Questions — AQA (3620 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 C4 2011 January Q1
6 marks Moderate -0.3
1
  1. Express \(2 \sin x + 5 \cos x\) in the form \(R \sin ( x + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give your value of \(\alpha\) to the nearest \(0.1 ^ { \circ }\).
    1. Write down the maximum value of \(2 \sin x + 5 \cos x\).
    2. Find the value of \(x\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\) at which this maximum occurs, giving the value of \(x\) to the nearest \(0.1 ^ { \circ }\).
AQA C4 2011 January Q2
10 marks Moderate -0.3
2
  1. The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 9 x ^ { 3 } + 18 x ^ { 2 } - x - 2\).
    1. Use the Factor Theorem to show that \(3 x + 1\) is a factor of \(\mathrm { f } ( x )\).
    2. Express \(\mathrm { f } ( x )\) as a product of three linear factors.
    3. Simplify \(\frac { 9 x ^ { 3 } + 21 x ^ { 2 } + 6 x } { \mathrm { f } ( x ) }\).
  2. When the polynomial \(9 x ^ { 3 } + p x ^ { 2 } - x - 2\) is divided by \(3 x - 2\), the remainder is - 4 . Find the value of the constant \(p\).
AQA C4 2011 January Q3
12 marks Standard +0.3
3
  1. Express \(\frac { 3 + 9 x } { ( 1 + x ) ( 3 + 5 x ) }\) in the form \(\frac { A } { 1 + x } + \frac { B } { 3 + 5 x }\), where \(A\) and \(B\) are integers.
  2. Hence, or otherwise, find the binomial expansion of \(\frac { 3 + 9 x } { ( 1 + x ) ( 3 + 5 x ) }\) up to and including the term in \(x ^ { 2 }\).
  3. Find the range of values of \(x\) for which the binomial expansion of \(\frac { 3 + 9 x } { ( 1 + x ) ( 3 + 5 x ) }\) is valid.
    (2 marks)
AQA C4 2011 January Q4
6 marks Standard +0.3
4 A curve is defined by the parametric equations $$x = 3 \mathrm { e } ^ { t } , \quad y = \mathrm { e } ^ { 2 t } - \mathrm { e } ^ { - 2 t }$$
    1. Find the gradient of the curve at the point where \(t = 0\).
    2. Find an equation of the tangent to the curve at the point where \(t = 0\).
  2. Show that the cartesian equation of the curve can be written in the form $$y = \frac { x ^ { 2 } } { k } - \frac { k } { x ^ { 2 } }$$ where \(k\) is an integer.
AQA C4 2011 January Q5
7 marks Moderate -0.3
5 A model for the radioactive decay of a form of iodine is given by $$m = m _ { 0 } 2 ^ { - \frac { 1 } { 8 } t }$$ The mass of the iodine after \(t\) days is \(m\) grams. Its initial mass is \(m _ { 0 }\) grams.
  1. Use the given model to find the mass that remains after 10 grams of this form of iodine have decayed for 14 days, giving your answer to the nearest gram.
  2. A mass of \(m _ { 0 }\) grams of this form of iodine decays to \(\frac { m _ { 0 } } { 16 }\) grams in \(d\) days. Find the value of \(d\).
  3. After \(n\) days, a mass of this form of iodine has decayed to less than \(1 \%\) of its initial mass. Find the minimum integer value of \(n\).
AQA C4 2011 January Q6
10 marks Standard +0.3
6
    1. Given that \(\tan 2 x + \tan x = 0\), show that \(\tan x = 0\) or \(\tan ^ { 2 } x = 3\).
    2. Hence find all solutions of \(\tan 2 x + \tan x = 0\) in the interval \(0 ^ { \circ } < x < 180 ^ { \circ }\).
      (l mark)
    1. Given that \(\cos x \neq 0\), show that the equation $$\sin 2 x = \cos x \cos 2 x$$ can be written in the form $$2 \sin ^ { 2 } x + 2 \sin x - 1 = 0$$
    2. Show that all solutions of the equation \(2 \sin ^ { 2 } x + 2 \sin x - 1 = 0\) are given by \(\sin x = \frac { \sqrt { 3 } - 1 } { p }\), where \(p\) is an integer.
AQA C4 2011 January Q7
10 marks Moderate -0.3
7
    1. Solve the differential equation \(\frac { \mathrm { d } x } { \mathrm {~d} t } = \sqrt { x } \sin \left( \frac { t } { 2 } \right)\) to find \(x\) in terms of \(t\).
    2. Given that \(x = 1\) when \(t = 0\), show that the solution can be written as $$x = ( a - \cos b t ) ^ { 2 }$$ where \(a\) and \(b\) are constants to be found.
  2. The height, \(x\) metres, above the ground of a car in a fairground ride at time \(t\) seconds is modelled by the differential equation \(\frac { \mathrm { d } x } { \mathrm {~d} t } = \sqrt { x } \sin \left( \frac { t } { 2 } \right)\). The car is 1 metre above the ground when \(t = 0\).
    1. Find the greatest height above the ground reached by the car during the ride.
    2. Find the value of \(t\) when the car is first 5 metres above the ground, giving your answer to one decimal place.
AQA C4 2011 January Q8
14 marks Standard +0.3
8 The coordinates of the points \(A\) and \(B\) are \(( 3 , - 2,4 )\) and \(( 6,0,3 )\) respectively.
The line \(l _ { 1 }\) has equation \(\mathbf { r } = \left[ \begin{array} { r } 3 \\ - 2 \\ 4 \end{array} \right] + \lambda \left[ \begin{array} { r } 2 \\ - 1 \\ 3 \end{array} \right]\).
    1. Find the vector \(\overrightarrow { A B }\).
    2. Calculate the acute angle between \(\overrightarrow { A B }\) and the line \(l _ { 1 }\), giving your answer to the nearest \(0.1 ^ { \circ }\).
  2. The point \(D\) lies on \(l _ { 1 }\) where \(\lambda = 2\). The line \(l _ { 2 }\) passes through \(D\) and is parallel to \(A B\).
    1. Find a vector equation of line \(l _ { 2 }\) with parameter \(\mu\).
    2. The diagram shows a symmetrical trapezium \(A B C D\), with angle \(D A B\) equal to angle \(A B C\). \includegraphics[max width=\textwidth, alt={}, center]{5fe2527a-33da-4076-b3fa-4cab545336ec-9_620_675_1197_726} The point \(C\) lies on line \(l _ { 2 }\). The length of \(A D\) is equal to the length of \(B C\). Find the coordinates of \(C\).
AQA C4 2012 January Q1
11 marks Standard +0.3
1
  1. Express \(\frac { 2 x + 3 } { 4 x ^ { 2 } - 1 }\) in the form \(\frac { A } { 2 x - 1 } + \frac { B } { 2 x + 1 }\), where \(A\) and \(B\) are integers. (3 marks)
  2. Express \(\frac { 12 x ^ { 3 } - 7 x - 6 } { 4 x ^ { 2 } - 1 }\) in the form \(C x + \frac { D ( 2 x + 3 ) } { 4 x ^ { 2 } - 1 }\), where \(C\) and \(D\) are integers.
    (3 marks)
  3. Evaluate \(\int _ { 1 } ^ { 2 } \frac { 12 x ^ { 3 } - 7 x - 6 } { 4 x ^ { 2 } - 1 } \mathrm {~d} x\), giving your answer in the form \(p + \ln q\), where \(p\) and \(q\) are rational numbers.
    (5 marks)
AQA C4 2012 January Q2
6 marks Moderate -0.3
2 Angle \(\alpha\) is acute and \(\cos \alpha = \frac { 3 } { 5 }\). Angle \(\beta\) is obtuse and \(\sin \beta = \frac { 1 } { 2 }\).
    1. Find the value of \(\tan \alpha\) as a fraction.
      (1 mark)
    2. Find the value of \(\tan \beta\) in surd form.
  2. Hence show that \(\tan ( \alpha + \beta ) = \frac { m \sqrt { 3 } - n } { n \sqrt { 3 } + m }\), where \(m\) and \(n\) are integers.
    (3 marks)