Questions — AQA C3 (172 questions)

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AQA C3 2010 January Q3
3
  1. Solve the equation $$\operatorname { cosec } x = 3$$ giving all values of \(x\) in radians to two decimal places, in the interval \(0 \leqslant x \leqslant 2 \pi\).
    (2 marks)
  2. By using a suitable trigonometric identity, solve the equation $$\cot ^ { 2 } x = 11 - \operatorname { cosec } x$$ giving all values of \(x\) in radians to two decimal places, in the interval \(0 \leqslant x \leqslant 2 \pi\).
    (6 marks)
AQA C3 2010 January Q4
4
  1. Sketch the graph of \(y = | 8 - 2 x |\).
  2. Solve the equation \(| 8 - 2 x | = 4\).
  3. Solve the inequality \(| 8 - 2 x | > 4\).
AQA C3 2010 January Q5
5
  1. Use the mid-ordinate rule with four strips to find an estimate for \(\int _ { 0 } ^ { 12 } \ln \left( x ^ { 2 } + 5 \right) \mathrm { d } x\), giving your answer to three significant figures.
  2. A curve has equation \(y = \ln \left( x ^ { 2 } + 5 \right)\).
    1. Show that this equation can be rewritten as \(x ^ { 2 } = \mathrm { e } ^ { y } - 5\).
    2. The region bounded by the curve, the lines \(y = 5\) and \(y = 10\) and the \(y\)-axis is rotated through \(360 ^ { \circ }\) about the \(y\)-axis. Find the exact value of the volume of the solid generated.
  3. The graph with equation \(y = \ln \left( x ^ { 2 } + 5 \right)\) is stretched with scale factor 4 parallel to the \(x\)-axis, and then translated through \(\left[ \begin{array} { l } 0
    3 \end{array} \right]\) to give the graph with equation \(y = \mathrm { f } ( x )\). Write down an expression for \(\mathrm { f } ( x )\).
AQA C3 2010 January Q6
6 The functions \(f\) and \(g\) are defined with their respective domains by $$\begin{array} { l l } \mathrm { f } ( x ) = \mathrm { e } ^ { 2 x } - 3 , & \text { for all real values of } x
\mathrm {~g} ( x ) = \frac { 1 } { 3 x + 4 } , & \text { for real values of } x , x \neq - \frac { 4 } { 3 } \end{array}$$
  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 \(\mathrm { f } ^ { - 1 } ( x ) = 0\).
    1. Find an expression for \(\operatorname { gf } ( x )\).
    2. Solve the equation \(\mathrm { gf } ( x ) = 1\), giving your answer in an exact form.
AQA C3 2010 January Q7
7 It is given that \(y = \tan 4 x\).
  1. By writing \(\tan 4 x\) as \(\frac { \sin 4 x } { \cos 4 x }\), use the quotient rule to show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = p \left( 1 + \tan ^ { 2 } 4 x \right)\), where \(p\) is a number to be determined.
  2. Show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = q y \left( 1 + y ^ { 2 } \right)\), where \(q\) is a number to be determined.
AQA C3 2010 January Q8
8
  1. Using integration by parts, find \(\int x \sin ( 2 x - 1 ) \mathrm { d } x\).
  2. Use the substitution \(u = 2 x - 1\) to find \(\int \frac { x ^ { 2 } } { 2 x - 1 } \mathrm {~d} x\), giving your answer in terms of \(x\).
    (6 marks)
AQA C3 2007 June Q1
1
  1. Differentiate \(\ln x\) with respect to \(x\).
  2. Given that \(y = ( x + 1 ) \ln x\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  3. Find an equation of the normal to the curve \(y = ( x + 1 ) \ln x\) at the point where \(x = 1\).
AQA C3 2007 June Q2
2
  1. Differentiate \(( x - 1 ) ^ { 4 }\) with respect to \(x\).
  2. The diagram shows the curve with equation \(y = 2 \sqrt { ( x - 1 ) ^ { 3 } }\) for \(x \geqslant 1\).
    \includegraphics[max width=\textwidth, alt={}, center]{9fd9fa54-b0e6-413d-8645-de34b99b859a-02_789_1180_1190_431} The shaded region \(R\) is bounded by the curve \(y = 2 \sqrt { ( x - 1 ) ^ { 3 } }\), the lines \(x = 2\) and \(x = 4\), and the \(x\)-axis. Find the exact value of the volume of the solid formed when the region \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
  3. Describe a sequence of two geometrical transformations that maps the graph of \(y = \sqrt { x ^ { 3 } }\) onto the graph of \(y = 2 \sqrt { ( x - 1 ) ^ { 3 } }\).
AQA C3 2007 June Q3
3
  1. Solve the equation \(\operatorname { cosec } x = 2\), giving all values of \(x\) in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\).
    (2 marks)
  2. The diagram shows the graph of \(y = \operatorname { cosec } x\) for \(0 ^ { \circ } < x < 360 ^ { \circ }\).
    \includegraphics[max width=\textwidth, alt={}, center]{9fd9fa54-b0e6-413d-8645-de34b99b859a-03_609_1045_559_479}
    1. The point \(A\) on the curve is where \(x = 90 ^ { \circ }\). State the \(y\)-coordinate of \(A\).
    2. Sketch the graph of \(y = | \operatorname { cosec } x |\) for \(0 ^ { \circ } < x < 360 ^ { \circ }\).
  3. Solve the equation \(| \operatorname { cosec } x | = 2\), giving all values of \(x\) in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\).
    (2 marks)
AQA C3 2007 June Q4
4 [Figure 1, printed on the insert, is provided for use in this question.]
  1. Use Simpson's rule with 5 ordinates (4 strips) to find an approximation to \(\int _ { 1 } ^ { 2 } 3 ^ { x } \mathrm {~d} x\), giving your answer to three significant figures.
  2. The curve \(y = 3 ^ { x }\) intersects the line \(y = x + 3\) at the point where \(x = \alpha\).
    1. Show that \(\alpha\) lies between 0.5 and 1.5.
    2. Show that the equation \(3 ^ { x } = x + 3\) can be rearranged into the form $$x = \frac { \ln ( x + 3 ) } { \ln 3 }$$
    3. Use the iteration \(x _ { n + 1 } = \frac { \ln \left( x _ { n } + 3 \right) } { \ln 3 }\) with \(x _ { 1 } = 0.5\) to find \(x _ { 3 }\) to two significant figures.
    4. The sketch on Figure 1 shows part of the graphs of \(y = \frac { \ln ( x + 3 ) } { \ln 3 }\) 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.
AQA C3 2007 June Q5
5 The functions \(f\) and \(g\) are defined with their respective domains by $$\begin{aligned} & \mathrm { f } ( x ) = \sqrt { x - 2 } \text { for } x \geqslant 2
& \mathrm {~g} ( x ) = \frac { 1 } { x } \quad \text { for real values of } x , x \neq 0 \end{aligned}$$
  1. State the range of f .
    1. Find fg(x).
    2. Solve the equation \(\operatorname { fg } ( x ) = 1\).
  3. The inverse of f is \(\mathrm { f } ^ { - 1 }\). Find \(\mathrm { f } ^ { - 1 } ( x )\).
AQA C3 2007 June Q6
6
  1. Use integration by parts to find \(\int x \mathrm { e } ^ { 5 x } \mathrm {~d} x\).
    1. Use the substitution \(u = \sqrt { x }\) to show that $$\int \frac { 1 } { \sqrt { x } ( 1 + \sqrt { x } ) } \mathrm { d } x = \int \frac { 2 } { 1 + u } \mathrm {~d} u$$
    2. Find the exact value of \(\int _ { 1 } ^ { 9 } \frac { 1 } { \sqrt { x } ( 1 + \sqrt { x } ) } \mathrm { d } x\).
AQA C3 2007 June Q7
7
  1. A curve has equation \(y = \left( x ^ { 2 } - 3 \right) \mathrm { e } ^ { x }\).
    1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
    2. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
    1. Find the \(x\)-coordinate of each of the stationary points of the curve.
    2. Using your answer to part (a)(ii), determine the nature of each of the stationary points.
AQA C3 2007 June Q8
8
  1. Write down \(\int \sec ^ { 2 } x \mathrm {~d} x\).
  2. Given that \(y = \frac { \cos x } { \sin x }\), use the quotient rule to show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \operatorname { cosec } ^ { 2 } x\).
  3. Prove the identity \(( \tan x + \cot x ) ^ { 2 } = \sec ^ { 2 } x + \operatorname { cosec } ^ { 2 } x\).
  4. Hence find \(\int _ { 0.5 } ^ { 1 } ( \tan x + \cot x ) ^ { 2 } \mathrm {~d} x\), giving your answer to two significant figures.
AQA C3 2015 June Q1
8 marks
1
  1. Use the mid-ordinate rule with four strips to find an estimate for \(\int _ { 1.5 } ^ { 5.5 } \mathrm { e } ^ { 2 - x } \ln ( 3 x - 2 ) \mathrm { d } x\), giving your answer to three decimal places.
    [0pt] [4 marks]
  2. Find the exact value of the gradient of the curve \(y = \mathrm { e } ^ { 2 - x } \ln ( 3 x - 2 )\) at the point on the curve where \(x = 2\).
    [0pt] [4 marks]
AQA C3 2015 June Q2
4 marks
2
  1. Sketch, on the axes below, the curve with equation \(y = 4 - | 2 x + 1 |\), indicating the coordinates where the curve crosses the axes.
  2. Solve the equation \(x = 4 - | 2 x + 1 |\).
  3. Solve the inequality \(x < 4 - | 2 x + 1 |\).
  4. Describe a sequence of two geometrical transformations that maps the graph of \(y = | 2 x + 1 |\) onto the graph of \(y = 4 - | 2 x + 1 |\).
    [0pt] [4 marks] \section*{Answer space for question 2}

  5. \includegraphics[max width=\textwidth, alt={}, center]{2df59047-3bfe-4b9c-a77f-142bc7506cbc-04_851_1459_1000_319}
AQA C3 2015 June Q3
3
  1. It is given that the curves with equations \(y = 6 \ln x\) and \(y = 8 x - x ^ { 2 } - 3\) intersect at a single point where \(x = \alpha\).
    1. Show that \(\alpha\) lies between 5 and 6 .
    2. Show that the equation \(x = 4 + \sqrt { 13 - 6 \ln x }\) can be rearranged into the form $$6 \ln x + x ^ { 2 } - 8 x + 3 = 0$$
    3. Use the iterative formula $$x _ { n + 1 } = 4 + \sqrt { 13 - 6 \ln x _ { n } }$$ with \(x _ { 1 } = 5\) to find the values of \(x _ { 2 }\) and \(x _ { 3 }\), giving your answers to three decimal places.
  2. A curve has equation \(y = \mathrm { f } ( x )\) where \(\mathrm { f } ( x ) = 6 \ln x + x ^ { 2 } - 8 x + 3\).
    1. Find the exact values of the coordinates of the stationary points of the curve.
    2. Hence, or otherwise, find the exact values of the coordinates of the stationary points of the curve with equation $$y = 2 \mathrm { f } ( x - 4 )$$
AQA C3 2015 June Q4
3 marks
4 The functions f and g are defined by $$\begin{array} { l l } \mathrm { f } ( x ) = 5 - \mathrm { e } ^ { 3 x } , & \text { for all real values of } x
\mathrm {~g} ( x ) = \frac { 1 } { 2 x - 3 } , & \text { for } x \neq 1.5 \end{array}$$
  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 \(\mathrm { f } ^ { - 1 } ( x ) = 0\).
  3. Find an expression for \(\operatorname { gg } ( x )\), giving your answer in the form \(\frac { a x + b } { c x + d }\), where \(a , b , c\) and \(d\) are integers.
    [0pt] [3 marks]
AQA C3 2015 June Q5
9 marks
5
  1. By writing \(\tan x\) as \(\frac { \sin x } { \cos x }\), use the quotient rule to show that \(\frac { \mathrm { d } } { \mathrm { d } x } ( \tan x ) = \sec ^ { 2 } x\).
    [0pt] [2 marks]
  2. Use integration by parts to find \(\int x \sec ^ { 2 } x \mathrm {~d} x\).
    [0pt] [4 marks]
  3. The region bounded by the curve \(y = ( 5 \sqrt { x } ) \sec x\), the \(x\)-axis from 0 to 1 and the line \(x = 1\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid. Find the value of the volume of the solid generated, giving your answer to two significant figures.
    [0pt] [3 marks]
AQA C3 2015 June Q6
6
  1. Sketch, on the axes below, the curve with equation \(y = \sin ^ { - 1 } ( 3 x )\), where \(y\) is in radians. State the exact values of the coordinates of the end points of the graph.
  2. Given that \(x = \frac { 1 } { 3 } \sin y\), write down \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) and hence find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(y\). \section*{Answer space for question 6}

  3. \includegraphics[max width=\textwidth, alt={}, center]{2df59047-3bfe-4b9c-a77f-142bc7506cbc-14_839_1451_813_324}
AQA C3 2015 June Q7
7 marks
7 Use the substitution \(u = 6 - x ^ { 2 }\) to find the value of \(\int _ { 1 } ^ { 2 } \frac { x ^ { 3 } } { \sqrt { 6 - x ^ { 2 } } } \mathrm {~d} x\), giving your answer in the form \(p \sqrt { 5 } + q \sqrt { 2 }\), where \(p\) and \(q\) are rational numbers.
[0pt] [7 marks]
AQA C3 2015 June Q8
5 marks
8
  1. Show that the equation \(4 \operatorname { cosec } ^ { 2 } \theta - \cot ^ { 2 } \theta = k\), where \(k \neq 4\), can be written in the form $$\sec ^ { 2 } \theta = \frac { k - 1 } { k - 4 }$$
  2. Hence, or otherwise, solve the equation $$4 \operatorname { cosec } ^ { 2 } \left( 2 x + 75 ^ { \circ } \right) - \cot ^ { 2 } \left( 2 x + 75 ^ { \circ } \right) = 5$$ giving all values of \(x\) in the interval \(0 ^ { \circ } < x < 180 ^ { \circ }\).
    [0pt] [5 marks]
    \includegraphics[max width=\textwidth, alt={}, center]{2df59047-3bfe-4b9c-a77f-142bc7506cbc-18_72_113_1055_159}
    \includegraphics[max width=\textwidth, alt={}]{2df59047-3bfe-4b9c-a77f-142bc7506cbc-20_2288_1707_221_153}