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 2009 January Q10
13 marks Standard +0.3
10 Fig. 10 shows a sketch of the graph of \(y = 7 x - x ^ { 2 } - 6\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{838d6b15-69a9-4e67-bc36-5bf60254a767-4_609_908_1338_621} \captionsetup{labelformat=empty} \caption{Fig. 10}
\end{figure}
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence find the equation of the tangent to the curve at the point on the curve where \(x = 2\). Show that this tangent crosses the \(x\)-axis where \(x = \frac { 2 } { 3 }\).
  2. Show that the curve crosses the \(x\)-axis where \(x = 1\) and find the \(x\)-coordinate of the other point of intersection of the curve with the \(x\)-axis.
  3. Find \(\int _ { 1 } ^ { 2 } \left( 7 x - x ^ { 2 } - 6 \right) \mathrm { d } x\). Hence find the area of the region bounded by the curve, the tangent and the \(x\)-axis, shown shaded on Fig. 10.
OCR MEI C2 2009 January Q11
11 marks Standard +0.3
11
  1. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{838d6b15-69a9-4e67-bc36-5bf60254a767-5_469_878_274_671} \captionsetup{labelformat=empty} \caption{Fig. 11.1}
    \end{figure} Fig. 11.1 shows the surface ABCD of a TV presenter's desk. AB and CD are arcs of circles with centre O and sector angle 2.5 radians. \(\mathrm { OC } = 60 \mathrm {~cm}\) and \(\mathrm { OB } = 140 \mathrm {~cm}\).
    (A) Calculate the length of the arc CD.
    (B) Calculate the area of the surface ABCD of the desk.
  2. The TV presenter is at point P , shown in Fig. 11.2. A TV camera can move along the track EF , which is of length 3.5 m . \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{838d6b15-69a9-4e67-bc36-5bf60254a767-5_378_877_1334_675} \captionsetup{labelformat=empty} \caption{Fig. 11.2}
    \end{figure} When the camera is at E , the TV presenter is 1.6 m away. When the camera is at F , the TV presenter is 2.8 m away.
    (A) Calculate, in degrees, the size of angle EFP.
    (B) Calculate the shortest possible distance between the camera and the TV presenter.
OCR MEI C2 2010 January Q1
3 marks Easy -1.2
1 Find \(\int \left( x - \frac { 3 } { x ^ { 2 } } \right) \mathrm { d } x\).
OCR MEI C2 2010 January Q2
3 marks Easy -1.2
2 A sequence begins $$\begin{array} { l l l l l l l l l l l } 1 & 3 & 5 & 3 & 1 & 3 & 5 & 3 & 1 & 3 & \ldots \end{array}$$ and continues in this pattern.
  1. Find the 55th term of this sequence, showing your method.
  2. Find the sum of the first 55 terms of the sequence.
OCR MEI C2 2010 January Q3
3 marks Moderate -0.8
3 You are given that \(\sin \theta = \frac { \sqrt { 2 } } { 3 }\) and that \(\theta\) is an acute angle. Find the exact value of \(\tan \theta\).
OCR MEI C2 2010 January Q4
3 marks Moderate -0.8
4 A sector of a circle has area \(8.45 \mathrm {~cm} ^ { 2 }\) and sector angle 0.4 radians. Calculate the radius of the sector.
OCR MEI C2 2010 January Q5
4 marks Moderate -0.8
5 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{053009a4-e88f-4711-ad97-cebb1740744b-2_547_991_1340_577} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure} Fig. 5 shows a sketch of the graph of \(y = \mathrm { f } ( x )\). On separate diagrams, sketch the graphs of the following, showing clearly the coordinates of the points corresponding to \(\mathrm { P } , \mathrm { Q }\) and R .
  1. \(y = \mathrm { f } ( 2 x )\)
  2. \(y = \frac { 1 } { 4 } \mathrm { f } ( x )\)
OCR MEI C2 2010 January Q6
5 marks Easy -1.3
6
  1. Find the 51 st term of the sequence given by $$\begin{aligned} u _ { 1 } & = 5 \\ u _ { n + 1 } & = u _ { n } + 4 \end{aligned}$$
  2. Find the sum to infinity of the geometric progression which begins $$5 \quad 2 \quad 0.8 \quad \ldots .$$ \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{053009a4-e88f-4711-ad97-cebb1740744b-3_531_969_744_589} \captionsetup{labelformat=empty} \caption{Fig. 7}
    \end{figure} Fig. 7 shows triangle ABC , with \(\mathrm { AB } = 8.4 \mathrm {~cm}\). D is a point on AC such that angle \(\mathrm { ADB } = 79 ^ { \circ }\), \(\mathrm { BD } = 5.6 \mathrm {~cm}\) and \(\mathrm { CD } = 7.8 \mathrm {~cm}\). Calculate
  3. angle BAD ,
  4. the length BC .
OCR MEI C2 2010 January Q8
5 marks Moderate -0.8
8 Find the equation of the tangent to the curve \(y = 6 \sqrt { x }\) at the point where \(x = 16\).
OCR MEI C2 2010 January Q9
5 marks Easy -1.2
9
  1. Sketch the graph of \(y = 3 ^ { x }\).
  2. Use logarithms to solve \(3 ^ { 2 x + 1 } = 10\), giving your answer correct to 2 decimal places.
OCR MEI C2 2010 January Q10
11 marks Moderate -0.8
10
  1. Differentiate \(x ^ { 3 } - 3 x ^ { 2 } - 9 x\). Hence find the \(x\)-coordinates of the stationary points on the curve \(y = x ^ { 3 } - 3 x ^ { 2 } - 9 x\), showing which is the maximum and which the minimum.
  2. Find, in exact form, the coordinates of the points at which the curve crosses the \(x\)-axis.
  3. Sketch the curve.
OCR MEI C2 2010 January Q11
12 marks Moderate -0.8
11 Fig. 11 shows the cross-section of a school hall, with measurements of the height in metres taken at 1.5 m intervals from O . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{053009a4-e88f-4711-ad97-cebb1740744b-4_579_1381_861_383} \captionsetup{labelformat=empty} \caption{Fig. 11}
\end{figure}
  1. Use the trapezium rule with 8 strips to calculate an estimate of the area of the cross-section.
  2. Use 8 rectangles to calculate a lower bound for the area of the cross-section. The curve of the roof may be modelled by \(y = - 0.013 x ^ { 3 } + 0.16 x ^ { 2 } - 0.082 x + 2.4\), where \(x\) metres is the horizontal distance from O across the hall, and \(y\) metres is the height.
  3. Use integration to find the area of the cross-section according to this model.
  4. Comment on the accuracy of this model for the height of the hall when \(x = 7.5\).
OCR MEI C2 2011 January Q1
2 marks Easy -1.2
1 Calculate \(\sum _ { r = 3 } ^ { 6 } \frac { 12 } { r }\).
OCR MEI C2 2011 January Q2
4 marks Easy -1.3
2 Find \(\int \left( 3 x ^ { 5 } + 2 x ^ { - \frac { 1 } { 2 } } \right) \mathrm { d } x\).
OCR MEI C2 2011 January Q3
3 marks Easy -1.2
3 At a place where a river is 7.5 m wide, its depth is measured every 1.5 m across the river. The table shows the results.
Distance across river \(( \mathrm { m } )\)01.534.567.5
Depth of river \(( \mathrm { m } )\)0.62.33.12.81.80.7
Use the trapezium rule with 5 strips to estimate the area of cross-section of the river.
OCR MEI C2 2011 January Q4
4 marks Easy -1.2
4 The curve \(y = \mathrm { f } ( x )\) has a minimum point at \(( 3,5 )\).
State the coordinates of the corresponding minimum point on the graph of
  1. \(y = 3 \mathrm { f } ( x )\),
  2. \(y = \mathrm { f } ( 2 x )\).
OCR MEI C2 2011 January Q5
5 marks Moderate -0.3
5 The second term of a geometric sequence is 6 and the fifth term is - 48 .
Find the tenth term of the sequence.
Find also, in simplified form, an expression for the sum of the first \(n\) terms of this sequence.
OCR MEI C2 2011 January Q6
5 marks Moderate -0.8
6 The third term of an arithmetic progression is 24 . The tenth term is 3 .
Find the first term and the common difference. Find also the sum of the 21st to 50th terms inclusive.
OCR MEI C2 2011 January Q7
4 marks Easy -1.3
7 Simplify
  1. \(\log _ { 10 } x ^ { 5 } + 3 \log _ { 10 } x ^ { 4 }\),
  2. \(\log _ { a } 1 - \log _ { a } a ^ { b }\).
OCR MEI C2 2011 January Q8
5 marks Moderate -0.3
8 Showing your method clearly, solve the equation $$5 \sin ^ { 2 } \theta = 5 + \cos \theta \quad \text { for } 0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ } .$$
OCR MEI C2 2011 January Q9
4 marks Standard +0.3
9 Charles has a slice of cake; its cross-section is a sector of a circle, as shown in Fig. 9. The radius is \(r \mathrm {~cm}\) and the sector angle is \(\frac { \pi } { 6 }\) radians. He wants to give half of the slice to Jan. He makes a cut across the sector as shown. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{aa8688bb-3608-45ab-85b7-bf84889dd189-3_420_657_497_744} \captionsetup{labelformat=empty} \caption{Fig. 9}
\end{figure} Show that when they each have half the slice, \(a = r \sqrt { \frac { \pi } { 6 } }\). Section B (36 marks)
OCR MEI C2 2011 January Q10
12 marks Standard +0.3
10 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{aa8688bb-3608-45ab-85b7-bf84889dd189-3_645_793_1377_676} \captionsetup{labelformat=empty} \caption{Fig. 10}
\end{figure} A is the point with coordinates \(( 1,4 )\) on the curve \(y = 4 x ^ { 2 }\). B is the point with coordinates \(( 0,1 )\), as shown in Fig. 10.
  1. The line through A and B intersects the curve again at the point C . Show that the coordinates of C are \(\left( - \frac { 1 } { 4 } , \frac { 1 } { 4 } \right)\).
  2. Use calculus to find the equation of the tangent to the curve at A and verify that the equation of the tangent at C is \(y = - 2 x - \frac { 1 } { 4 }\).
  3. The two tangents intersect at the point D . Find the \(y\)-coordinate of D . \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{aa8688bb-3608-45ab-85b7-bf84889dd189-4_773_1027_255_557} \captionsetup{labelformat=empty} \caption{Fig. 11}
    \end{figure} Fig. 11 shows the curve \(y = x ^ { 3 } - 3 x ^ { 2 } - x + 3\).
OCR MEI C2 2011 January Q12
13 marks Moderate -0.3
12 The table shows the size of a population of house sparrows from 1980 to 2005.
Year198019851990199520002005
Population250002200018750162501350012000
The 'red alert' category for birds is used when a population has decreased by at least \(50 \%\) in the previous 25 years.
  1. Show that the information for this population is consistent with the house sparrow being on red alert in 2005. The size of the population may be modelled by a function of the form \(P = a \times 10 ^ { - k t }\), where \(P\) is the population, \(t\) is the number of years after 1980, and \(a\) and \(k\) are constants.
  2. Write the equation \(P = a \times 10 ^ { - k t }\) in logarithmic form using base 10, giving your answer as simply as possible.
  3. Complete the table and draw the graph of \(\log _ { 10 } P\) against \(t\), drawing a line of best fit by eye.
  4. Use your graph to find the values of \(a\) and \(k\) and hence the equation for \(P\) in terms of \(t\).
  5. Find the size of the population in 2015 as predicted by this model. Would the house sparrow still be on red alert? Give a reason for your answer.
OCR MEI C2 2012 January Q1
2 marks Easy -1.2
1 Find \(\sum _ { r = 3 } ^ { 6 } r ( r + 2 )\).
OCR MEI C2 2012 January Q2
4 marks Easy -1.3
2 Find \(\int \left( x ^ { 5 } + 10 x ^ { \frac { 3 } { 2 } } \right) \mathrm { d } x\).