Questions — AQA C2 (184 questions)

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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 2011 January Q6
6 A geometric series has third term 36 and sixth term 972.
    1. Show that the common ratio of the series is 3 .
    2. Find the first term of the series.
  2. The \(n\)th term of the series is \(u _ { n }\).
    1. Show that \(\sum _ { n = 1 } ^ { 20 } u _ { n } = 2 \left( 3 ^ { 20 } - 1 \right)\).
    2. Find the least value of \(n\) such that \(u _ { n } > 4 \times 10 ^ { 15 }\).
      \(7 \quad\) A curve \(C\) is defined for \(x > 0\) by the equation \(y = x + 3 + \frac { 8 } { x ^ { 4 } }\) and is sketched below.
      \includegraphics[max width=\textwidth, alt={}, center]{1c06ba04-575c-4eb8-b4aa-0a7510838cd2-08_602_799_447_632}
AQA C2 2011 January Q8
8
  1. Given that \(2 \log _ { k } x - \log _ { k } 5 = 1\), express \(k\) in terms of \(x\). Give your answer in a form not involving logarithms.
  2. Given that \(\log _ { a } y = \frac { 3 } { 2 }\) and that \(\log _ { 4 } a = b + 2\), show that \(y = 2 ^ { p }\), where \(p\) is an expression in terms of \(b\).
    \includegraphics[max width=\textwidth, alt={}]{1c06ba04-575c-4eb8-b4aa-0a7510838cd2-09_2102_1717_605_150}
AQA C2 2011 January Q9
9
  1. Solve the equation \(\tan x = - 3\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\), giving your answers to the nearest degree.
    1. Given that $$7 \sin ^ { 2 } \theta + \sin \theta \cos \theta = 6$$ show that $$\tan ^ { 2 } \theta + \tan \theta - 6 = 0$$
    2. Hence solve the equation \(7 \sin ^ { 2 } \theta + \sin \theta \cos \theta = 6\) in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\), giving your answers to the nearest degree.
      (4 marks)
AQA C2 2012 January Q1
1 The diagram shows a sector \(O A B\) of a circle with centre \(O\) and radius 6 cm .
\includegraphics[max width=\textwidth, alt={}, center]{02e5dfac-18d7-480d-ac23-dfd2ca348cba-2_358_332_358_829} The angle between the radii \(O A\) and \(O B\) is \(\theta\) radians.
The area of the sector \(O A B\) is \(21.6 \mathrm {~cm} ^ { 2 }\).
  1. Find the value of \(\theta\).
  2. Find the length of the \(\operatorname { arc } A B\).
AQA C2 2012 January Q2
2
  1. Use the trapezium rule with five ordinates (four strips) to find an approximate value for $$\int _ { 0 } ^ { 4 } \frac { 2 ^ { x } } { x + 1 } \mathrm {~d} x$$ giving your answer to three significant figures.
  2. State how you could obtain a better approximation to the value of the integral using the trapezium rule.
AQA C2 2012 January Q3
3
  1. Write \(\sqrt [ 4 ] { x ^ { 3 } }\) in the form \(x ^ { k }\).
  2. Write \(\frac { 1 - x ^ { 2 } } { \sqrt [ 4 ] { x ^ { 3 } } }\) in the form \(x ^ { p } - x ^ { q }\).
AQA C2 2012 January Q4
4 The triangle \(A B C\), shown in the diagram, is such that \(A B\) is 10 metres and angle \(B A C\) is \(150 ^ { \circ }\).
\includegraphics[max width=\textwidth, alt={}, center]{02e5dfac-18d7-480d-ac23-dfd2ca348cba-3_323_746_406_648} The area of triangle \(A B C\) is \(40 \mathrm {~m} ^ { 2 }\).
  1. Show that the length of \(A C\) is 16 metres.
  2. Calculate the length of \(B C\), giving your answer, in metres, to two decimal places.
  3. Calculate the smallest angle of triangle \(A B C\), giving your answer to the nearest \(0.1 ^ { \circ }\).
AQA C2 2012 January Q5
5
    1. Describe the geometrical transformation that maps the graph of \(y = \left( 1 + \frac { x } { 3 } \right) ^ { 6 }\) onto the graph of \(y = ( 1 + 2 x ) ^ { 6 }\).
    2. The curve \(y = \left( 1 + \frac { x } { 3 } \right) ^ { 6 }\) is translated by the vector \(\left[ \begin{array} { l } 3
      0 \end{array} \right]\) to give the curve \(y = \mathrm { g } ( x )\). Find an expression for \(\mathrm { g } ( x )\), simplifying your answer.
  2. The first four terms in the binomial expansion of \(\left( 1 + \frac { x } { 3 } \right) ^ { 6 }\) are \(1 + a x + b x ^ { 2 } + c x ^ { 3 }\). Find the values of the constants \(a , b\) and \(c\), giving your answers in their simplest form.
AQA C2 2012 January Q6
6 An arithmetic series has first term \(a\) and common difference \(d\). The sum of the first 25 terms of the series is 3500 .
  1. Show that \(a + 12 d = 140\).
  2. The fifth term of this series is 100 . Find the value of \(d\) and the value of \(a\).
  3. The \(n\)th term of this series is \(u _ { n }\). Given that $$33 \left( \sum _ { n = 1 } ^ { 25 } u _ { n } - \sum _ { n = 1 } ^ { k } u _ { n } \right) = 67 \sum _ { n = 1 } ^ { k } u _ { n }$$ find the value of \(\sum _ { n = 1 } ^ { k } u _ { n }\).
    (3 marks)
AQA C2 2012 January Q7
7
  1. Sketch the graph of \(y = \frac { 1 } { 2 ^ { x } }\), indicating the value of the intercept on the \(y\)-axis.
  2. Use logarithms to solve the equation \(\frac { 1 } { 2 ^ { x } } = \frac { 5 } { 4 }\), giving your answer to three significant figures.
  3. Given that $$\log _ { a } \left( b ^ { 2 } \right) + 3 \log _ { a } y = 3 + 2 \log _ { a } \left( \frac { y } { a } \right)$$ express \(y\) in terms of \(a\) and \(b\).
    Give your answer in a form not involving logarithms.
AQA C2 2012 January Q8
8
  1. Given that \(2 \sin \theta = 7 \cos \theta\), find the value of \(\tan \theta\).
    1. Use an appropriate identity to show that the equation $$6 \sin ^ { 2 } x = 4 + \cos x$$ can be written as $$6 \cos ^ { 2 } x + \cos x - 2 = 0$$
    2. Hence solve the equation \(6 \sin ^ { 2 } x = 4 + \cos x\) in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\), giving your answers to the nearest degree.
AQA C2 2012 January Q9
9 The diagram shows part of a curve crossing the \(x\)-axis at the origin \(O\) and at the point \(A ( 8,0 )\). Tangents to the curve at \(O\) and \(A\) meet at the point \(P\), as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{02e5dfac-18d7-480d-ac23-dfd2ca348cba-5_547_536_497_760} The curve has equation $$y = 12 x - 3 x ^ { \frac { 5 } { 3 } }$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
    1. Find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at the point \(O\) and hence write down an equation of the tangent at \(O\).
    2. Show that the equation of the tangent at \(A ( 8,0 )\) is \(y + 8 x = 64\).
  3. Find \(\int \left( 12 x - 3 x ^ { \frac { 5 } { 3 } } \right) \mathrm { d } x\).
  4. Calculate the area of the shaded region bounded by the curve from \(O\) to \(A\) and the tangents \(O P\) and \(A P\).
AQA C2 2013 January Q1
1 The diagram shows a sector \(O A B\) of a circle with centre \(O\) and radius \(r \mathrm {~cm}\).
\includegraphics[max width=\textwidth, alt={}, center]{bfe96138-9587-4efb-95c5-84c4d5eadfbe-2_382_351_379_826} The angle \(A O B\) is 1.25 radians. The perimeter of the sector is 39 cm .
  1. Show that \(r = 12\).
  2. Calculate the area of the sector \(O A B\).
AQA C2 2013 January Q2
2
  1. Use the trapezium rule with five ordinates (four strips) to find an approximate value for $$\int _ { 1 } ^ { 5 } \frac { 1 } { x ^ { 2 } + 1 } \mathrm {~d} x$$ giving your answer to three significant figures.
    1. Find \(\int \left( x ^ { - \frac { 3 } { 2 } } + 6 x ^ { \frac { 1 } { 2 } } \right) \mathrm { d } x\), giving the coefficient of each term in its simplest form.
    2. Hence find the value of \(\int _ { 1 } ^ { 4 } \left( x ^ { - \frac { 3 } { 2 } } + 6 x ^ { \frac { 1 } { 2 } } \right) \mathrm { d } x\).
AQA C2 2013 January Q3
3 The diagram shows a triangle \(A B C\).
\includegraphics[max width=\textwidth, alt={}, center]{bfe96138-9587-4efb-95c5-84c4d5eadfbe-3_273_622_356_708} The lengths of \(A C\) and \(B C\) are 5 cm and 6 cm respectively.
The area of triangle \(A B C\) is \(12.5 \mathrm {~cm} ^ { 2 }\), and angle \(A C B\) is obtuse.
  1. Find the size of angle \(A C B\), giving your answer to the nearest \(0.1 ^ { \circ }\).
  2. Find the length of \(A B\), giving your answer to two significant figures.
AQA C2 2013 January Q4
4 Given that $$\log _ { a } N - \log _ { a } x = \frac { 3 } { 2 }$$ express \(x\) in terms of \(a\) and \(N\), giving your answer in a form not involving logarithms.
(3 marks)
AQA C2 2013 January Q5
5 The point \(P ( 2,8 )\) lies on a curve, and the point \(M\) is the only stationary point of the curve. The curve has equation \(y = 6 + 2 x - \frac { 8 } { x ^ { 2 } }\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Show that the normal to the curve at the point \(P ( 2,8 )\) has equation \(x + 4 y = 34\).
    1. Show that the stationary point \(M\) lies on the \(x\)-axis.
    2. Hence write down the equation of the tangent to the curve at \(M\).
  4. The tangent to the curve at \(M\) and the normal to the curve at \(P\) intersect at the point \(T\). Find the coordinates of \(T\).
AQA C2 2013 January Q6
6
  1. A geometric series begins \(420 + 294 + 205.8 + \ldots\).
    1. Show that the common ratio of the series is 0.7 .
    2. Find the sum to infinity of the series.
    3. Write the \(n\)th term of the series in the form \(p \times q ^ { n }\), where \(p\) and \(q\) are constants.
  2. The first term of an arithmetic series is 240 and the common difference of the series is - 8 . The \(n\)th term of the series is \(u _ { n }\).
    1. Write down an expression for \(u _ { n }\).
    2. Given that \(u _ { k } = 0\), find the value of \(\sum _ { n = 1 } ^ { k } u _ { n }\).
AQA C2 2013 January Q7
7
  1. Describe a geometrical transformation that maps the graph of \(y = 4 ^ { x }\) onto the graph of \(y = 3 \times 4 ^ { x }\).
  2. Sketch the curve with equation \(y = 3 \times 4 ^ { x }\), indicating the value of the intercept on the \(y\)-axis.
  3. The curve with equation \(y = 4 ^ { - x }\) intersects the curve \(y = 3 \times 4 ^ { x }\) at the point \(P\). Use logarithms to find the \(x\)-coordinate of \(P\), giving your answer to three significant figures.
AQA C2 2013 January Q8
8
  1. Expand \(\left( 1 + \frac { 4 } { x } \right) ^ { 2 }\).
  2. The first four terms of the binomial expansion of \(\left( 1 + \frac { x } { 4 } \right) ^ { 8 }\) in ascending powers of \(x\) are \(1 + a x + b x ^ { 2 } + c x ^ { 3 }\). Find the values of the constants \(a , b\) and \(c\).
  3. Hence find the coefficient of \(x\) in the expansion of \(\left( 1 + \frac { 4 } { x } \right) ^ { 2 } \left( 1 + \frac { x } { 4 } \right) ^ { 8 }\).
AQA C2 2013 January Q9
9
  1. Write down the two solutions of the equation \(\tan \left( x + 30 ^ { \circ } \right) = \tan 79 ^ { \circ }\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
    (2 marks)
  2. Describe a single geometrical transformation that maps the graph of \(y = \tan x\) onto the graph of \(y = \tan \left( x + 30 ^ { \circ } \right)\).
    1. Given that \(5 + \sin ^ { 2 } \theta = ( 5 + 3 \cos \theta ) \cos \theta\), show that \(\cos \theta = \frac { 3 } { 4 }\).
    2. Hence solve the equation \(5 + \sin ^ { 2 } 2 x = ( 5 + 3 \cos 2 x ) \cos 2 x\) in the interval \(0 < x < 2 \pi\), giving your values of \(x\) in radians to three significant figures.
AQA C2 2005 June Q1
1 The diagram shows a triangle \(A B C\).
\includegraphics[max width=\textwidth, alt={}, center]{37627fc4-a90b-4f3b-9b10-0a9e200f8485-2_423_707_612_657} The lengths of \(A C\) and \(B C\) are 5 cm and 4.8 cm respectively.
The size of the angle \(B C A\) is \(30 ^ { \circ }\).
  1. Calculate the area of the triangle \(A B C\).
  2. Calculate the length of \(A B\), giving your answer to three significant figures.
AQA C2 2005 June Q2
2 The diagram shows a sector \(O A B\) of a circle with centre \(O\) and radius \(r \mathrm {~cm}\).
\includegraphics[max width=\textwidth, alt={}, center]{37627fc4-a90b-4f3b-9b10-0a9e200f8485-2_486_381_1686_739} The angle \(A O B\) is 1.5 radians. The perimeter of the sector is 56 cm .
  1. Show that \(r = 16\).
  2. Find the area of the sector.
AQA C2 2005 June Q3
3 The \(n\)th term of an arithmetic sequence is \(u _ { n }\), where $$u _ { n } = 90 - 3 n$$
  1. Find the value of \(u _ { 1 }\) and the value of \(u _ { 2 }\).
  2. Write down the common difference of the arithmetic sequence.
  3. Given that \(\sum _ { n = 1 } ^ { k } u _ { n } = 0\), find the value of \(k\).
AQA C2 2005 June Q4
4 The diagram shows a curve \(C\) with equation \(y = \sqrt { x }\). The point \(O\) is the origin \(( 0,0 )\).
\includegraphics[max width=\textwidth, alt={}, center]{37627fc4-a90b-4f3b-9b10-0a9e200f8485-3_488_1136_1009_443} The region bounded by the curve \(C\), the \(x\)-axis and the vertical lines \(x = 1\) and \(x = 4\) is shown shaded in the diagram.
    1. Write \(\sqrt { x }\) in the form \(x ^ { p }\), where \(p\) is a constant.
    2. Find \(\int \sqrt { x } \mathrm {~d} x\).
    3. Hence find the area of the shaded region.
  2. The point on \(C\) for which \(x = 4\) is \(P\). The tangent to \(C\) at the point \(P\) intersects the \(x\)-axis and the \(y\)-axis at the points \(A\) and \(B\) respectively.
    1. Find an equation for the tangent to the curve \(C\) at the point \(P\).
    2. Find the area of the triangle \(A O B\).
  3. Describe the single geometrical transformation by which the curve with equation \(y = \sqrt { x - 1 }\) can be obtained from the curve \(C\).
  4. Use the trapezium rule with four ordinates (three strips) to find an approximation for \(\int _ { 1 } ^ { 4 } \sqrt { x - 1 } \mathrm {~d} x\), giving your answer to three significant figures.