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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
AQA C2 2015 June Q9
5 marks Moderate -0.3
9
  1. Use logarithms to solve the equation \(2 ^ { 3 x } = 5\), giving your value of \(x\) to three significant figures.
  2. Given that \(\log _ { a } k - \log _ { a } 2 = \frac { 2 } { 3 }\), express \(a\) in terms of \(k\).
    1. By using the binomial expansion, or otherwise, express \(( 1 + 2 x ) ^ { 3 }\) in ascending powers of \(x\).
    2. It is given that $$\log _ { 2 } \left[ ( 1 + 2 n ) ^ { 3 } - 8 n \right] = \log _ { 2 } ( 1 + 2 n ) + \log _ { 2 } \left[ 4 \left( 1 + n ^ { 2 } \right) \right]$$ By forming and solving a suitable quadratic equation, find the possible values of \(n\). [5 marks] \includegraphics[max width=\textwidth, alt={}, center]{24641e66-b73b-4323-98c8-349727151aba-20_1581_1714_1126_153}
      \includegraphics[max width=\textwidth, alt={}, center]{24641e66-b73b-4323-98c8-349727151aba-24_2488_1728_219_141}
AQA C2 2016 June Q1
2 marks Moderate -0.8
1
  1. Find \(\int \left( \frac { 36 } { x ^ { 2 } } + a x \right) \mathrm { d } x\), where \(a\) is a constant.
  2. Hence, given that \(\int _ { 1 } ^ { 3 } \left( \frac { 36 } { x ^ { 2 } } + a x \right) \mathrm { d } x = 16\), find the value of the constant \(a\).
    [0pt] [2 marks]
AQA C2 2016 June Q2
1 marks Moderate -0.8
2
  1. Sketch the graph of \(y = ( 0.2 ) ^ { x }\), indicating the value of the intercept on the \(y\)-axis.
  2. Use logarithms to solve the equation \(( 0.2 ) ^ { x } = 4\), giving your answer to three significant figures.
  3. Describe the geometrical transformation that maps the graph of \(y = ( 0.2 ) ^ { x }\) onto the graph of \(y = 5 ^ { x }\).
    [0pt] [1 mark]
AQA C2 2016 June Q3
3 marks Standard +0.3
3 The diagram shows a curve with a maximum point \(M\).
\includegraphics[max width=\textwidth, alt={}, center]{e183578a-29a8-4112-b941-06c8894ed078-06_512_867_354_589} The curve is defined for \(x > 0\) by the equation $$y = 6 x ^ { \frac { 1 } { 2 } } - x - 3$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Hence find the \(y\)-coordinate of the maximum point \(M\).
  3. Find an equation of the normal to the curve at the point \(P ( 4,5 )\).
  4. It is given that the normal to the curve at \(P\), when translated by the vector \(\left[ \begin{array} { l } k \\ 0 \end{array} \right]\), passes through the point \(M\). Find the value of the constant \(k\).
    [0pt] [3 marks]
AQA C2 2016 June Q4
Moderate -0.3
4 An arithmetic series has first term \(a\) and common difference \(d\).
The sum of the first 21 terms is 168 .
  1. Show that \(a + 10 d = 8\).
  2. The sum of the second term and the third term is 50 . The \(n\)th term of the series is \(u _ { n }\).
    1. Find the value of \(u _ { 12 }\).
    2. Find the value of \(\sum _ { n = 4 } ^ { 21 } u _ { n }\).
AQA C2 2016 June Q5
Moderate -0.8
5
  1. Use the trapezium rule with four ordinates (three strips) to find an approximate value for \(\int _ { 2 } ^ { 11 } \sqrt { x ^ { 2 } + 9 } \mathrm {~d} x\). Give your answer to one decimal place.
  2. Describe the geometrical transformation that maps the graph of \(y = \sqrt { x ^ { 2 } + 9 }\) onto the graph of :
    1. \(y = 5 + \sqrt { x ^ { 2 } + 9 }\);
    2. \(y = 3 \sqrt { x ^ { 2 } + 1 }\).
AQA C2 2016 June Q6
6 marks Standard +0.3
6 The diagram shows a triangle \(A B C\). The lengths of \(A B , B C\) and \(A C\) are \(8 \mathrm {~cm} , 5 \mathrm {~cm}\) and 9 cm respectively.
Angle \(B A C\) is \(\theta\) radians.
  1. Show that \(\theta = 0.586\), correct to three significant figures.
  2. Find the area of triangle \(A B C\), giving your answer, in \(\mathrm { cm } ^ { 2 }\), to three significant figures.
  3. A circular sector, centre \(A\) and radius \(r \mathrm {~cm}\), is removed from triangle \(A B C\). The remaining shape is shown shaded in the diagram below.
    \includegraphics[max width=\textwidth, alt={}, center]{e183578a-29a8-4112-b941-06c8894ed078-14_467_677_1462_685} Given that the area of the sector removed is equal to the area of the shaded shape, find the perimeter of the shaded shape. Give your answer in cm to three significant figures.
    [0pt] [6 marks]
AQA C2 2016 June Q7
5 marks Standard +0.3
7
  1. The expression \(( 1 - 2 x ) ^ { 5 }\) can be written in the form $$1 + p x + q x ^ { 2 } + r x ^ { 3 } + 80 x ^ { 4 } - 32 x ^ { 5 }$$ By using the binomial expansion, or otherwise, find the values of the coefficients \(p , q\) and \(r\).
  2. Find the value of the coefficient of \(x ^ { 10 }\) in the expansion of \(( 1 - 2 x ) ^ { 5 } ( 2 + x ) ^ { 7 }\).
    [0pt] [5 marks]
AQA C2 2016 June Q8
4 marks Standard +0.3
8
    1. Given that \(4 \sin x + 5 \cos x = 0\), find the value of \(\tan x\).
    2. Hence solve the equation \(( 1 - \tan x ) ( 4 \sin x + 5 \cos x ) = 0\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\), giving your values of \(x\) to the nearest degree.
  2. By first showing that \(\frac { 16 + 9 \sin ^ { 2 } \theta } { 5 - 3 \cos \theta }\) can be expressed in the form \(p + q \cos \theta\), where \(p\) and \(q\) are integers, find the least possible value of \(\frac { 16 + 9 \sin ^ { 2 } \theta } { 5 - 3 \cos \theta }\). State the exact value of \(\theta\), in radians in the interval \(0 \leqslant \theta < 2 \pi\), at which this least value occurs.
    [0pt] [4 marks]
AQA C2 2016 June Q9
4 marks Standard +0.3
9
  1. Given that \(\log _ { 3 } c = m\) and \(\log _ { 27 } d = n\), express \(\frac { \sqrt { c } } { d ^ { 2 } }\) in the form \(3 ^ { y }\), where \(y\) is an expression in terms of \(m\) and \(n\).
  2. Show that the equation $$\log _ { 4 } ( 2 x + 3 ) + \log _ { 4 } ( 2 x + 15 ) = 1 + \log _ { 4 } ( 14 x + 5 )$$ has only one solution and state its value.
    [0pt] [4 marks]
AQA C3 2007 January Q1
Moderate -0.8
1 Use the mid-ordinate rule with four strips of equal width to find an estimate for \(\int _ { 1 } ^ { 5 } \frac { 1 } { 1 + \ln x } \mathrm {~d} x\), giving your answer to three significant figures.
(4 marks)
AQA C3 2007 January Q2
Standard +0.3
2 Describe a sequence of two geometrical transformations that maps the graph of \(y = \sec x\) onto the graph of \(y = 1 + \sec 3 x\).
AQA C3 2007 January Q3
Moderate -0.3
3 The functions \(f\) and \(g\) are defined with their respective domains by $$\begin{array} { l l } \mathrm { f } ( x ) = 3 - x ^ { 2 } , & \text { for all real values of } x \\ \mathrm {~g} ( x ) = \frac { 2 } { x + 1 } , & \text { for real values of } x , x \neq - 1 \end{array}$$
  1. Find the range of f.
  2. The inverse of g is \(\mathrm { g } ^ { - 1 }\).
    1. Find \(\mathrm { g } ^ { - 1 } ( x )\).
    2. State the range of \(\mathrm { g } ^ { - 1 }\).
  3. The composite function gf is denoted by h .
    1. Find \(\mathrm { h } ( x )\), simplifying your answer.
    2. State the greatest possible domain of h .
AQA C3 2007 January Q4
Moderate -0.3
4
  1. Use integration by parts to find \(\int x \sin x \mathrm {~d} x\).
  2. Using the substitution \(u = x ^ { 2 } + 5\), or otherwise, find \(\int x \sqrt { x ^ { 2 } + 5 } \mathrm {~d} x\).
  3. The diagram shows the curve \(y = x ^ { 2 } - 9\) for \(x \geqslant 0\).
    \includegraphics[max width=\textwidth, alt={}, center]{6890a681-2b7f-4853-a5f0-f88b7b435367-3_844_663_685_694} The shaded region \(R\) is bounded by the curve, the lines \(y = 1\) and \(y = 2\), and the \(y\)-axis. Find the exact value of the volume of the solid generated when the region \(R\) is rotated through \(360 ^ { \circ }\) about the \(\boldsymbol { y }\)-axis.
AQA C3 2007 January Q5
Moderate -0.3
5
    1. Show that the equation $$2 \cot ^ { 2 } x + 5 \operatorname { cosec } x = 10$$ can be written in the form \(2 \operatorname { cosec } ^ { 2 } x + 5 \operatorname { cosec } x - 12 = 0\).
    2. Hence show that \(\sin x = - \frac { 1 } { 4 }\) or \(\sin x = \frac { 2 } { 3 }\).
  2. Hence, or otherwise, solve the equation $$2 \cot ^ { 2 } ( \theta - 0.1 ) + 5 \operatorname { cosec } ( \theta - 0.1 ) = 10$$ giving all values of \(\theta\) in radians to two decimal places in the interval \(- \pi < \theta < \pi\).
    (3 marks)
AQA C3 2007 January Q6
Moderate -0.3
6
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when:
    1. \(y = \left( 4 x ^ { 2 } + 3 x + 2 \right) ^ { 10 }\);
    2. \(y = x ^ { 2 } \tan x\).
    1. Find \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) when \(x = 2 y ^ { 3 } + \ln y\).
    2. Hence find an equation of the tangent to the curve \(x = 2 y ^ { 3 } + \ln y\) at the point \(( 2,1 )\).
AQA C3 2007 January Q7
Moderate -0.3
7
  1. Sketch the graph of \(y = | 2 x |\).
  2. On a separate diagram, sketch the graph of \(y = 4 - | 2 x |\), indicating the coordinates of the points where the graph crosses the coordinate axes.
  3. Solve \(4 - | 2 x | = x\).
  4. Hence, or otherwise, solve the inequality \(4 - | 2 x | > x\).
AQA C3 2007 January Q8
Standard +0.3
8 The diagram shows the curve \(y = \cos ^ { - 1 } x\) for \(- 1 \leqslant x \leqslant 1\).
\includegraphics[max width=\textwidth, alt={}, center]{6890a681-2b7f-4853-a5f0-f88b7b435367-4_492_698_1640_671}
  1. Write down the exact coordinates of the points \(A\) and \(B\).
  2. The equation \(\cos ^ { - 1 } x = 3 x + 1\) has only one root. Given that the root of this equation is \(\alpha\), show that \(0.1 \leqslant \alpha \leqslant 0.2\).
  3. Use the iteration \(x _ { n + 1 } = \frac { 1 } { 3 } \left( \cos ^ { - 1 } x _ { n } - 1 \right)\) with \(x _ { 1 } = 0.1\) to find the values of \(x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\), giving your answers to three decimal places.
AQA C3 2007 January Q9
Standard +0.3
9 The sketch shows the graph of \(y = 4 - \mathrm { e } ^ { 2 x }\). The curve crosses the \(y\)-axis at the point \(A\) and the \(x\)-axis at the point \(B\).
\includegraphics[max width=\textwidth, alt={}, center]{6890a681-2b7f-4853-a5f0-f88b7b435367-5_711_921_466_557}
    1. Find \(\int \left( 4 - \mathrm { e } ^ { 2 x } \right) \mathrm { d } x\).
      (2 marks)
    2. Hence show that \(\int _ { 0 } ^ { \ln 2 } \left( 4 - \mathrm { e } ^ { 2 x } \right) \mathrm { d } x = 4 \ln 2 - \frac { 3 } { 2 }\).
    1. Write down the \(y\)-coordinate of \(A\).
    2. Show that \(x = \ln 2\) at \(B\).
  3. Find the equation of the normal to the curve \(y = 4 - \mathrm { e } ^ { 2 x }\) at the point \(B\).
  4. Find the area of the region enclosed by the curve \(y = 4 - \mathrm { e } ^ { 2 x }\), the normal to the curve at \(B\) and the \(y\)-axis.
AQA C3 2008 January Q1
Moderate -0.8
1
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when:
    1. \(y = \left( 2 x ^ { 2 } - 5 x + 1 \right) ^ { 20 }\);
    2. \(y = x \cos x\).
  2. Given that $$y = \frac { x ^ { 3 } } { x - 2 }$$ show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { k x ^ { 2 } ( x - 3 ) } { ( x - 2 ) ^ { 2 } }$$ where \(k\) is a positive integer.
AQA C3 2008 January Q2
Moderate -0.3
2
  1. Solve the equation \(\cot x = 2\), giving all values of \(x\) in the interval \(0 \leqslant x \leqslant 2 \pi\) in radians to two decimal places.
  2. Show that the equation \(\operatorname { cosec } ^ { 2 } x = \frac { 3 \cot x + 4 } { 2 }\) can be written as $$2 \cot ^ { 2 } x - 3 \cot x - 2 = 0$$
  3. Solve the equation \(\operatorname { cosec } ^ { 2 } x = \frac { 3 \cot x + 4 } { 2 }\), giving all values of \(x\) in the interval \(0 \leqslant x \leqslant 2 \pi\) in radians to two decimal places.
AQA C3 2008 January Q3
Standard +0.3
3 The equation $$x + ( 1 + 3 x ) ^ { \frac { 1 } { 4 } } = 0$$ has a single root, \(\alpha\).
  1. Show that \(\alpha\) lies between - 0.33 and - 0.32 .
  2. Show that the equation \(x + ( 1 + 3 x ) ^ { \frac { 1 } { 4 } } = 0\) can be rearranged into the form $$x = \frac { 1 } { 3 } \left( x ^ { 4 } - 1 \right)$$
  3. Use the iteration \(x _ { n + 1 } = \frac { \left( x _ { n } ^ { 4 } - 1 \right) } { 3 }\) with \(x _ { 1 } = - 0.3\) to find \(x _ { 4 }\), giving your answer to three significant figures.
AQA C3 2008 January Q4
Moderate -0.8
4 The functions f and g are defined with their respective domains by $$\begin{array} { l l } \mathrm { f } ( x ) = x ^ { 3 } , & \text { for all real values of } x \\ \mathrm {~g} ( x ) = \frac { 1 } { x - 3 } , & \text { for real values of } x , x \neq 3 \end{array}$$
  1. State the range of f.
    1. Find fg(x).
    2. Solve the equation \(\operatorname { fg } ( x ) = 64\).
    1. The inverse of g is \(\mathrm { g } ^ { - 1 }\). Find \(\mathrm { g } ^ { - 1 } ( x )\).
    2. State the range of \(\mathrm { g } ^ { - 1 }\).
AQA C3 2008 January Q5
Standard +0.3
5
    1. Given that \(y = 2 x ^ { 2 } - 8 x + 3\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
    2. Hence, or otherwise, find $$\int _ { 4 } ^ { 6 } \frac { x - 2 } { 2 x ^ { 2 } - 8 x + 3 } d x$$ giving your answer in the form \(k \ln 3\), where \(k\) is a rational number.
  2. Use the substitution \(u = 3 x - 1\) to find \(\int x \sqrt { 3 x - 1 } \mathrm {~d} x\), giving your answer in terms of \(x\).
AQA C3 2008 January Q6
Moderate -0.8
6
  1. Sketch the curve with equation \(y = \operatorname { cosec } x\) for \(0 < x < \pi\).
  2. Use the mid-ordinate rule with four strips to find an estimate for \(\int _ { 0.1 } ^ { 0.5 } \operatorname { cosec } x \mathrm {~d} x\), giving your answer to three significant figures.