Questions — AQA C3 (173 questions)

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AQA C3 2014 June Q3
10 marks Standard +0.3
3
    1. Differentiate \(\left( x ^ { 2 } + 1 \right) ^ { \frac { 5 } { 2 } }\) with respect to \(x\).
    2. Given that \(y = \mathrm { e } ^ { 2 x } \left( x ^ { 2 } + 1 \right) ^ { \frac { 5 } { 2 } }\), find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(x = 0\).
  2. A curve has equation \(y = \frac { 4 x - 3 } { x ^ { 2 } + 1 }\). Use the quotient rule to find the \(x\)-coordinates of the stationary points of the curve.
    [0pt] [5 marks]
    \includegraphics[max width=\textwidth, alt={}]{57412ec0-ad97-4418-8ba8-93f1f7d8aac1-06_1855_1709_852_153}
AQA C3 2014 June Q4
11 marks Standard +0.3
4 The sketch shows part of the curve with equation \(y = \mathrm { f } ( x )\). \includegraphics[max width=\textwidth, alt={}, center]{57412ec0-ad97-4418-8ba8-93f1f7d8aac1-08_536_1054_367_539}
  1. On Figure 2 below, sketch the curve with equation \(y = - | \mathrm { f } ( x ) |\).
  2. On Figure 3 on the page opposite, sketch the curve with equation \(y = \mathrm { f } ( | 2 x | )\).
    1. Describe a sequence of two geometrical transformations that maps the graph of \(y = \mathrm { f } ( x )\) onto the graph of \(y = \mathrm { f } ( 2 x + 2 )\).
    2. Find the coordinates of the image of the point \(P ( 4 , - 3 )\) under the sequence of transformations given in part (c)(i). \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{57412ec0-ad97-4418-8ba8-93f1f7d8aac1-09_778_1032_424_529}
      \end{figure}
AQA C3 2014 June Q5
11 marks Standard +0.3
5 The functions \(f\) and \(g\) are defined with their respective domains by $$\begin{array} { l l } \mathrm { f } ( x ) = x ^ { 2 } - 6 x + 5 , & \text { for } x \geqslant 3 \\ \mathrm {~g} ( x ) = | x - 6 | , & \text { for all real values of } x \end{array}$$
  1. Find the range of f.
  2. The inverse of f is \(\mathrm { f } ^ { - 1 }\). Find \(\mathrm { f } ^ { - 1 } ( x )\). Give your answer in its simplest form.
    1. Find \(\mathrm { gf } ( x )\).
    2. Solve the equation \(\operatorname { gf } ( x ) = 6\).
AQA C3 2014 June Q6
9 marks Standard +0.8
6
  1. By using integration by parts twice, find $$\int x ^ { 2 } \sin 2 x d x$$
  2. A curve has equation \(y = x \sqrt { \sin 2 x }\), for \(0 \leqslant x \leqslant \frac { \pi } { 2 }\). The region bounded by the curve and the \(x\)-axis is rotated through \(2 \pi\) radians about the \(x\)-axis to generate a solid. Find the exact value of the volume of the solid generated.
    [0pt] [3 marks]
AQA C3 2014 June Q7
6 marks Standard +0.8
7 Use the substitution \(u = 3 - x ^ { 3 }\) to find the exact value of \(\int _ { 0 } ^ { 1 } \frac { x ^ { 5 } } { 3 - x ^ { 3 } } \mathrm {~d} x\).
[0pt] [6 marks]
AQA C3 2014 June Q8
12 marks Standard +0.3
8
  1. Show that the expression \(\frac { 1 - \sin x } { \cos x } + \frac { \cos x } { 1 - \sin x }\) can be written as \(2 \sec x\).
    [0pt] [4 marks]
  2. Hence solve the equation $$\frac { 1 - \sin x } { \cos x } + \frac { \cos x } { 1 - \sin x } = \tan ^ { 2 } x - 2$$ giving the values of \(x\) to the nearest degree in the interval \(0 ^ { \circ } \leqslant x < 360 ^ { \circ }\).
    [0pt] [6 marks]
  3. Hence solve the equation $$\frac { 1 - \sin \left( 2 \theta - 30 ^ { \circ } \right) } { \cos \left( 2 \theta - 30 ^ { \circ } \right) } + \frac { \cos \left( 2 \theta - 30 ^ { \circ } \right) } { 1 - \sin \left( 2 \theta - 30 ^ { \circ } \right) } = \tan ^ { 2 } \left( 2 \theta - 30 ^ { \circ } \right) - 2$$ giving the values of \(\theta\) to the nearest degree in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
    [0pt] [2 marks]
    \includegraphics[max width=\textwidth, alt={}]{57412ec0-ad97-4418-8ba8-93f1f7d8aac1-16_1517_1709_1190_153}
    \includegraphics[max width=\textwidth, alt={}]{57412ec0-ad97-4418-8ba8-93f1f7d8aac1-20_2489_1730_221_139}
AQA C3 2016 June Q1
6 marks Moderate -0.3
1
  1. Given that \(y = ( 4 x + 1 ) ^ { 3 } \sin 2 x\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Given that \(y = \frac { 2 x ^ { 2 } + 3 } { 3 x ^ { 2 } + 4 }\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { p x } { \left( 3 x ^ { 2 } + 4 \right) ^ { 2 } }\), where \(p\) is a constant.
  3. Given that \(y = \ln \left( \frac { 2 x ^ { 2 } + 3 } { 3 x ^ { 2 } + 4 } \right)\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
AQA C3 2016 June Q2
15 marks Standard +0.3
2 The curve with equation \(y = x ^ { x }\), where \(x > 0\), intersects the line \(y = 5\) at a single point, where \(x = \alpha\).
  1. Show that \(\alpha\) lies between 2 and 3 .
  2. Show that the equation \(x ^ { x } = 5\) can be rearranged into the form $$x = \mathrm { e } ^ { \left( \frac { \ln 5 } { x } \right) }$$
  3. Use the iterative formula $$x _ { n + 1 } = \mathrm { e } ^ { \left( \frac { \ln 5 } { x _ { n } } \right) }$$ with \(x _ { 1 } = 2\) to find the values of \(x _ { 2 }\) and \(x _ { 3 }\), giving your answers to three decimal places.
    1. Use Simpson's rule with 7 ordinates ( 6 strips) to find an approximation to $$\int _ { 0.5 } ^ { 1.7 } \left( 5 - x ^ { x } \right) \mathrm { d } x$$ giving your answer to three significant figures.
    2. Hence find an approximation to \(\int _ { 0.5 } ^ { 1.7 } x ^ { x } \mathrm {~d} x\).
AQA C3 2016 June Q3
5 marks Standard +0.8
3 Solve $$x ^ { 2 } \geqslant | 5 x - 6 |$$ [5 marks]
AQA C3 2016 June Q4
10 marks Moderate -0.3
4
  1. Describe a sequence of two geometrical transformations that maps the graph of \(y = \mathrm { e } ^ { x }\) onto the graph of \(y = \mathrm { e } ^ { 2 x - 5 }\).
  2. The normal to the curve \(y = \mathrm { e } ^ { 2 x - 5 }\) at the point \(P \left( 2 , \mathrm { e } ^ { - 1 } \right)\) intersects the \(x\)-axis at the point \(A\) and the \(y\)-axis at the point \(B\). Show that the area of the triangle \(O A B\) is \(\frac { \left( \mathrm { e } ^ { 2 } + 1 \right) ^ { m } } { \mathrm { e } ^ { n } }\), where \(m\) and \(n\) are integers.
    [0pt] [6 marks]
AQA C3 2016 June Q5
7 marks Standard +0.8
5 The function f is defined by $$\mathrm { f } ( x ) = 16 x - \mathrm { e } ^ { 2 x } , \text { for all real } x$$ The graph of \(y = \mathrm { f } ( x )\) is sketched below. \includegraphics[max width=\textwidth, alt={}, center]{bf427498-f1ee-4167-a6f2-ddaa2ff5ef81-12_789_1349_534_347}
  1. Find the range of f.
  2. The composite function fg is defined by $$\operatorname { fg } ( x ) = \frac { 16 } { x } - \mathrm { e } ^ { \frac { 2 } { x } } , \text { for real } x , x \neq 0$$ Find an expression for \(\operatorname { gg } ( x )\).
AQA C3 2016 June Q6
11 marks Standard +0.8
6
  1. Use integration by parts to find \(\int \frac { \ln ( 3 x ) } { x ^ { 2 } } \mathrm {~d} x\).
  2. The region bounded by the curve \(y = \frac { \ln ( 3 x ) } { x }\), the \(x\)-axis from \(\frac { 1 } { 3 }\) to 1 , and the line \(x = 1\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid. Find the exact value of the volume of the solid generated.
    [0pt] [7 marks]
AQA C3 2016 June Q7
8 marks Moderate -0.3
7
  1. By writing \(\sec x = ( \cos x ) ^ { - 1 }\), use the chain rule to show that, if \(y = \sec x\), then $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \sec x \tan x$$
  2. The function f is defined by $$\mathrm { f } ( x ) = 2 \tan x - 3 \sec x , \text { for } 0 < x < \frac { \pi } { 2 }$$ Find the value of the \(y\)-coordinate of the stationary point of the graph of \(y = \mathrm { f } ( x )\), giving your answer in the form \(p \sqrt { q }\), where \(p\) and \(q\) are integers.
    [0pt] [6 marks]
AQA C3 2016 June Q8
7 marks Moderate -0.3
8 Use the substitution \(u = 4 x - 1\) to find the exact value of $$\int _ { \frac { 1 } { 4 } } ^ { \frac { 1 } { 2 } } ( 5 - 2 x ) ( 4 x - 1 ) ^ { \frac { 1 } { 3 } } \mathrm {~d} x$$
\includegraphics[max width=\textwidth, alt={}]{bf427498-f1ee-4167-a6f2-ddaa2ff5ef81-18_2104_1712_603_153}
AQA C3 2016 June Q9
8 marks Standard +0.3
9
  1. It is given that \(\sec x - \tan x = - 5\).
    1. Show that \(\sec x + \tan x = - 0.2\).
    2. Hence find the exact value of \(\cos x\).
  2. Hence solve the equation $$\sec \left( 2 x - 70 ^ { \circ } \right) - \tan \left( 2 x - 70 ^ { \circ } \right) = - 5$$ giving all values of \(x\), to one decimal place, in the interval \(- 90 ^ { \circ } \leqslant x \leqslant 90 ^ { \circ }\).
    [0pt] [3 marks] \section*{DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED}
AQA C3 Q2
Moderate -0.3
2 Use Simpson's rule with 5 ordinates ( 4 strips) to find an approximation to $$\int _ { 1 } ^ { 3 } \frac { 1 } { \sqrt { 1 + x ^ { 3 } } } \mathrm {~d} x$$ giving your answer to three significant figures.
AQA C3 Q5
Standard +0.3
5 The diagram shows part of the graph of \(y = \mathrm { e } ^ { 2 x } - 9\). The graph cuts the coordinate axes at ( \(0 , a\) ) and ( \(b , 0\) ). \includegraphics[max width=\textwidth, alt={}, center]{9aac4ee4-2435-4315-a87d-fe9fa8e15665-004_817_908_479_550}
  1. State the value of \(a\), and show that \(b = \ln 3\).
  2. Show that \(y ^ { 2 } = \mathrm { e } ^ { 4 x } - 18 \mathrm { e } ^ { 2 x } + 81\).
  3. The shaded region \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Find the volume of the solid formed, giving your answer in the form \(\pi ( p \ln 3 + q )\), where \(p\) and \(q\) are integers.
  4. Sketch the curve with equation \(y = \left| \mathrm { e } ^ { 2 x } - 9 \right|\) for \(x \geqslant 0\).
AQA C3 Q6
Moderate -0.5
6 [Figure 1, printed on the insert, is provided for use in this question.]
The curve \(y = x ^ { 3 } + 4 x - 3\) intersects the \(x\)-axis at the point \(A\) where \(x = \alpha\).
  1. Show that \(\alpha\) lies between 0.5 and 1.0.
  2. Show that the equation \(x ^ { 3 } + 4 x - 3 = 0\) can be rearranged into the form \(x = \frac { 3 - x ^ { 3 } } { 4 }\).
    (1 mark)
    1. Use the iteration \(x _ { n + 1 } = \frac { 3 - x _ { n } { } ^ { 3 } } { 4 }\) with \(x _ { 1 } = 0.5\) to find \(x _ { 3 }\), giving your answer to two decimal places.
    2. The sketch on Figure 1 shows parts of the graphs of \(y = \frac { 3 - x ^ { 3 } } { 4 }\) 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.
      (3 marks)
AQA C3 Q7
Standard +0.3
7
  1. The sketch shows the graph of \(y = \sin ^ { - 1 } x\). \includegraphics[max width=\textwidth, alt={}, center]{9aac4ee4-2435-4315-a87d-fe9fa8e15665-006_819_824_456_591} Write down the coordinates of the points \(P\) and \(Q\), the end-points of the graph.
  2. Sketch the graph of \(y = - \sin ^ { - 1 } ( x - 1 )\).
AQA C3 Q8
Moderate -0.3
8 The functions \(f\) and \(g\) are defined with their respective domains by $$\begin{array} { l l } \mathrm { f } ( x ) = x ^ { 2 } & \text { for all real values of } x \\ \mathrm {~g} ( x ) = \frac { 1 } { x + 2 } & \text { for real values of } x , \quad x \neq - 2 \end{array}$$
  1. State the range of f.
    1. Find fg(x).
    2. Solve the equation \(\operatorname { fg } ( x ) = 4\).
    1. Explain why the function f does not have an inverse.
    2. The inverse of g is \(\mathrm { g } ^ { - 1 }\). Find \(\mathrm { g } ^ { - 1 } ( x )\).
AQA C3 Q9
Standard +0.3
9
  1. Given that \(y = x ^ { - 2 } \ln x\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 - 2 \ln x } { x ^ { 3 } }\).
  2. Using integration by parts, find \(\int x ^ { - 2 } \ln x \mathrm {~d} x\).
  3. The sketch shows the graph of \(y = x ^ { - 2 } \ln x\). \includegraphics[max width=\textwidth, alt={}, center]{9aac4ee4-2435-4315-a87d-fe9fa8e15665-007_593_1034_696_543}
    1. Using the answer to part (a), find, in terms of e, the \(x\)-coordinate of the stationary point \(A\).
    2. The region \(R\) is bounded by the curve, the \(x\)-axis and the line \(x = 5\). Using your answer to part (b), show that the area of \(R\) is $$\frac { 1 } { 5 } ( 4 - \ln 5 )$$
AQA C3 Q10
Standard +0.3
10
    1. By writing \(\ln x\) as \(( \ln x ) \times 1\), use integration by parts to find \(\int \ln x \mathrm {~d} x\).
    2. Find \(\int ( \ln x ) ^ { 2 } \mathrm {~d} x\).
  2. Use the substitution \(u = \sqrt { x }\) to find the exact value of $$\int _ { 1 } ^ { 4 } \frac { 1 } { x + \sqrt { x } } \mathrm {~d} x$$ (7 marks)
AQA C3 2006 January Q1
5 marks Moderate -0.8
1
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(y = \tan 3 x\).
    (2 marks)
  2. Given that \(y = \frac { 3 x + 1 } { 2 x + 1 }\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { ( 2 x + 1 ) ^ { 2 } }\).
    (3 marks)
AQA C3 2006 January Q2
4 marks Moderate -0.5
2 Use Simpson's rule with 5 ordinates (4 strips) to find an approximation to $$\int _ { 1 } ^ { 3 } \frac { 1 } { \sqrt { 1 + x ^ { 3 } } } \mathrm {~d} x$$ giving your answer to three significant figures.
AQA C3 2006 January Q3
10 marks Standard +0.3
3
    1. Given that \(\mathrm { f } ( x ) = x ^ { 4 } + 2 x\), find \(\mathrm { f } ^ { \prime } ( x )\).
    2. Hence, or otherwise, find \(\int \frac { 2 x ^ { 3 } + 1 } { x ^ { 4 } + 2 x } \mathrm {~d} x\).
    1. Use the substitution \(u = 2 x + 1\) to show that $$\int x \sqrt { 2 x + 1 } \mathrm {~d} x = \frac { 1 } { 4 } \int \left( u ^ { \frac { 3 } { 2 } } - u ^ { \frac { 1 } { 2 } } \right) \mathrm { d } u$$
    2. Hence show that \(\int _ { 0 } ^ { 4 } x \sqrt { 2 x + 1 } \mathrm {~d} x = 19.9\) correct to three significant figures.