Questions C2 (1410 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
OCR MEI C2 2016 June Q1
5 marks Easy -1.8
1
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(y = 6 \sqrt { x }\).
  2. Find \(\int \frac { 12 } { x ^ { 2 } } \mathrm {~d} x\).
OCR MEI C2 2016 June Q2
3 marks Moderate -0.8
2 A sequence is defined as follows.
\(u _ { 1 } = a\), where \(a > 0\)
To obtain \(u _ { r + 1 }\)
  • find the remainder when \(u _ { r }\) is divided by 3 ,
  • multiply the remainder by 5 ,
  • the result is \(u _ { r + 1 }\).
Find \(\sum _ { r = 2 } ^ { 4 } u _ { r }\) in each of the following cases.
  1. \(a = 5\)
  2. \(a = 6\)
OCR MEI C2 2016 June Q3
5 marks Moderate -0.8
3 An arithmetic progression (AP) and a geometric progression (GP) have the same first and fourth terms as each other. The first term of both is 1.5 and the fourth term of both is 12 . Calculate the difference between the tenth terms of the AP and the GP. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{50ebbd77-39da-4a48-993a-bcf99ada9dcd-2_581_855_1644_587} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} Fig. 4 shows triangle ABC , where \(\mathrm { AB } = 7.2 \mathrm {~cm} , \mathrm { AC } = 5.6 \mathrm {~cm}\) and angle \(\mathrm { BAC } = 68 ^ { \circ }\).
Calculate the size of angle ACB .
OCR MEI C2 2016 June Q5
4 marks Moderate -0.8
5
  1. Fig. 5 shows the graph of a sine function. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{50ebbd77-39da-4a48-993a-bcf99ada9dcd-3_534_1154_312_450} \captionsetup{labelformat=empty} \caption{Fig. 5}
    \end{figure} State the equation of this curve.
  2. Sketch the graph of \(y = \sin x - 3\) for \(0 ^ { \circ } \leqslant x \leqslant 450 ^ { \circ }\).
OCR MEI C2 2016 June Q6
4 marks Standard +0.3
6 A sector of a circle has radius \(r \mathrm {~cm}\) and sector angle \(\theta\) radians. It is divided into two regions, A and B . Region A is an isosceles triangle with the equal sides being of length \(a \mathrm {~cm}\), as shown in Fig. 6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{50ebbd77-39da-4a48-993a-bcf99ada9dcd-3_407_469_1343_612} \captionsetup{labelformat=empty} \caption{Fig. 6}
\end{figure} \section*{Not to scale}
  1. Express the area of B in terms of \(a , r\) and \(\theta\).
  2. Given that \(r = 12\) and \(\theta = 0.8\), find the value of \(a\) for which the areas of A and B are equal. Give your answer correct to 3 significant figures.
OCR MEI C2 2016 June Q7
5 marks Moderate -0.8
7
  1. Show that, when \(x\) is an acute angle, \(\tan x \sqrt { 1 - \sin ^ { 2 } x } = \sin x\).
  2. Solve \(4 \sin ^ { 2 } y = \sin y\) for \(0 ^ { \circ } \leqslant y \leqslant 360 ^ { \circ }\).
OCR MEI C2 2016 June Q8
5 marks Moderate -0.8
8
  1. Simplify \(\log _ { a } 1 - \log _ { a } \left( a ^ { m } \right) ^ { 3 }\).
  2. Use logarithms to solve the equation \(3 ^ { 2 x + 1 } = 1000\). Give your answer correct to 3 significant figures.
OCR MEI C2 2016 June Q9
11 marks Standard +0.3
9 Fig. 9 shows the cross-section of a straight, horizontal tunnel. The \(x\)-axis from 0 to 6 represents the floor of the tunnel. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{50ebbd77-39da-4a48-993a-bcf99ada9dcd-4_668_734_456_662} \captionsetup{labelformat=empty} \caption{Fig. 9}
\end{figure} With axes as shown, and units in metres, the roof of the tunnel passes through the points shown in the table.
\(x\)0123456
\(y\)04.04.95.04.94.00
The length of the tunnel is 50 m .
  1. Use the trapezium rule with 6 strips to estimate the area of cross-section of the tunnel. Hence estimate the volume of earth removed in digging the tunnel.
  2. An engineer models the height of the roof of the tunnel using the curve \(y = \frac { 5 } { 81 } \left( 108 x - 54 x ^ { 2 } + 12 x ^ { 3 } - x ^ { 4 } \right)\). This curve is symmetrical about \(x = 3\).
    (A) Show that, according to this model, a vehicle of rectangular cross-section which is 3.6 m wide and 4.4 m high would not be able to pass through the tunnel.
    (B) Use integration to calculate the area of the cross-section given by this model. Hence obtain another estimate of the volume of earth removed in digging the tunnel.
OCR MEI C2 2016 June Q10
13 marks Moderate -0.5
10
  1. Calculate the gradient of the chord of the curve \(y = x ^ { 2 } - 2 x\) joining the points at which the values of \(x\) are 5 and 5.1.
  2. Given that \(\mathrm { f } ( x ) = x ^ { 2 } - 2 x\), find and simplify \(\frac { \mathrm { f } ( 5 + h ) - \mathrm { f } ( 5 ) } { h }\).
  3. Use your result in part (ii) to find the gradient of the curve \(y = x ^ { 2 } - 2 x\) at the point where \(x = 5\), showing your reasoning.
  4. Find the equation of the tangent to the curve \(y = x ^ { 2 } - 2 x\) at the point where \(x = 5\). Find the area of the triangle formed by this tangent and the coordinate axes.
OCR MEI C2 2016 June Q11
12 marks Moderate -0.3
11 There are many different flu viruses. The numbers of flu viruses detected in the first few weeks of the 2012-2013 flu epidemic in the UK were as follows.
Week12345678910
Number of flu viruses710243240386396234480
These data may be modelled by an equation of the form \(y = a \times 10 ^ { b t }\), where \(y\) is the number of flu viruses detected in week \(t\) of the epidemic, and \(a\) and \(b\) are constants to be determined.
  1. Explain why this model leads to a straight-line graph of \(\log _ { 10 } y\) against \(t\). State the gradient and intercept of this graph in terms of \(a\) and \(b\).
  2. Complete the values of \(\log _ { 10 } y\) in the table, draw the graph of \(\log _ { 10 } y\) against \(t\), and draw by eye a line of best fit for the data. Hence determine the values of \(a\) and \(b\) and the equation for \(y\) in terms of \(t\) for this model. During the decline of the epidemic, an appropriate model was $$y = 921 \times 10 ^ { - 0.137 w }$$ where \(y\) is the number of flu viruses detected in week \(w\) of the decline.
  3. Use this to find the number of viruses detected in week 4 of the decline.
AQA C2 Q4
Standard +0.3
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
Moderate -0.3
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
Moderate -0.8
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
Standard +0.3
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
8 marks Moderate -0.8
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
10 marks Moderate -0.8
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
6 marks Moderate -0.3
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
9 marks Moderate -0.8
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
7 marks Easy -1.2
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
10 marks Moderate -0.8
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
11 marks Moderate -0.8
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
12 marks Moderate -0.8
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
5 marks Moderate -0.5
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
5 marks Moderate -0.8
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
9 marks Moderate -0.8
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 .