Questions C3 (1200 questions)

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AQA C3 2006 January Q3
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.
AQA C3 2006 January Q4
4 It is given that \(2 \operatorname { cosec } ^ { 2 } x = 5 - 5 \cot x\).
  1. Show that the equation \(2 \operatorname { cosec } ^ { 2 } x = 5 - 5 \cot x\) can be written in the form $$2 \cot ^ { 2 } x + 5 \cot x - 3 = 0$$
  2. Hence show that \(\tan x = 2\) or \(\tan x = - \frac { 1 } { 3 }\).
  3. Hence, or otherwise, solve the equation \(2 \operatorname { cosec } ^ { 2 } x = 5 - 5 \cot x\), giving all values of \(x\) in radians to one decimal place in the interval \(- \pi < x \leqslant \pi\).
AQA C3 2006 January Q5
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]{908f530c-076d-47b1-90dd-38dbfe44f898-03_826_924_477_541}
  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 2006 January Q6
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.
      (3 marks)
    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 2006 January Q7
7
  1. The sketch shows the graph of \(y = \sin ^ { - 1 } x\).
    \includegraphics[max width=\textwidth, alt={}, center]{908f530c-076d-47b1-90dd-38dbfe44f898-05_835_834_447_587} 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 2006 January Q8
8 The functions \(f\) and \(g\) are defined with their respective domains by $$\begin{array} { l l } \mathrm { f } ( x ) = x ^ { 2 } & \text { for all real values of } x
\mathrm {~g} ( x ) = \frac { 1 } { x + 2 } & \text { for real values of } x , x \neq - 2 \end{array}$$
  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 2006 January Q9
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]{908f530c-076d-47b1-90dd-38dbfe44f898-06_604_1045_687_536}
    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 2009 January Q1
1 Use Simpson's rule with 5 ordinates (4 strips) to find an approximation to \(\int _ { 1 } ^ { 9 } \frac { 1 } { 1 + \sqrt { x } } \mathrm {~d} x\), giving your answer to three significant figures.
AQA C3 2009 January Q2
2 The diagram shows the curve with equation \(y = \sqrt { ( x - 2 ) ^ { 5 } }\) for \(x \geqslant 2\).
\includegraphics[max width=\textwidth, alt={}, center]{59b896ae-60ce-49ea-9c70-0f76fc5fffae-2_885_1125_854_461} The shaded region \(R\) is bounded by the curve \(y = \sqrt { ( x - 2 ) ^ { 5 } }\), the \(x\)-axis and the lines \(x = 3\) and \(x = 4\). Find the exact value of the volume of the solid formed when the region \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
AQA C3 2009 January Q3
3 [Figure 1, printed on the insert, is provided for use in this question.]
The curve with equation \(y = x ^ { 3 } + 5 x - 4\) intersects the \(x\)-axis at the point \(A\), where \(x = \alpha\).
  1. Show that \(\alpha\) lies between 0.5 and 1 .
  2. Show that the equation \(x ^ { 3 } + 5 x - 4 = 0\) can be rearranged into the form $$x = \frac { 1 } { 5 } \left( 4 - x ^ { 3 } \right)$$
  3. Use the iteration \(x _ { n + 1 } = \frac { 1 } { 5 } \left( 4 - x _ { n } { } ^ { 3 } \right)\) with \(x _ { 1 } = 0.5\) to find \(x _ { 3 }\), giving your answer to three decimal places.
  4. The sketch on Figure 1 shows parts of the graphs of \(y = \frac { 1 } { 5 } \left( 4 - x ^ { 3 } \right)\) 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.
AQA C3 2009 January Q4
4
  1. Solve the equation \(\sec x = \frac { 3 } { 2 }\), giving all values of \(x\) to the nearest degree in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\).
  2. By using a suitable trigonometrical identity, solve the equation $$2 \tan ^ { 2 } x = 10 - 5 \sec x$$ giving all values of \(x\) to the nearest degree in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\).
AQA C3 2009 January Q5
5 The functions \(f\) and \(g\) are defined with their respective domains by $$\begin{array} { l l } \mathrm { f } ( x ) = 2 - x ^ { 4 } & \text { for all real values of } x
\mathrm {~g} ( x ) = \frac { 1 } { x - 4 } & \text { for real values of } x , x \neq 4 \end{array}$$
  1. State the range of f .
  2. Explain why the function f does not have an inverse.
    1. Write down an expression for fg(x).
    2. Solve the equation \(\operatorname { fg } ( x ) = - 14\).
AQA C3 2009 January Q6
6 A curve has equation \(y = \mathrm { e } ^ { 2 x } \left( x ^ { 2 } - 4 x - 2 \right)\).
  1. Find the value of the \(x\)-coordinate of each of the stationary points of the curve.
    1. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
    2. Determine the nature of each of the stationary points of the curve.
AQA C3 2009 January Q7
7
  1. Given that \(3 \mathrm { e } ^ { x } = 4\), find the exact value of \(x\).
    1. By substituting \(y = \mathrm { e } ^ { x }\), show that the equation \(3 \mathrm { e } ^ { x } + 20 \mathrm { e } ^ { - x } = 19\) can be written as \(3 y ^ { 2 } - 19 y + 20 = 0\).
    2. Hence solve the equation \(3 \mathrm { e } ^ { x } + 20 \mathrm { e } ^ { - x } = 19\), giving your answers as exact values. (3 marks)
AQA C3 2009 January Q8
8 The sketch shows the graph of \(y = \cos ^ { - 1 } x\).
\includegraphics[max width=\textwidth, alt={}, center]{59b896ae-60ce-49ea-9c70-0f76fc5fffae-5_593_686_383_683}
  1. Write down the coordinates of \(P\) and \(Q\), the end points of the graph.
  2. Describe a sequence of two geometrical transformations that maps the graph of \(y = \cos ^ { - 1 } x\) onto the graph of \(y = 2 \cos ^ { - 1 } ( x - 1 )\).
  3. Sketch the graph of \(y = 2 \cos ^ { - 1 } ( x - 1 )\).
    1. Write the equation \(y = 2 \cos ^ { - 1 } ( x - 1 )\) in the form \(x = \mathrm { f } ( y )\).
    2. Hence find the value of \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) when \(y = 2\).
AQA C3 2009 January Q9
9
  1. Given that \(y = \frac { 4 x } { 4 x - 3 }\), use the quotient rule to show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { k } { ( 4 x - 3 ) ^ { 2 } }\), where \(k\) is an integer.
    1. Given that \(y = x \ln ( 4 x - 3 )\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
    2. Find an equation of the tangent to the curve \(y = x \ln ( 4 x - 3 )\) at the point where \(x = 1\).
    1. Use the substitution \(u = 4 x - 3\) to find \(\int \frac { 4 x } { 4 x - 3 } \mathrm {~d} x\), giving your answer in terms of \(x\).
    2. By using integration by parts, or otherwise, find \(\int \ln ( 4 x - 3 ) \mathrm { d } x\).
AQA C3 2010 January Q1
1 A curve has equation \(y = \mathrm { e } ^ { - 4 x } \left( x ^ { 2 } + 2 x - 2 \right)\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 \mathrm { e } ^ { - 4 x } \left( 5 - 3 x - 2 x ^ { 2 } \right)\).
  2. Find the exact values of the coordinates of the stationary points of the curve.
AQA C3 2010 January Q2
2 [Figure 1, printed on the insert, is provided for use in this question.]
    1. Sketch the graph of \(y = \sin ^ { - 1 } x\), where \(y\) is in radians. State the coordinates of the end points of the graph.
    2. By drawing a suitable straight line on your sketch, show that the equation $$\sin ^ { - 1 } x = \frac { 1 } { 4 } x + 1$$ has only one solution.
  2. The root of the equation \(\sin ^ { - 1 } x = \frac { 1 } { 4 } x + 1\) is \(\alpha\). Show that \(0.5 < \alpha < 1\).
  3. The equation \(\sin ^ { - 1 } x = \frac { 1 } { 4 } x + 1\) can be rewritten as \(x = \sin \left( \frac { 1 } { 4 } x + 1 \right)\).
    1. Use the iteration \(x _ { n + 1 } = \sin \left( \frac { 1 } { 4 } x _ { n } + 1 \right)\) with \(x _ { 1 } = 0.5\) to find the values of \(x _ { 2 }\) and \(x _ { 3 }\), giving your answers to three decimal places.
    2. The sketch on Figure 1 shows parts of the graphs of \(y = \sin \left( \frac { 1 } { 4 } x + 1 \right)\) 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.
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 \(x = 1\).