Questions C2 (1550 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 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 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 PURE 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 PURE S1 S2 S3 S4 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 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 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
Sort by: Default | Easiest first | Hardest first
OCR MEI C2 2012 January Q11
12 marks Standard +0.3
11 The point A has \(x\)-coordinate 5 and lies on the curve \(y = x ^ { 2 } - 4 x + 3\).
  1. Sketch the curve.
  2. Use calculus to find the equation of the tangent to the curve at A .
  3. Show that the equation of the normal to the curve at A is \(x + 6 y = 53\). Find also, using an algebraic method, the \(x\)-coordinate of the point at which this normal crosses the curve again.
OCR MEI C2 2012 January Q12
12 marks Moderate -0.3
12 The equation of a curve is \(y = 9 x ^ { 2 } - x ^ { 4 }\).
  1. Show that the curve meets the \(x\)-axis at the origin and at \(x = \pm a\), stating the value of \(a\).
  2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\). Hence show that the origin is a minimum point on the curve. Find the \(x\)-coordinates of the maximum points.
  3. Use calculus to find the area of the region bounded by the curve and the \(x\)-axis between \(x = 0\) and \(x = a\), using the value you found for \(a\) in part (i).
OCR MEI C2 2012 January Q13
12 marks Moderate -0.3
13 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ba9f2fd1-7a36-4749-86ec-40c93d23a84b-4_709_709_262_303} \captionsetup{labelformat=empty} \caption{Fig. 13.1}
\end{figure}
\includegraphics[max width=\textwidth, alt={}]{ba9f2fd1-7a36-4749-86ec-40c93d23a84b-4_392_544_415_1197}
In a concert hall, seats are arranged along arcs of concentric circles, as shown in Fig. 13.1. As shown in Fig. 13.2, the stage is part of a sector ABO of radius 11 m . Fig. 13.2 also gives the dimensions of the stage.
  1. Show that angle \(\mathrm { COD } = 1.55\) radians, correct to 2 decimal places. Hence find the area of the stage.
  2. There are four rows of seats, with their backs along arcs, with centre O, of radii \(7.4 \mathrm {~m} , 8.6 \mathrm {~m} , 9.8 \mathrm {~m}\) and 11 m . Each seat takes up 80 cm of the arc.
    (A) Calculate how many seats can fit in the front row.
    (B) Calculate how many more seats can fit in the back row than the front row.
OCR MEI C2 2011 June Q1
3 marks Easy -1.2
1 Find \(\int _ { 2 } ^ { 5 } \left( 2 x ^ { 3 } + 3 \right) \mathrm { d } x\).
OCR MEI C2 2011 June Q2
3 marks Moderate -0.8
2 A sequence is defined by $$\begin{aligned} u _ { 1 } & = 10 \\ u _ { r + 1 } & = \frac { 5 } { u _ { r } ^ { 2 } } \end{aligned}$$ Calculate the values of \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\).
What happens to the terms of the sequence as \(r\) tends to infinity?
OCR MEI C2 2011 June Q3
5 marks Moderate -0.8
3 The equation of a curve is \(y = \sqrt { 1 + 2 x }\).
  1. Calculate the gradient of the chord joining the points on the curve where \(x = 4\) and \(x = 4.1\). Give your answer correct to 4 decimal places.
  2. Showing the points you use, calculate the gradient of another chord of the curve which is a closer approximation to the gradient of the curve when \(x = 4\).
OCR MEI C2 2011 June Q4
3 marks Moderate -0.8
4 The graph of \(y = a b ^ { x }\) passes through the points \(( 1,6 )\) and \(( 2,3.6 )\). Find the values of \(a\) and \(b\).
OCR MEI C2 2011 June Q5
5 marks Moderate -0.8
5 Find the equation of the normal to the curve \(y = 8 x ^ { 4 } + 4\) at the point where \(x = \frac { 1 } { 2 }\).
OCR MEI C2 2011 June Q6
5 marks Moderate -0.8
6 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 \sqrt { x } - 2\). Given also that the curve passes through the point \(( 9,4 )\), find the equation of the curve.
OCR MEI C2 2011 June Q7
4 marks Moderate -0.3
7 Solve the equation \(\tan \theta = 2 \sin \theta\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
OCR MEI C2 2011 June Q8
3 marks Easy -1.2
8 Using logarithms, rearrange \(p = s t ^ { n }\) to make \(n\) the subject.
OCR MEI C2 2011 June Q9
3 marks Moderate -0.8
9 You are given that $$\log _ { a } x = \frac { 1 } { 2 } \log _ { a } 16 + \log _ { a } 75 - 2 \log _ { a } 5 .$$ Find the value of \(x\).
OCR MEI C2 2011 June Q10
2 marks Easy -1.2
10 The \(n\)th term, \(t _ { n }\), of a sequence is given by $$t _ { n } = \sin ( \theta + 180 n ) ^ { \circ } .$$ Express \(t _ { 1 }\) and \(t _ { 2 }\) in terms of \(\sin \theta ^ { \circ }\).
OCR MEI C2 2011 June Q11
11 marks Moderate -0.3
11
  1. The standard formulae for the volume \(V\) and total surface area \(A\) of a solid cylinder of radius \(r\) and height \(h\) are $$V = \pi r ^ { 2 } h \quad \text { and } \quad A = 2 \pi r ^ { 2 } + 2 \pi r h .$$ Use these to show that, for a cylinder with \(A = 200\), $$V = 100 r - \pi r ^ { 3 }$$
  2. Find \(\frac { \mathrm { d } V } { \mathrm {~d} r }\) and \(\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} r ^ { 2 } }\).
  3. Use calculus to find the value of \(r\) that gives a maximum value for \(V\) and hence find this maximum value, giving your answers correct to 3 significant figures.
OCR MEI C2 2011 June Q12
17 marks Moderate -0.3
12 Jim and Mary are each planning monthly repayments for money they want to borrow.
  1. Jim's first payment is \(\pounds 500\), and he plans to pay \(\pounds 10\) less each month, so that his second payment is \(\pounds 490\), his third is \(\pounds 480\), and so on.
    (A) Calculate his 12th payment.
    (B) He plans to make 24 payments altogether. Show that he pays \(\pounds 9240\) in total.
  2. Mary's first payment is \(\pounds 460\) and she plans to pay \(2 \%\) less each month than the previous month, so that her second payment is \(\pounds 450.80\), her third is \(\pounds 441.784\), and so on.
    (A) Calculate her 12th payment.
    (B) Show that Jim's 20th payment is less than Mary's 20th payment but that his 19th is not less than her 19th.
    (C) Mary plans to make 24 payments altogether. Calculate how much she pays in total.
    (D) How much would Mary's first payment need to be if she wishes to pay \(2 \%\) less each month as before, but to pay the same in total as Jim, \(\pounds 9240\), over the 24 months?
OCR MEI C2 2011 June Q13
12 marks Moderate -0.3
13 Fig. 13.1 shows a greenhouse which is built against a wall. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{97ed9d1d-b9e5-47d6-a451-b14757c0e19d-4_606_828_347_358} \captionsetup{labelformat=empty} \caption{Fig. 13.1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{97ed9d1d-b9e5-47d6-a451-b14757c0e19d-4_401_350_529_1430} \captionsetup{labelformat=empty} \caption{Fig. 13.2}
\end{figure} The greenhouse is a prism of length 5.5 m . The curve AC is an arc of a circle with centre B and radius 2.1 m , as shown in Fig. 13.2. The sector angle ABC is 1.8 radians and ABD is a straight line. The curved surface of the greenhouse is covered in polythene.
  1. Find the length of the arc AC and hence find the area of polythene required for the curved surface of the greenhouse.
  2. Calculate the length BD .
  3. Calculate the volume of the greenhouse.
OCR MEI C2 2012 June Q1
3 marks Easy -1.2
1 Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(y = \sqrt { x } + \frac { 3 } { x }\).
OCR MEI C2 2012 June Q2
4 marks Easy -1.8
2 Find the second and third terms in the sequence given by $$\begin{aligned} & u _ { 1 } = 5 \\ & u _ { n + 1 } = u _ { n } + 3 . \end{aligned}$$ Find also the sum of the first 50 terms of this sequence.
OCR MEI C2 2012 June Q3
5 marks Moderate -0.3
3 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8f7413d8-2814-4d5c-bec0-ce118fec80eb-2_592_693_845_502} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} \section*{Not to scale} In Fig. 3, BCD is a straight line. \(\mathrm { AC } = 9.8 \mathrm {~cm} , \mathrm { BC } = 7.3 \mathrm {~cm}\) and \(\mathrm { CD } = 6.4 \mathrm {~cm}\); angle \(\mathrm { ACD } = 53.4 ^ { \circ }\).
  1. Calculate the length AD .
  2. Calculate the area of triangle ABC .
OCR MEI C2 2012 June Q4
4 marks Easy -1.2
4 The point \(\mathrm { P } ( 6,3 )\) lies on the curve \(y = \mathrm { f } ( x )\). State the coordinates of the image of P after the transformation which maps \(y = \mathrm { f } ( x )\) onto
  1. \(y = 3 \mathrm { f } ( x )\),
  2. \(y = \mathrm { f } ( 4 x )\).
OCR MEI C2 2012 June Q5
5 marks Moderate -0.3
5 A sector of a circle has angle 1.6 radians and area \(45 \mathrm {~cm} ^ { 2 }\). Find the radius and perimeter of the sector.
OCR MEI C2 2012 June Q6
5 marks Moderate -0.3
6 Fig. 6 shows the relationship between \(\log _ { 10 } x\) and \(\log _ { 10 } y\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8f7413d8-2814-4d5c-bec0-ce118fec80eb-3_497_787_287_644} \captionsetup{labelformat=empty} \caption{Fig. 6}
\end{figure} Find \(y\) in terms of \(x\).
OCR MEI C2 2012 June Q7
5 marks Moderate -0.8
7 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x ^ { \frac { 1 } { 2 } } - 5\). Given also that the curve passes through the point (4, 20), find the equation of the curve.
OCR MEI C2 2012 June Q8
5 marks Moderate -0.8
8 Solve the equation \(\sin 2 \theta = 0.7\) for values of \(\theta\) between 0 and \(2 \pi\), giving your answers in radians correct to 3 significant figures.
OCR MEI C2 2012 June Q9
12 marks Moderate -0.3
9 A farmer digs ditches for flood relief. He experiments with different cross-sections. Assume that the surface of the ground is horizontal.
  1. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{8f7413d8-2814-4d5c-bec0-ce118fec80eb-4_437_640_470_715} \captionsetup{labelformat=empty} \caption{Fig. 9.1}
    \end{figure} Fig. 9.1 shows the cross-section of one ditch, with measurements in metres. The width of the ditch is 1.2 m and Fig. 9.1 shows the depth every 0.2 m across the ditch. Use the trapezium rule with six intervals to estimate the area of cross-section. Hence estimate the volume of water that can be contained in a 50-metre length of this ditch.
  2. Another ditch is 0.9 m wide, with cross-section as shown in Fig. 9.2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{8f7413d8-2814-4d5c-bec0-ce118fec80eb-4_574_808_1402_632} \captionsetup{labelformat=empty} \caption{Fig. 9.2}
    \end{figure} With \(x\) - and \(y\)-axes as shown in Fig. 9.2, the curve of the ditch may be modelled closely by \(y = 3.8 x ^ { 4 } - 6.8 x ^ { 3 } + 7.7 x ^ { 2 } - 4.2 x\).
    (A) The actual ditch is 0.6 m deep when \(x = 0.2\). Calculate the difference between the depth given by the model and the true depth for this value of \(x\).
    (B) Find \(\int \left( 3.8 x ^ { 4 } - 6.8 x ^ { 3 } + 7.7 x ^ { 2 } - 4.2 x \right) \mathrm { d } x\). Hence estimate the volume of water that can be contained in a 50 -metre length of this ditch.