15 marks Moderate -0.3
5 The function f is defined by $$\mathrm { f } ( x ) = \frac { x ^ { 2 } - 4 } { 3 } , \text { for real values of } x , \text { where } \boldsymbol { x } \leqslant \mathbf { 0 }$$
  1. State the range of f.
  2. The inverse of f is \(\mathrm { f } ^ { - 1 }\).
    1. Write down the domain of \(\mathrm { f } ^ { - 1 }\).
    2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
  3. The function g is defined by $$\mathrm { g } ( x ) = \ln | 3 x - 1 | , \quad \text { for real values of } x , \text { where } x \neq \frac { 1 } { 3 }$$ The curve with equation \(y = \mathrm { g } ( x )\) is sketched below. \includegraphics[max width=\textwidth, alt={}, center]{b8614dd6-2197-40c3-a673-5bef3e3653a5-6_469_819_1254_612}
    1. The curve \(y = \mathrm { g } ( x )\) intersects the \(x\)-axis at the origin and at the point \(P\). Find the \(x\)-coordinate of \(P\).
    2. State whether the function \(g\) has an inverse. Give a reason for your answer.
    3. Show that \(\operatorname { gf } ( x ) = \ln \left| x ^ { 2 } - k \right|\), stating the value of the constant \(k\).
    4. Solve the equation \(\mathrm { gf } ( x ) = 0\).
AQA C3 2013 January Q6
11 marks Standard +0.3
6
  1. Show that $$\frac { \sec ^ { 2 } x } { ( \sec x + 1 ) ( \sec x - 1 ) }$$ can be written as \(\operatorname { cosec } ^ { 2 } x\).
  2. Hence solve the equation $$\frac { \sec ^ { 2 } x } { ( \sec x + 1 ) ( \sec x - 1 ) } = \operatorname { cosec } x + 3$$ giving the values of \(x\) to the nearest degree in the interval \(- 180 ^ { \circ } < x < 180 ^ { \circ }\).
  3. Hence solve the equation $$\frac { \sec ^ { 2 } \left( 2 \theta - 60 ^ { \circ } \right) } { \left( \sec \left( 2 \theta - 60 ^ { \circ } \right) + 1 \right) \left( \sec \left( 2 \theta - 60 ^ { \circ } \right) - 1 \right) } = \operatorname { cosec } \left( 2 \theta - 60 ^ { \circ } \right) + 3$$ giving the values of \(\theta\) to the nearest degree in the interval \(0 ^ { \circ } < \theta < 90 ^ { \circ }\).
AQA C3 2013 January Q7
10 marks Standard +0.3
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
12 marks Challenging +1.2
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
8 marks Moderate -0.8
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
6 marks Moderate -0.8
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 \(\mathrm { h } ^ { - 1 }\).