Questions — AQA C2 (184 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure 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 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 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 Stats 1 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
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 Q4
4 A curve is defined for \(x > 0\). The gradient of the curve at the point \(( x , y )\) is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 } { x ^ { 2 } } - \frac { x } { 4 }$$
  1. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
  2. The curve has a stationary point \(M\) whose \(y\)-coordinate is \(\frac { 5 } { 2 }\).
    1. Find the \(x\)-coordinate of \(M\).
    2. Use your answers to parts (a) and (b)(i) to show that \(M\) is a maximum point.
    3. Find the equation of the curve.
AQA C2 2015 June Q5
5 The \(n\)th term of a sequence is \(u _ { n }\).
The sequence is defined by \(u _ { n + 1 } = p u _ { n } + q\), where \(p\) and \(q\) are constants.
The second term of the sequence is 160 . The third term of the sequence is 132 .
The limit of \(u _ { n }\) as \(n\) tends to infinity is 20 .
  1. Find the value of \(p\) and the value of \(q\).
  2. Hence find the value of the first term of the sequence.
AQA C2 2015 June Q6
6
  1. Solve the equation \(\sin ( x + 0.7 ) = 0.6\) in the interval \(- \pi < x < \pi\), giving your answers in radians to two significant figures.
  2. It is given that \(5 \cos ^ { 2 } \theta - \cos \theta = \sin ^ { 2 } \theta\).
    1. By forming and solving a suitable quadratic equation, find the possible values of \(\cos \theta\).
    2. Hence show that a possible value of \(\tan \theta\) is \(2 \sqrt { 2 }\).
AQA C2 2015 June Q7
3 marks
7 The diagram shows a sketch of two curves.
\includegraphics[max width=\textwidth, alt={}, center]{24641e66-b73b-4323-98c8-349727151aba-14_448_527_370_762} The equations of the two curves are \(y = 1 + \sqrt { x }\) and \(y = 4 ^ { \frac { x } { 9 } }\).
The curves meet at the points \(P ( 0,1 )\) and \(Q ( 9,4 )\).
    1. Describe the geometrical transformation that maps the graph of \(y = \sqrt { x }\) onto the graph of \(y = 1 + \sqrt { x }\).
    2. Describe the geometrical transformation that maps the graph of \(y = 4 ^ { x }\) onto the graph of \(y = 4 ^ { \frac { x } { 9 } }\).
    1. Given that \(\int _ { 0 } ^ { 9 } \sqrt { x } \mathrm {~d} x = 18\), find the value of \(\int _ { 0 } ^ { 9 } ( 1 + \sqrt { x } ) \mathrm { d } x\).
    2. Use the trapezium rule with five ordinates (four strips) to find an approximate value for \(\int _ { 0 } ^ { 9 } 4 ^ { \frac { x } { 9 } } \mathrm {~d} x\). Give your answer to one decimal place.
    3. Hence find an approximate value for the area of the shaded region bounded by the two curves and state, with an explanation, whether your approximation will be an overestimate or an underestimate of the true value for the area of the shaded region.
      [0pt] [3 marks]
AQA C2 2015 June Q8
5 marks
8 The point \(A\) lies on the curve with equation \(y = x ^ { \frac { 1 } { 2 } }\). The tangent to this curve at \(A\) is parallel to the line \(3 y - 2 x = 1\). Find an equation of this tangent at \(A\).
[0pt] [5 marks]
AQA C2 2015 June Q9
5 marks
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
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
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
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
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
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
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
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
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
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 C2 2011 January Q7
  1. Given that \(y = x + 3 + \frac { 8 } { x ^ { 4 } }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Find an equation of the tangent at the point on the curve \(C\) where \(x = 1\).
  3. The curve \(C\) has a minimum point \(M\). Find the coordinates of \(M\).
    1. Find \(\int \left( x + 3 + \frac { 8 } { x ^ { 4 } } \right) \mathrm { d } x\).
    2. Hence find the area of the region bounded by the curve \(C\), the \(x\)-axis and the lines \(x = 1\) and \(x = 2\).
  5. The curve \(C\) is translated by \(\left[ \begin{array} { l } 0
    k \end{array} \right]\) to give the curve \(y = \mathrm { f } ( x )\). Given that the \(x\)-axis is a tangent to the curve \(y = \mathrm { f } ( x )\), state the value of the constant \(k\).
    (1 mark)
AQA C2 2011 June Q6
  1. The area of the shaded region is given by \(\int _ { 0 } ^ { 2 } \sin x \mathrm {~d} x\), where \(x\) is in radians. Use the trapezium rule with five ordinates (four strips) to find an approximate value for the area of the shaded region, giving your answer to three significant figures.
  2. Describe the geometrical transformation that maps the graph of \(y = \sin x\) onto the graph of \(y = 2 \sin x\).
  3. Use a trigonometrical identity to solve the equation $$2 \sin x = \cos x$$ in the interval \(0 \leqslant x \leqslant 2 \pi\), giving your solutions in radians to three significant figures.
AQA C2 2007 January Q1
1 The diagram shows a sector \(O A B\) of a circle with centre \(O\). The radius of the circle is 6 cm and the angle \(A O B\) is 1.2 radians.
  1. Find the area of the sector \(O A B\).
  2. Find the perimeter of the sector \(O A B\).
AQA C2 2007 January Q2
2 Use the trapezium rule with four ordinates (three strips) to find an approximate value for $$\int _ { 0 } ^ { 3 } \sqrt { 2 ^ { x } } \mathrm {~d} x$$ giving your answer to three decimal places.
AQA C2 2007 January Q3
3
  1. Write down the values of \(p , q\) and \(r\) given that:
    1. \(64 = 8 ^ { p }\);
    2. \(\frac { 1 } { 64 } = 8 ^ { q }\);
    3. \(\sqrt { 8 } = 8 ^ { r }\).
  2. Find the value of \(x\) for which $$\frac { 8 ^ { x } } { \sqrt { 8 } } = \frac { 1 } { 64 }$$
AQA C2 2007 January Q4
4 The triangle \(A B C\), shown in the diagram, is such that \(B C = 6 \mathrm {~cm} , A C = 5 \mathrm {~cm}\) and \(A B = 4 \mathrm {~cm}\). The angle \(B A C\) is \(\theta\).
\includegraphics[max width=\textwidth, alt={}, center]{c16d94a6-52f2-4bf3-acee-0b227ae55a1a-3_442_652_452_678}
  1. Use the cosine rule to show that \(\cos \theta = \frac { 1 } { 8 }\).
  2. Hence use a trigonometrical identity to show that \(\sin \theta = \frac { 3 \sqrt { 7 } } { 8 }\).
  3. Hence find the area of the triangle \(A B C\).
AQA C2 2007 January Q5
5 The second term of a geometric series is 48 and the fourth term is 3 .
  1. Show that one possible value for the common ratio, \(r\), of the series is \(- \frac { 1 } { 4 }\) and state the other value.
  2. In the case when \(r = - \frac { 1 } { 4 }\), find:
    1. the first term;
    2. the sum to infinity of the series.
AQA C2 2007 January Q6
6 A curve \(C\) is defined for \(x > 0\) by the equation \(y = x + 1 + \frac { 4 } { x ^ { 2 } }\) and is sketched below.
\includegraphics[max width=\textwidth, alt={}, center]{c16d94a6-52f2-4bf3-acee-0b227ae55a1a-4_545_784_420_628}
    1. Given that \(y = x + 1 + \frac { 4 } { x ^ { 2 } }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
    2. The curve \(C\) has a minimum point \(M\). Find the coordinates of \(M\).
    3. Find an equation of the normal to \(C\) at the point ( 1,6 ).
    1. Find \(\int \left( x + 1 + \frac { 4 } { x ^ { 2 } } \right) \mathrm { d } x\).
    2. Hence find the area of the region bounded by the curve \(C\), the lines \(x = 1\) and \(x = 4\) and the \(x\)-axis.
AQA C2 2007 January Q7
7
  1. The first four terms of the binomial expansion of \(( 1 + 2 x ) ^ { 8 }\) in ascending powers of \(x\) are \(1 + a x + b x ^ { 2 } + c x ^ { 3 }\). Find the values of the integers \(a , b\) and \(c\).
  2. Hence find the coefficient of \(x ^ { 3 }\) in the expansion of \(\left( 1 + \frac { 1 } { 2 } x \right) ( 1 + 2 x ) ^ { 8 }\).
AQA C2 2007 January Q8
8
  1. Solve the equation \(\cos x = 0.3\) in the interval \(0 \leqslant x \leqslant 2 \pi\), giving your answers in radians to three significant figures.
  2. The diagram shows the graph of \(y = \cos x\) for \(0 \leqslant x \leqslant 2 \pi\) and the line \(y = k\).
    \includegraphics[max width=\textwidth, alt={}, center]{c16d94a6-52f2-4bf3-acee-0b227ae55a1a-5_524_805_559_648} The line \(y = k\) intersects the curve \(y = \cos x , 0 \leqslant x \leqslant 2 \pi\), at the points \(P\) and \(Q\). The point \(M\) is the minimum point of the curve.
    1. Write down the coordinates of the point \(M\).
    2. The \(x\)-coordinate of \(P\) is \(\alpha\). Write down the \(x\)-coordinate of \(Q\) in terms of \(\pi\) and \(\alpha\).
  3. Describe the geometrical transformation that maps the graph of \(y = \cos x\) onto the graph of \(y = \cos 2 x\).
  4. Solve the equation \(\cos 2 x = \cos \frac { 4 \pi } { 5 }\) in the interval \(0 \leqslant x \leqslant 2 \pi\), giving the values of \(x\) in terms of \(\pi\).
    (4 marks)