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 C2 Specimen Q4
4 Records are kept of the number of copies of a certain book that are sold each week. In the first week after publication 3000 copies were sold, and in the second week 2400 copies were sold. The publisher forecasts future sales by assuming that the number of copies sold each week will form a geometric progression with first two terms 3000 and 2400. Calculate the publisher’s forecasts for
  1. the number of copies that will be sold in the 20th week after publication,
  2. the total number of copies sold during the first 20 weeks after publication,
  3. the total number of copies that will ever be sold.
OCR C2 Specimen Q5
5
  1. Show that the equation \(15 \cos ^ { 2 } \theta ^ { \circ } = 13 + \sin \theta ^ { \circ }\) may be written as a quadratic equation in \(\sin \theta ^ { \circ }\).
  2. Hence solve the equation, giving all values of \(\theta\) such that \(0 \leqslant \theta \leqslant 360\).
OCR C2 Specimen Q6
6 The diagram shows triangle \(A B C\), in which \(A B = 3 \mathrm {~cm} , A C = 5 \mathrm {~cm}\) and angle \(A B C = 2.1\) radians. Calculate
  1. angle \(A C B\), giving your answer in radians,
  2. the area of the triangle. An arc of a circle with centre \(A\) and radius 3 cm is drawn, cutting \(A C\) at the point \(D\).
  3. Calculate the perimeter and the area of the sector \(A B D\).
OCR C2 Specimen Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{73d67b39-3611-4afb-9470-2f813115abb5-3_460_709_1114_708} The diagram shows the curves \(y = - 3 x ^ { 2 } - 9 x + 30\) and \(y = x ^ { 2 } + 3 x - 10\).
  1. Verify that the curves intersect at the points \(A ( - 5,0 )\) and \(B ( 2,0 )\).
  2. Show that the area of the shaded region between the curves is given by \(\int _ { - 5 } ^ { 2 } \left( - 4 x ^ { 2 } - 12 x + 40 \right) \mathrm { d } x\).
  3. Hence or otherwise show that the area of the shaded region between the curves is \(228 \frac { 2 } { 3 }\).
OCR C2 Specimen Q8
8
\includegraphics[max width=\textwidth, alt={}, center]{73d67b39-3611-4afb-9470-2f813115abb5-4_415_714_287_678} The diagram shows the curve \(y = 1.25 ^ { x }\).
  1. A point on the curve has \(y\)-coordinate 2. Calculate its \(x\)-coordinate.
  2. Use the trapezium rule with 4 intervals to estimate the area of the shaded region, bounded by the curve, the axes, and the line \(x = 4\).
  3. State, with a reason, whether the estimate found in part (ii) is an overestimate or an underestimate.
  4. Explain briefly how the trapezium rule could be used to find a more accurate estimate of the area of the shaded region.
OCR C2 Specimen Q9
9 The cubic polynomial \(x ^ { 3 } + a x ^ { 2 } + b x - 6\) is denoted by \(\mathrm { f } ( x )\).
  1. The remainder when \(\mathrm { f } ( x )\) is divided by ( \(x - 2\) ) is equal to the remainder when \(\mathrm { f } ( x )\) is divided by \(( x + 2 )\). Show that \(b = - 4\).
  2. Given also that ( \(x - 1\) ) is a factor of \(\mathrm { f } ( x )\), find the value of \(a\).
  3. With these values of \(a\) and \(b\), express \(\mathrm { f } ( x )\) as a product of a linear factor and a quadratic factor.
  4. Hence determine the number of real roots of the equation \(\mathrm { f } ( x ) = 0\), explaining your reasoning.
OCR MEI C2 Q7
7 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1e43ddbe-ae95-467b-a527-351ab8a4c4fe-003_305_897_310_795} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure} Fig. 7 shows a sector of a circle of radius 5 cm which has angle \(\theta\) radians. The sector has area \(30 \mathrm {~cm} ^ { 2 }\).
  1. Find \(\theta\).
  2. Hence find the perimeter of the sector.
OCR MEI C2 Q9
9
  1. A tunnel is 100 m long. Its cross-section, shown in Fig. 9.1, is modelled by the curve $$y = \frac { 1 } { 4 } \left( 10 x - x ^ { 2 } \right) ,$$ where \(x\) and \(y\) are horizontal and vertical distances in metres. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{1e43ddbe-ae95-467b-a527-351ab8a4c4fe-004_506_812_676_653} \captionsetup{labelformat=empty} \caption{Figure 9.1}
    \end{figure} Using this model,
    (A) find the greatest height of the tunnel,
    (B) explain why \(100 \int _ { 0 } ^ { 10 } y \mathrm {~d} x\) gives the volume, in cubic metres, of earth removed to make the tunnel. Calculate this volume.
  2. The roof of the tunnel is re-shaped to allow for larger vehicles. Fig. 9.2 shows the new crosssection. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{1e43ddbe-ae95-467b-a527-351ab8a4c4fe-004_513_1256_1894_575} \captionsetup{labelformat=empty} \caption{Fig. 9.2}
    \end{figure} Use the trapezium rule with 5 strips to estimate the new cross-sectional area.
    Hence estimate the volume of earth removed when the tunnel is re-shaped.
OCR MEI C2 Q11
11 Answer part (iii) of this question on the insert provided.
A hot drink is made and left to cool. The table shows its temperature at ten-minute intervals after it is made.
Time (minutes)1020304050
Temperature \(\left( { } ^ { \circ } \mathrm { C } \right)\)6853423631
The room temperature is \(22 ^ { \circ } \mathrm { C }\). The difference between the temperature of the drink and room temperature at time \(t\) minutes is \(z ^ { \circ } \mathrm { C }\). The relationship between \(z\) and \(t\) is modelled by $$z = z _ { 0 } 10 ^ { - k t }$$ where \(z _ { 0 }\) and \(k\) are positive constants.
  1. Give a physical interpretation for the constant \(z _ { 0 }\).
  2. Show that \(\log _ { 10 } z = - k t + \log _ { 10 } z _ { 0 }\).
  3. On the insert, complete the table and draw the graph of \(\log _ { 10 } z\) against \(t\). Use your graph to estimate the values of \(k\) and \(z _ { 0 }\).
    Hence estimate the temperature of the drink 70 minutes after it is made. \section*{OXFORD CAMBRIDGE AND RSA EXAMINATIONS} Advanced Subsidiary General Certificate of Education Advanced General Certificate of Education \section*{MEI STRUCTURED MATHEMATICS} Concepts for Advanced Mathematics (C2)
    INSERT
    Wednesday
OCR MEI C2 2005 January Q1
1 Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(y = x ^ { 6 } + \sqrt { x }\).
OCR MEI C2 2005 January Q2
2 Find \(\int \left( x ^ { 3 } + \frac { 1 } { x ^ { 3 } } \right) \mathrm { d } x\).
OCR MEI C2 2005 January Q3
3 Sketch the graph of \(y = \sin x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
Solve the equation \(\sin x = - 0.2\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
OCR MEI C2 2005 January Q4
4 Fig. 4 For triangle ABC shown in Fig. 4, calculate
  1. the length of BC ,
  2. the area of triangle ABC .
OCR MEI C2 2005 January Q5
5 The first three terms of a geometric progression are 4, 2, 1.
Find the twentieth term, expressing your answer as a power of 2.
Find also the sum to infinity of this progression.
OCR MEI C2 2005 January Q6
6 A sequence is given by $$\begin{gathered} a _ { 1 } = 4
a _ { r + 1 } = a _ { r } + 3 \end{gathered}$$ Write down the first 4 terms of this sequence.
Find the sum of the first 100 terms of the sequence.
OCR MEI C2 2005 January Q7
7 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{72b4624f-e716-4a37-96f3-01b46e0bd0fd-4_305_897_310_795} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure} Fig. 7 shows a sector of a circle of radius 5 cm which has angle \(\theta\) radians. The sector has area \(30 \mathrm {~cm} ^ { 2 }\).
  1. Find \(\theta\).
  2. Hence find the perimeter of the sector.
OCR MEI C2 2005 January Q8
8
  1. Solve the equation \(10 ^ { x } = 316\).
  2. Simplify \(\log _ { a } \left( a ^ { 2 } \right) - 4 \log _ { a } \left( \frac { 1 } { a } \right)\).
OCR MEI C2 2005 January Q9
9
  1. A tunnel is 100 m long. Its cross-section, shown in Fig. 9.1, is modelled by the curve $$y = \frac { 1 } { 4 } \left( 10 x - x ^ { 2 } \right) ,$$ where \(x\) and \(y\) are horizontal and vertical distances in metres. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{72b4624f-e716-4a37-96f3-01b46e0bd0fd-5_506_812_676_653} \captionsetup{labelformat=empty} \caption{Figure 9.1}
    \end{figure} Using this model,
    (A) find the greatest height of the tunnel,
    (B) explain why \(100 \int _ { 0 } ^ { 10 } y \mathrm {~d} x\) gives the volume, in cubic metres, of earth removed to make the tunnel. Calculate this volume.
  2. The roof of the tunnel is re-shaped to allow for larger vehicles. Fig. 9.2 shows the new crosssection. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{72b4624f-e716-4a37-96f3-01b46e0bd0fd-5_513_1256_1894_575} \captionsetup{labelformat=empty} \caption{Fig. 9.2}
    \end{figure} Use the trapezium rule with 5 strips to estimate the new cross-sectional area.
    Hence estimate the volume of earth removed when the tunnel is re-shaped.
OCR MEI C2 2005 January Q10
10 A curve has equation \(y = x ^ { 3 } - 6 x ^ { 2 } + 12\).
  1. Use calculus to find the coordinates of the turning points of this curve. Determine also the nature of these turning points.
  2. Find, in the form \(y = m x + c\), the equation of the normal to the curve at the point \(( 2 , - 4 )\).
OCR MEI C2 2005 January Q11
11 Answer part (iii) of this question on the insert provided.
A hot drink is made and left to cool. The table shows its temperature at ten-minute intervals after it is made.
Time (minutes)1020304050
Temperature \(\left( { } ^ { \circ } \mathrm { C } \right)\)6853423631
The room temperature is \(22 ^ { \circ } \mathrm { C }\). The difference between the temperature of the drink and room temperature at time \(t\) minutes is \(z ^ { \circ } \mathrm { C }\). The relationship between \(z\) and \(t\) is modelled by $$z = z _ { 0 } 10 ^ { - k t }$$ where \(z _ { 0 }\) and \(k\) are positive constants.
  1. Give a physical interpretation for the constant \(z _ { 0 }\).
  2. Show that \(\log _ { 10 } z = - k t + \log _ { 10 } z _ { 0 }\).
  3. On the insert, complete the table and draw the graph of \(\log _ { 10 } z\) against \(t\). Use your graph to estimate the values of \(k\) and \(z _ { 0 }\).
    Hence estimate the temperature of the drink 70 minutes after it is made.
OCR MEI C2 2006 January Q1
1 Given that \(140 ^ { \circ } = k \pi\) radians, find the exact value of \(k\).
OCR MEI C2 2006 January Q2
2 Find the numerical value of \(\sum _ { k = 2 } ^ { 5 } k ^ { 3 }\).
OCR MEI C2 2006 January Q3
3 Fig. 3 Beginning with the triangle shown in Fig. 3, prove that \(\sin 60 ^ { \circ } = \frac { \sqrt { 3 } } { 2 }\).
OCR MEI C2 2006 January Q4
4 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{15b8f97b-c058-409f-907f-cb0a6102abc4-2_615_971_1457_539} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} Fig. 4 shows a curve which passes through the points shown in the following table.
\(x\)11.522.533.54
\(y\)8.26.45.55.04.74.44.2
Use the trapezium rule with 6 strips to estimate the area of the region bounded by the curve, the lines \(x = 1\) and \(x = 4\), and the \(x\)-axis. State, with a reason, whether the trapezium rule gives an overestimate or an underestimate of the area of this region.
OCR MEI C2 2006 January Q5
5
  1. Sketch the graph of \(y = \tan x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
  2. Solve the equation \(4 \sin x = 3 \cos x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).