Questions — AQA C3 (172 questions)

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AQA C3 2010 June Q5
5
  1. Show that the equation $$10 \operatorname { cosec } ^ { 2 } x = 16 - 11 \cot x$$ can be written in the form $$10 \cot ^ { 2 } x + 11 \cot x - 6 = 0$$
  2. Hence, given that \(10 \operatorname { cosec } ^ { 2 } x = 16 - 11 \cot x\), find the possible values of \(\tan x\).
AQA C3 2010 June Q6
6 The diagram shows the curve \(y = \frac { \ln x } { x }\).
\includegraphics[max width=\textwidth, alt={}, center]{33ca7e6d-b9eb-46be-b5b0-c5685212d7ff-4_586_1034_1612_513} The curve crosses the \(x\)-axis at \(A\) and has a stationary point at \(B\).
  1. State the coordinates of \(A\).
  2. Find the coordinates of the stationary point, \(B\), of the curve, giving your answer in an exact form.
  3. Find the exact value of the gradient of the normal to the curve at the point where \(x = \mathrm { e } ^ { 3 }\).
AQA C3 2010 June Q7
7
  1. Use integration by parts to find:
    1. \(\quad \int x \cos 4 x \mathrm {~d} x\);
      (4 marks)
    2. \(\int x ^ { 2 } \sin 4 x d x\).
      (4 marks)
  2. The region bounded by the curve \(y = 8 x \sqrt { ( \sin 4 x ) }\) and the lines \(x = 0\) and \(x = 0.2\) is rotated through \(2 \pi\) radians about the \(x\)-axis. Find the value of the volume of the solid generated, giving your answer to three significant figures.
    (3 marks)
AQA C3 2010 June Q8
8 The diagram shows the curves \(y = \mathrm { e } ^ { 2 x } - 1\) and \(y = 4 \mathrm { e } ^ { - 2 x } + 2\).
\includegraphics[max width=\textwidth, alt={}, center]{33ca7e6d-b9eb-46be-b5b0-c5685212d7ff-6_958_1492_372_242} The curve \(y = 4 \mathrm { e } ^ { - 2 x } + 2\) crosses the \(y\)-axis at the point \(A\) and the curves intersect at the point \(B\).
  1. Describe a sequence of two geometrical transformations that maps the graph of \(y = \mathrm { e } ^ { x }\) onto the graph of \(y = \mathrm { e } ^ { 2 x } - 1\).
  2. Write down the coordinates of the point \(A\).
    1. Show that the \(x\)-coordinate of the point \(B\) satisfies the equation $$\left( \mathrm { e } ^ { 2 x } \right) ^ { 2 } - 3 \mathrm { e } ^ { 2 x } - 4 = 0$$
    2. Hence find the exact value of the \(x\)-coordinate of the point \(B\).
  4. Find the exact value of the area of the shaded region bounded by the curves \(y = \mathrm { e } ^ { 2 x } - 1\) and \(y = 4 \mathrm { e } ^ { - 2 x } + 2\) and the \(y\)-axis.
AQA C3 2011 June Q1
1 The diagram shows the curve with equation \(y = \ln ( 6 x )\).
\includegraphics[max width=\textwidth, alt={}, center]{7148f43d-dc7d-43e2-b96e-ed1fb94073bf-2_448_501_370_790}
  1. State the \(x\)-coordinate of the point of intersection of the curve with the \(x\)-axis.
    (1 mark)
  2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
    (2 marks)
  3. Use Simpson's rule with 6 strips (7 ordinates) to find an estimate for \(\int _ { 1 } ^ { 7 } \ln ( 6 x ) \mathrm { d } x\), giving your answer to three significant figures.
AQA C3 2011 June Q2
2
    1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(y = x \mathrm { e } ^ { 2 x }\).
    2. Find an equation of the tangent to the curve \(y = x \mathrm { e } ^ { 2 x }\) at the point \(\left( 1 , \mathrm { e } ^ { 2 } \right)\).
  2. Given that \(y = \frac { 2 \sin 3 x } { 1 + \cos 3 x }\), use the quotient rule to show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { k } { 1 + \cos 3 x }$$ where \(k\) is an integer.
AQA C3 2011 June Q3
3 The curve \(y = \cos ^ { - 1 } ( 2 x - 1 )\) intersects the curve \(y = \mathrm { e } ^ { x }\) at a single point where \(x = \alpha\).
  1. Show that \(\alpha\) lies between 0.4 and 0.5 .
  2. Show that the equation \(\cos ^ { - 1 } ( 2 x - 1 ) = \mathrm { e } ^ { x }\) can be written as \(x = \frac { 1 } { 2 } + \frac { 1 } { 2 } \cos \left( \mathrm { e } ^ { x } \right)\).
  3. Use the iteration \(x _ { n + 1 } = \frac { 1 } { 2 } + \frac { 1 } { 2 } \cos \left( \mathrm { e } ^ { x _ { n } } \right)\) with \(x _ { 1 } = 0.4\) to find the values of \(x _ { 2 }\) and \(x _ { 3 }\), giving your answers to three decimal places.
AQA C3 2011 June Q4
4
    1. Solve the equation \(\operatorname { cosec } \theta = - 4\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\), giving your answers to the nearest \(0.1 ^ { \circ }\).
    2. Solve the equation $$2 \cot ^ { 2 } \left( 2 x + 30 ^ { \circ } \right) = 2 - 7 \operatorname { cosec } \left( 2 x + 30 ^ { \circ } \right)$$ for \(0 ^ { \circ } < x < 180 ^ { \circ }\), giving your answers to the nearest \(0.1 ^ { \circ }\).
  2. Describe a sequence of two geometrical transformations that maps the graph of \(y = \operatorname { cosec } x\) onto the graph of \(y = \operatorname { cosec } \left( 2 x + 30 ^ { \circ } \right)\).
AQA C3 2011 June Q5
5 The functions f and g are defined with their respective domains by $$\begin{array} { l l } \mathrm { f } ( x ) = x ^ { 2 } & \text { for all real values of } x
\mathrm {~g} ( x ) = \frac { 1 } { 2 x + 1 } & \text { for real values of } x , \quad x \neq - 0.5 \end{array}$$
  1. Explain why f does not have an inverse.
  2. The inverse of g is \(\mathrm { g } ^ { - 1 }\). Find \(\mathrm { g } ^ { - 1 } ( x )\).
  3. State the range of \(\mathrm { g } ^ { - 1 }\).
  4. Solve the equation \(\mathrm { fg } ( x ) = \mathrm { g } ( x )\).
AQA C3 2011 June Q6
6
  1. Given that \(3 \ln x = 4\), find the exact value of \(x\).
  2. By forming a quadratic equation in \(\ln x\), solve \(3 \ln x + \frac { 20 } { \ln x } = 19\), giving your answers for \(x\) in an exact form.
AQA C3 2011 June Q7
7
  1. On separate diagrams:
    1. sketch the curve with equation \(y = | 3 x + 3 |\);
    2. sketch the curve with equation \(y = \left| x ^ { 2 } - 1 \right|\).
    1. Solve the equation \(| 3 x + 3 | = \left| x ^ { 2 } - 1 \right|\).
    2. Hence solve the inequality \(| 3 x + 3 | < \left| x ^ { 2 } - 1 \right|\).
      \(8 \quad\) Use the substitution \(u = 1 + 2 \tan x\) to find $$\int \frac { 1 } { ( 1 + 2 \tan x ) ^ { 2 } \cos ^ { 2 } x } d x$$
AQA C3 2011 June Q9
9
  1. Use integration by parts to find \(\int x \ln x \mathrm {~d} x\).
  2. Given that \(y = ( \ln x ) ^ { 2 }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
    (2 marks)
  3. The diagram shows part of the curve with equation \(y = \sqrt { x } \ln x\).
    \includegraphics[max width=\textwidth, alt={}, center]{7148f43d-dc7d-43e2-b96e-ed1fb94073bf-5_406_645_696_719} The shaded region \(R\) is bounded by the curve \(y = \sqrt { x } \ln x\), the line \(x = \mathrm { e }\) and the \(x\)-axis from \(x = 1\) to \(x = \mathrm { e }\). Find the volume of the solid generated when the region \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis, giving your answer in an exact form.
    (6 marks)
AQA C3 2012 June Q1
1 Use the mid-ordinate rule with four strips to find an estimate for \(\int _ { 0.4 } ^ { 1.2 } \cot \left( x ^ { 2 } \right) \mathrm { d } x\), giving your answer to three decimal places.
AQA C3 2012 June Q2
2 For \(0 < x \leqslant 2\), the curves with equations \(y = 4 \ln x\) and \(y = \sqrt { x }\) intersect at a single point where \(x = \alpha\).
  1. Show that \(\alpha\) lies between 0.5 and 1.5.
  2. Show that the equation \(4 \ln x = \sqrt { x }\) can be rearranged into the form $$x = \mathrm { e } ^ { \left( \frac { \sqrt { x } } { 4 } \right) }$$
  3. Use the iterative formula $$x _ { n + 1 } = \mathrm { e } ^ { \left( \frac { \sqrt { x _ { n } } } { 4 } \right) }$$ with \(x _ { 1 } = 0.5\) to find the values of \(x _ { 2 }\) and \(x _ { 3 }\), giving your answers to three decimal places.
  4. Figure 1, on the page 3, shows a sketch of parts of the graphs of \(y = \mathrm { e } ^ { \left( \frac { \sqrt { x } } { 4 } \right) }\) 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. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{d3c66c34-b09c-4223-8383-cf0a68419bf9-3_1285_1543_356_296}
    \end{figure}
AQA C3 2012 June Q3
3 A curve has equation \(y = x ^ { 3 } \ln x\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
    1. Find an equation of the tangent to the curve \(y = x ^ { 3 } \ln x\) at the point on the curve where \(x = \mathrm { e }\).
    2. This tangent intersects the \(x\)-axis at the point \(A\). Find the exact value of the \(x\)-coordinate of the point \(A\).
AQA C3 2012 June Q4
4
  1. By using integration by parts, find \(\int x \mathrm { e } ^ { 6 x } \mathrm {~d} x\).
    (4 marks)
  2. The diagram shows part of the curve with equation \(y = \sqrt { x } \mathrm { e } ^ { 3 x }\).
    \includegraphics[max width=\textwidth, alt={}, center]{d3c66c34-b09c-4223-8383-cf0a68419bf9-4_547_846_536_591} The shaded region \(R\) is bounded by the curve \(y = \sqrt { x } \mathrm { e } ^ { 3 x }\), the line \(x = 1\) and the \(x\)-axis from \(x = 0\) to \(x = 1\). 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 \(\pi \left( p \mathrm { e } ^ { 6 } + q \right)\), where \(p\) and \(q\) are rational numbers.
    (3 marks)
AQA C3 2012 June Q5
5 The functions f and g are defined with their respective domains by $$\begin{array} { l l } \mathrm { f } ( x ) = \sqrt { 2 x - 5 } , & \text { for } x \geqslant 2.5
\mathrm {~g} ( x ) = \frac { 10 } { x } , & \text { for real values of } x , \quad x \neq 0 \end{array}$$
  1. State the range of f .
    1. Find \(\mathrm { fg } ( x )\).
    2. Solve the equation \(\operatorname { fg } ( x ) = 5\).
  3. The inverse of f is \(\mathrm { f } ^ { - 1 }\).
    1. Find \(\mathrm { f } ^ { - 1 } ( x )\).
    2. Solve the equation \(\mathrm { f } ^ { - 1 } ( x ) = 7\).
AQA C3 2012 June Q6
6 Use the substitution \(u = x ^ { 4 } + 2\) to find the value of \(\int _ { 0 } ^ { 1 } \frac { x ^ { 7 } } { \left( x ^ { 4 } + 2 \right) ^ { 2 } } \mathrm {~d} x\), giving your answer in the form \(p \ln q + r\), where \(p , q\) and \(r\) are rational numbers.
AQA C3 2012 June Q7
7 The sketch shows part of the curve with equation \(y = \mathrm { f } ( x )\).
\includegraphics[max width=\textwidth, alt={}, center]{d3c66c34-b09c-4223-8383-cf0a68419bf9-5_632_1029_712_541}
  1. On Figure 2 on page 6, sketch the curve with equation \(y = | \mathrm { f } ( x ) |\).
  2. On Figure 3 on page 6, sketch the curve with equation \(y = \mathrm { f } ( | x | )\).
  3. Describe a sequence of two geometrical transformations that maps the graph of \(y = \mathrm { f } ( x )\) onto the graph of \(y = \frac { 1 } { 2 } \mathrm { f } ( x + 1 )\).
  4. The maximum point of the curve with equation \(y = \mathrm { f } ( x )\) has coordinates \(( - 1,10 )\). Find the coordinates of the maximum point of the curve with equation \(y = \frac { 1 } { 2 } \mathrm { f } ( x + 1 )\).
    (2 marks)
  5. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{d3c66c34-b09c-4223-8383-cf0a68419bf9-6_785_1022_358_548}
    \end{figure}
  6. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{d3c66c34-b09c-4223-8383-cf0a68419bf9-6_776_1022_1395_548}
    \end{figure}
AQA C3 2012 June Q8
8
  1. Show that the equation $$\frac { 1 } { 1 + \cos \theta } + \frac { 1 } { 1 - \cos \theta } = 32$$ can be written in the form $$\operatorname { cosec } ^ { 2 } \theta = 16$$
  2. Hence, or otherwise, solve the equation $$\frac { 1 } { 1 + \cos ( 2 x - 0.6 ) } + \frac { 1 } { 1 - \cos ( 2 x - 0.6 ) } = 32$$ giving all values of \(x\) in radians to two decimal places in the interval \(0 < x < \pi\).
    (5 marks)
AQA C3 2012 June Q9
9
  1. Given that \(x = \frac { \sin y } { \cos y }\), use the quotient rule to show that $$\frac { \mathrm { d } x } { \mathrm {~d} y } = \sec ^ { 2 } y$$ (3 marks)
  2. Given that \(\tan y = x - 1\), use a trigonometrical identity to show that $$\sec ^ { 2 } y = x ^ { 2 } - 2 x + 2$$
  3. Show that, if \(y = \tan ^ { - 1 } ( x - 1 )\), then $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { x ^ { 2 } - 2 x + 2 }$$ (l mark)
  4. A curve has equation \(y = \tan ^ { - 1 } ( x - 1 ) - \ln x\).
    1. Find the value of the \(x\)-coordinate of each of the stationary points of the curve.
    2. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
    3. Hence show that the curve has a minimum point which lies on the \(x\)-axis.
AQA C3 2013 June Q1
1 The diagram below shows the graphs of \(y = | 2 x - 3 |\) and \(y = | x |\).
\includegraphics[max width=\textwidth, alt={}, center]{063bbfa5-df49-44a1-8143-5e076397f63f-02_579_1150_351_482}
  1. Find the \(x\)-coordinates of the points of intersection of the graphs of \(y = | 2 x - 3 |\) and \(y = | x |\).
    (3 marks)
  2. Hence, or otherwise, solve the inequality $$| 2 x - 3 | \geqslant | x |$$ (2 marks)
AQA C3 2013 June Q2
2
  1. Given that \(y = x ^ { 4 } \tan 2 x\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
    (3 marks)
  2. Find the gradient of the curve with equation \(y = \frac { x ^ { 2 } } { x - 1 }\) at the point where \(x = 3\).
    (3 marks)
AQA C3 2013 June Q3
3
  1. The equation \(\mathrm { e } ^ { - x } - 2 + \sqrt { x } = 0\) has a single root, \(\alpha\).
    Show that \(\alpha\) lies between 3 and 4 .
  2. Use the recurrence relation \(x _ { n + 1 } = \left( 2 - e ^ { - x _ { n } } \right) ^ { 2 }\), with \(x _ { 1 } = 3.5\), to find \(x _ { 2 }\) and \(x _ { 3 }\), giving your answers to three decimal places.
  3. The diagram below shows parts of the graphs of \(y = \left( 2 - \mathrm { e } ^ { - x } \right) ^ { 2 }\) and \(y = x\), and a position of \(x _ { 1 }\). On the diagram, draw a staircase or cobweb diagram to show how convergence takes place, indicating the positions of \(x _ { 2 }\) and \(x _ { 3 }\) on the \(x\)-axis.
    \includegraphics[max width=\textwidth, alt={}, center]{063bbfa5-df49-44a1-8143-5e076397f63f-03_1100_1402_881_367}
AQA C3 2013 June Q4
4 By forming and solving a quadratic equation, solve the equation $$8 \sec x - 2 \sec ^ { 2 } x = \tan ^ { 2 } x - 2$$ in the interval \(0 < x < 2 \pi\), giving the values of \(x\) in radians to three significant figures.