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 Q4
4 The triangle \(A B C\), shown in the diagram, is such that \(A C = 8 \mathrm {~cm} , C B = 12 \mathrm {~cm}\) and angle \(A C B = \theta\) radians. The area of triangle \(A B C = 20 \mathrm {~cm} ^ { 2 }\).
  1. Show that \(\theta = 0.430\) correct to three significant figures.
  2. Use the cosine rule to calculate the length of \(A B\), giving your answer to two significant figures.
  3. The point \(D\) lies on \(C B\) such that \(A D\) is an arc of a circle centre \(C\) and radius 8 cm . The region bounded by the arc \(A D\) and the straight lines \(D B\) and \(A B\) is shaded in the diagram.
    \includegraphics[max width=\textwidth, alt={}, center]{48c5470e-6489-4b25-98a6-1b4e101ab01c-004_417_883_1436_557} Calculate, to two significant figures:
    1. the length of the \(\operatorname { arc } A D\);
    2. the area of the shaded region.
AQA C2 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 first three terms of the sequence are given by $$u _ { 1 } = 200 \quad u _ { 2 } = 150 \quad u _ { 3 } = 120$$
  1. Show that \(p = 0.6\) and find the value of \(q\).
  2. Find the value of \(u _ { 4 }\).
  3. The limit of \(u _ { n }\) as \(n\) tends to infinity is \(L\). Write down an equation for \(L\) and hence find the value of \(L\).
AQA C2 Q6
6
  1. Describe the geometrical transformation that maps the curve with equation \(y = \sin x\) onto the curve with equation:
    1. \(y = 2 \sin x\);
    2. \(y = - \sin x\);
    3. \(\quad y = \sin \left( x - 30 ^ { \circ } \right)\).
  2. Solve the equation \(\sin \left( \theta - 30 ^ { \circ } \right) = 0.7\), giving your answers to the nearest \(0.1 ^ { \circ }\) in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
  3. Prove that \(( \cos x + \sin x ) ^ { 2 } + ( \cos x - \sin x ) ^ { 2 } = 2\).
AQA C2 Q8
8 A curve, drawn from the origin \(O\), crosses the \(x\)-axis at the point \(A ( 9,0 )\). Tangents to the curve at \(O\) and \(A\) meet at the point \(P\), as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{48c5470e-6489-4b25-98a6-1b4e101ab01c-006_763_879_466_577} The curve, defined for \(x \geqslant 0\), has equation $$y = x ^ { \frac { 3 } { 2 } } - 3 x$$
  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 \(A ( 9,0 )\) is \(2 y = 3 x - 27\).
    3. Hence find the coordinates of the point \(P\) where the two tangents meet.
  3. Find \(\int \left( x ^ { \frac { 3 } { 2 } } - 3 x \right) \mathrm { d } x\).
  4. Calculate the area of the shaded region bounded by the curve and the tangents \(O P\) and \(A P\).
AQA C2 2005 January Q1
1 A curve is defined for \(x > 0\) by the equation \(y = x + \frac { 2 } { x }\).
    1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
    2. Hence show that the gradient of the curve at the point \(P\) where \(x = 2\) is \(\frac { 1 } { 2 }\).
  2. Find an equation of the normal to the curve at this point \(P\).
AQA C2 2005 January Q2
2 The diagram shows a triangle \(A B C\) and the arc \(A B\) of a circle whose centre is \(C\) and whose radius is 24 cm .
\includegraphics[max width=\textwidth, alt={}, center]{4a4d4dcd-4137-427d-834f-ac2fe83f8aeb-2_506_403_1187_781} The length of the side \(A B\) of the triangle is 32 cm . The size of the angle \(A C B\) is \(\theta\) radians.
  1. Show that \(\theta = 1.46\) correct to three significant figures.
  2. Calculate the length of the \(\operatorname { arc } A B\) to the nearest cm .
    1. Calculate the area of the sector \(A B C\) to the nearest \(\mathrm { cm } ^ { 2 }\).
    2. Hence calculate the area of the shaded segment to the nearest \(\mathrm { cm } ^ { 2 }\).
AQA C2 2005 January Q3
3 An arithmetic series has fifth term 46 and twentieth term 181.
    1. Show that the common difference is 9 .
    2. Find the first term.
  2. Find the sum of the first 20 terms of the series.
  3. The \(n\)th term of the series is \(u _ { n }\). Given that the sum of the first 50 terms of the series is 11525 , find the value of $$\sum _ { n = 21 } ^ { 50 } u _ { n }$$
AQA C2 2005 January Q4
4
  1. Write \(\sqrt { x }\) in the form \(x ^ { k }\), where \(k\) is a fraction.
  2. Hence express \(\sqrt { x } ( x - 1 )\) in the form \(x ^ { p } - x ^ { q }\).
  3. Find \(\int \sqrt { x } ( x - 1 ) \mathrm { d } x\).
  4. Hence show that \(\int _ { 1 } ^ { 2 } \sqrt { x } ( x - 1 ) \mathrm { d } x = \frac { 4 } { 15 } ( \sqrt { 2 } + 1 )\).
AQA C2 2005 January Q5
5
  1. Given that $$\log _ { a } x = 3 \log _ { a } 6 - \log _ { a } 8$$ where \(a\) is a positive constant, show that \(x = 27\).
  2. Write down the value of:
    1. \(\quad \log _ { 4 } 1\);
    2. \(\log _ { 4 } 4\);
    3. \(\log _ { 4 } 2\);
    4. \(\quad \log _ { 4 } 8\).
AQA C2 2005 January Q6
6
    1. Using the binomial expansion, or otherwise, express \(( 2 + x ) ^ { 3 }\) in the form \(8 + a x + b x ^ { 2 } + x ^ { 3 }\), where \(a\) and \(b\) are integers. (3 marks)
    2. Write down the expansion of \(( 2 - x ) ^ { 3 }\).
  2. Hence show that \(( 2 + x ) ^ { 3 } - ( 2 - x ) ^ { 3 } = 24 x + 2 x ^ { 3 }\).
  3. Hence show that the curve with equation $$y = ( 2 + x ) ^ { 3 } - ( 2 - x ) ^ { 3 }$$ has no stationary points.
AQA C2 2005 January Q7
7 The diagram shows the graph of \(y = \cos 2 x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
\includegraphics[max width=\textwidth, alt={}, center]{4a4d4dcd-4137-427d-834f-ac2fe83f8aeb-4_518_906_1098_552}
  1. Write down the coordinates of the points \(A , B\) and \(C\) marked on the diagram.
  2. Describe the single geometrical transformation by which the curve with equation \(y = \cos 2 x\) can be obtained from the curve with equation \(y = \cos x\).
  3. Solve the equation $$\cos 2 x = 0.37$$ giving all solutions to the nearest \(0.1 ^ { \circ }\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\). (No credit will be given for simply reading values from a graph.)
    (5 marks)
AQA C2 2005 January Q8
8 The diagram shows a sketch of the curve with equation \(y = 3 ^ { x } + 1\).
\includegraphics[max width=\textwidth, alt={}, center]{4a4d4dcd-4137-427d-834f-ac2fe83f8aeb-5_535_1011_411_513} The curve intersects the \(y\)-axis at the point \(A\).
  1. Write down the \(y\)-coordinate of point \(A\).
    1. Use the trapezium rule with five ordinates (four strips) to find an approximation for \(\int _ { 0 } ^ { 1 } \left( 3 ^ { x } + 1 \right) \mathrm { d } x\), giving your answer to three significant figures.
      (4 marks)
    2. By considering the graph of \(y = 3 ^ { x } + 1\), explain with the aid of a diagram whether your approximation will be an overestimate or an underestimate of the true value of \(\int _ { 0 } ^ { 1 } \left( 3 ^ { x } + 1 \right) \mathrm { d } x\).
      (2 marks)
  3. The line \(y = 5\) intersects the curve \(y = 3 ^ { x } + 1\) at the point \(P\). By solving a suitable equation, find the \(x\)-coordinate of the point \(P\). Give your answer to four decimal places.
    (4 marks)
  4. The curve \(y = 3 ^ { x } + 1\) is reflected in the \(y\)-axis to give the curve with equation \(y = \mathrm { f } ( x )\). Write down an expression for \(\mathrm { f } ( x )\).
    (1 mark)
AQA C2 2006 January Q1
1 Given that \(y = 16 x + x ^ { - 1 }\), find the two values of \(x\) for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\).
(5 marks)
AQA C2 2006 January Q2
2
  1. Use the trapezium rule with five ordinates (four strips) to find an approximate value for $$\int _ { 0 } ^ { 4 } \frac { 1 } { x ^ { 2 } + 1 } \mathrm {~d} x$$ giving your answer to four significant figures.
  2. State how you could obtain a better approximation to the value of the integral using the trapezium rule.
AQA C2 2006 January Q3
3
  1. Use logarithms to solve the equation \(0.8 ^ { x } = 0.05\), giving your answer to three decimal places.
  2. An infinite geometric series has common ratio \(r\). The sum to infinity of the series is five times the first term of the series.
    1. Show that \(r = 0.8\).
    2. Given that the first term of the series is 20 , find the least value of \(n\) such that the \(n\)th term of the series is less than 1 .
AQA C2 2006 January Q4
4 The triangle \(A B C\), shown in the diagram, is such that \(A C = 8 \mathrm {~cm} , C B = 12 \mathrm {~cm}\) and angle \(A C B = \theta\) radians. The area of triangle \(A B C = 20 \mathrm {~cm} ^ { 2 }\).
  1. Show that \(\theta = 0.430\) correct to three significant figures.
  2. Use the cosine rule to calculate the length of \(A B\), giving your answer to two significant figures.
  3. The point \(D\) lies on \(C B\) such that \(A D\) is an arc of a circle centre \(C\) and radius 8 cm . The region bounded by the arc \(A D\) and the straight lines \(D B\) and \(A B\) is shaded in the diagram.
    \includegraphics[max width=\textwidth, alt={}, center]{9fee4b6f-06e2-4ed8-8835-33ef33b98c94-3_424_894_1434_555} Calculate, to two significant figures:
    1. the length of the \(\operatorname { arc } A D\);
    2. the area of the shaded region.
AQA C2 2006 January 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 first three terms of the sequence are given by $$u _ { 1 } = 200 \quad u _ { 2 } = 150 \quad u _ { 3 } = 120$$
  1. Show that \(p = 0.6\) and find the value of \(q\).
  2. Find the value of \(u _ { 4 }\).
  3. The limit of \(u _ { n }\) as \(n\) tends to infinity is \(L\). Write down an equation for \(L\) and hence find the value of \(L\).
AQA C2 2006 January Q6
6
  1. Describe the geometrical transformation that maps the curve with equation \(y = \sin x\) onto the curve with equation:
    1. \(y = 2 \sin x\);
    2. \(y = - \sin x\);
    3. \(y = \sin \left( x - 30 ^ { \circ } \right)\).
  2. Solve the equation \(\sin \left( \theta - 30 ^ { \circ } \right) = 0.7\), giving your answers to the nearest \(0.1 ^ { \circ }\) in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
  3. Prove that \(( \cos x + \sin x ) ^ { 2 } + ( \cos x - \sin x ) ^ { 2 } = 2\).
AQA C2 2006 January Q7
7 It is given that \(n\) satisfies the equation $$2 \log _ { a } n - \log _ { a } ( 5 n - 24 ) = \log _ { a } 4$$
  1. Show that \(n ^ { 2 } - 20 n + 96 = 0\).
  2. Hence find the possible values of \(n\).
AQA C2 2006 January Q8
8 A curve, drawn from the origin \(O\), crosses the \(x\)-axis at the point \(A ( 9,0 )\). Tangents to the curve at \(O\) and \(A\) meet at the point \(P\), as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{9fee4b6f-06e2-4ed8-8835-33ef33b98c94-5_778_901_461_571} The curve, defined for \(x \geqslant 0\), has equation $$y = x ^ { \frac { 3 } { 2 } } - 3 x$$
  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 \(A ( 9,0 )\) is \(2 y = 3 x - 27\).
    3. Hence find the coordinates of the point \(P\) where the two tangents meet.
  3. Find \(\int \left( x ^ { \frac { 3 } { 2 } } - 3 x \right) \mathrm { d } x\).
  4. Calculate the area of the shaded region bounded by the curve and the tangents \(O P\) and \(A P\).
AQA C2 2008 January Q1
1 The diagrams show a rectangle of length 6 cm and width 3 cm , and a sector of a circle of radius 6 cm and angle \(\theta\) radians.
\includegraphics[max width=\textwidth, alt={}, center]{14c2acbb-5f3e-40e2-8b88-162341ab9f71-2_266_1128_589_424} The area of the rectangle is twice the area of the sector.
  1. Show that \(\theta = 0.5\).
  2. Find the perimeter of the sector.
AQA C2 2008 January Q2
2 The arithmetic series $$51 + 58 + 65 + 72 + \ldots + 1444$$ has 200 terms.
  1. Write down the common difference of the series.
  2. Find the 101st term of the series.
  3. Find the sum of the last 100 terms of the series.
AQA C2 2008 January Q3
3 The diagram shows a triangle \(A B C\). The length of \(A C\) is 18.7 cm , and the sizes of angles \(B A C\) and \(A B C\) are \(72 ^ { \circ }\) and \(50 ^ { \circ }\) respectively.
  1. Show that the length of \(B C = 23.2 \mathrm {~cm}\), correct to the nearest 0.1 cm .
  2. Calculate the area of triangle \(A B C\), giving your answer to the nearest \(\mathrm { cm } ^ { 2 }\).
AQA C2 2008 January Q4
4 Use the trapezium rule with four ordinates (three strips) to find an approximate value for $$\int _ { 0 } ^ { 3 } \sqrt { x ^ { 2 } + 3 } d x$$ giving your answer to three decimal places.
AQA C2 2008 January Q5
5 A curve, drawn from the origin \(O\), crosses the \(x\)-axis at the point \(P ( 4,0 )\).
The normal to the curve at \(P\) meets the \(y\)-axis at the point \(Q\), as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{14c2acbb-5f3e-40e2-8b88-162341ab9f71-3_526_629_916_813} The curve, defined for \(x \geqslant 0\), has equation $$y = 4 x ^ { \frac { 1 } { 2 } } - x ^ { \frac { 3 } { 2 } }$$
    1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
      (3 marks)
    2. Show that the gradient of the curve at \(P ( 4,0 )\) is - 2 .
    3. Find an equation of the normal to the curve at \(P ( 4,0 )\).
    4. Find the \(y\)-coordinate of \(Q\) and hence find the area of triangle \(O P Q\).
    5. The curve has a maximum point \(M\). Find the \(x\)-coordinate of \(M\).
    1. Find \(\int \left( 4 x ^ { \frac { 1 } { 2 } } - x ^ { \frac { 3 } { 2 } } \right) \mathrm { d } x\).
    2. Find the total area of the region bounded by the curve and the lines \(P Q\) and \(Q O\).