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 2009 January Q1
6 marks Easy -1.2
1 Find
  1. \(\int \left( x ^ { 3 } + 8 x - 5 \right) \mathrm { d } x\),
  2. \(\int 12 \sqrt { x } \mathrm {~d} x\).
OCR C2 2009 January Q2
6 marks Moderate -0.8
2
\includegraphics[max width=\textwidth, alt={}, center]{bbee5a50-4a32-4171-8713-8eb38914a511-2_311_521_651_810} The diagram shows a sector \(O A B\) of a circle, centre \(O\) and radius 7 cm . The angle \(A O B\) is \(140 ^ { \circ }\).
  1. Express \(140 ^ { \circ }\) in radians, giving your answer in an exact form as simply as possible.
  2. Find the perimeter of the segment shaded in the diagram, giving your answer correct to 3 significant figures.
OCR C2 2009 January Q3
7 marks Moderate -0.8
3 A sequence of terms \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { n } = 24 - \frac { 2 } { 3 } n$$
  1. Write down the exact values of \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\).
  2. Find the value of \(k\) such that \(u _ { k } = 0\).
  3. Find \(\sum _ { n = 1 } ^ { 20 } u _ { n }\).
OCR C2 2009 January Q4
7 marks Moderate -0.5
4
\includegraphics[max width=\textwidth, alt={}, center]{bbee5a50-4a32-4171-8713-8eb38914a511-3_570_853_269_644} The diagram shows the curve \(y = x ^ { 4 } + 3\) and the line \(y = 19\) which intersect at \(( - 2,19 )\) and \(( 2,19 )\). Use integration to find the exact area of the shaded region enclosed by the curve and the line.
OCR C2 2009 January Q5
7 marks Standard +0.3
5
\includegraphics[max width=\textwidth, alt={}, center]{bbee5a50-4a32-4171-8713-8eb38914a511-3_623_355_1123_897} Some walkers see a tower, \(T\), in the distance and want to know how far away it is. They take a bearing from a point \(A\) and then walk for 50 m in a straight line before taking another bearing from a point \(B\). They find that angle \(T A B\) is \(70 ^ { \circ }\) and angle \(T B A\) is \(107 ^ { \circ }\) (see diagram).
  1. Find the distance of the tower from \(A\).
  2. They continue walking in the same direction for another 100 m to a point \(C\), so that \(A C\) is 150 m . What is the distance of the tower from \(C\) ?
  3. Find the shortest distance of the walkers from the tower as they walk from \(A\) to \(C\).
OCR C2 2009 January Q6
8 marks Moderate -0.8
6 A geometric progression has first term 20 and common ratio 0.9.
  1. Find the sum to infinity.
  2. Find the sum of the first 30 terms.
  3. Use logarithms to find the smallest value of \(p\) such that the \(p\) th term is less than 0.4 .
OCR C2 2009 January Q7
9 marks Moderate -0.3
7 In the binomial expansion of \(( k + a x ) ^ { 4 }\) the coefficient of \(x ^ { 2 }\) is 24 .
  1. Given that \(a\) and \(k\) are both positive, show that \(a k = 2\).
  2. Given also that the coefficient of \(x\) in the expansion is 128 , find the values of \(a\) and \(k\).
  3. Hence find the coefficient of \(x ^ { 3 }\) in the expansion.
OCR C2 2009 January Q8
10 marks Moderate -0.8
8
  1. Given that \(\log _ { a } x = p\) and \(\log _ { a } y = q\), express the following in terms of \(p\) and \(q\).
    1. \(\log _ { a } ( x y )\)
    2. \(\log _ { a } \left( \frac { a ^ { 2 } x ^ { 3 } } { y } \right)\)
    1. Express \(\log _ { 10 } \left( x ^ { 2 } - 10 \right) - \log _ { 10 } x\) as a single logarithm.
    2. Hence solve the equation \(\log _ { 10 } \left( x ^ { 2 } - 10 \right) - \log _ { 10 } x = 2 \log _ { 10 } 3\).
OCR C2 2009 January Q9
12 marks Standard +0.3
9
  1. The polynomial \(\mathrm { f } ( x )\) is defined by $$\mathrm { f } ( x ) = x ^ { 3 } - x ^ { 2 } - 3 x + 3$$ Show that \(x = 1\) is a root of the equation \(\mathrm { f } ( x ) = 0\), and hence find the other two roots.
  2. Hence solve the equation $$\tan ^ { 3 } x - \tan ^ { 2 } x - 3 \tan x + 3 = 0$$ for \(0 \leqslant x \leqslant 2 \pi\). Give each solution for \(x\) in an exact form.
OCR C2 2010 January Q1
6 marks Moderate -0.3
1
  1. Show that the equation $$2 \sin ^ { 2 } x = 5 \cos x - 1$$ can be expressed in the form $$2 \cos ^ { 2 } x + 5 \cos x - 3 = 0$$
  2. Hence solve the equation $$2 \sin ^ { 2 } x = 5 \cos x - 1$$ giving all values of \(x\) between \(0 ^ { \circ }\) and \(360 ^ { \circ }\).
OCR C2 2010 January Q2
7 marks Moderate -0.8
2 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x - 4\). The curve passes through the distinct points ( 2,5 ) and ( \(p , 5\) ).
  1. Find the equation of the curve.
  2. Find the value of \(p\).
OCR C2 2010 January Q3
6 marks Moderate -0.8
3
  1. Find and simplify the first four terms in the expansion of \(( 2 - x ) ^ { 7 }\) in ascending powers of \(x\).
  2. Hence find the coefficient of \(w ^ { 6 }\) in the expansion of \(\left( 2 - \frac { 1 } { 4 } w ^ { 2 } \right) ^ { 7 }\).
OCR C2 2010 January Q4
6 marks Moderate -0.3
4
  1. Use the trapezium rule, with 4 strips each of width 0.5 , to find an approximate value for $$\int _ { 3 } ^ { 5 } \log _ { 10 } ( 2 + x ) d x$$ giving your answer correct to 3 significant figures.
  2. Use your answer to part (i) to deduce an approximate value for \(\int _ { 3 } ^ { 5 } \log _ { 10 } \sqrt { 2 + x } \mathrm {~d} x\), showing your method clearly.
OCR C2 2010 January Q5
7 marks Standard +0.3
5
\includegraphics[max width=\textwidth, alt={}, center]{9362eb16-88c9-4279-97aa-907b4916b965-3_646_839_255_653} The diagram shows parts of the curves \(y = x ^ { 2 } + 1\) and \(y = 11 - \frac { 9 } { x ^ { 2 } }\), which intersect at \(( 1,2 )\) and \(( 3,10 )\). Use integration to find the exact area of the shaded region enclosed between the two curves.
OCR C2 2010 January Q6
9 marks Moderate -0.3
6 The cubic polynomial \(\mathrm { f } ( x )\) is given by $$\mathrm { f } ( x ) = 2 x ^ { 3 } + a x ^ { 2 } + b x + 15$$ where \(a\) and \(b\) are constants. It is given that ( \(x + 3\) ) is a factor of \(\mathrm { f } ( x )\) and that, when \(\mathrm { f } ( x )\) is divided by ( \(x - 2\) ), the remainder is 35 .
  1. Find the values of \(a\) and \(b\).
  2. Using these values of \(a\) and \(b\), divide \(\mathrm { f } ( x )\) by ( \(x + 3\) ).
OCR C2 2010 January Q7
10 marks Standard +0.3
7
\includegraphics[max width=\textwidth, alt={}, center]{9362eb16-88c9-4279-97aa-907b4916b965-3_469_673_1720_737} The diagram shows triangle \(A B C\), with \(A B = 10 \mathrm {~cm} , B C = 13 \mathrm {~cm}\) and \(C A = 14 \mathrm {~cm} . E\) and \(F\) are points on \(A B\) and \(A C\) respectively such that \(A E = A F = 4 \mathrm {~cm}\). The sector \(A E F\) of a circle with centre \(A\) is removed to leave the shaded region \(E B C F\).
  1. Show that angle \(C A B\) is 1.10 radians, correct to 3 significant figures.
  2. Find the perimeter of the shaded region \(E B C F\).
  3. Find the area of the shaded region \(E B C F\).
OCR C2 2010 January Q8
10 marks Moderate -0.8
8 A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { 1 } = 8 \quad \text { and } \quad u _ { n + 1 } = u _ { n } + 3 .$$
  1. Show that \(u _ { 5 } = 20\).
  2. The \(n\)th term of the sequence can be written in the form \(u _ { n } = p n + q\). State the values of \(p\) and \(q\).
  3. State what type of sequence it is.
  4. Find the value of \(N\) such that \(\sum _ { n = 1 } ^ { 2 N } u _ { n } - \sum _ { n = 1 } ^ { N } u _ { n } = 1256\).
OCR C2 2010 January Q9
11 marks Standard +0.3
9
  1. Sketch the curve \(y = 6 \times 5 ^ { x }\), stating the coordinates of any points of intersection with the axes.
  2. The point \(P\) on the curve \(y = 9 ^ { x }\) has \(y\)-coordinate equal to 150 . Use logarithms to find the \(x\)-coordinate of \(P\), correct to 3 significant figures.
  3. The curves \(y = 6 \times 5 ^ { x }\) and \(y = 9 ^ { x }\) intersect at the point \(Q\). Show that the \(x\)-coordinate of \(Q\) can be written as \(x = \frac { 1 + \log _ { 3 } 2 } { 2 - \log _ { 3 } 5 }\).
OCR C2 2011 January Q1
6 marks Moderate -0.8
1
  1. Find and simplify the first three terms, in ascending powers of \(x\), in the binomial expansion of \(( 1 + 2 x ) ^ { 7 }\).
  2. Hence find the coefficient of \(x ^ { 2 }\) in the expansion of \(( 2 - 5 x ) ( 1 + 2 x ) ^ { 7 }\).
OCR C2 2011 January Q2
6 marks Moderate -0.8
2 A sequence \(S\) has terms \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) defined by \(u _ { n } = 3 n + 2\) for \(n \geqslant 1\).
  1. Write down the values of \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\).
  2. State what type of sequence \(S\) is.
  3. Find \(\sum _ { n = 101 } ^ { 200 } u _ { n }\).
OCR C2 2011 January Q3
6 marks Moderate -0.8
3
\includegraphics[max width=\textwidth, alt={}, center]{c52fe7e9-0442-4b3e-b924-2e5e4b3e98f5-02_510_791_991_678} The diagram shows the curve \(y = \sqrt { x - 3 }\).
  1. Use the trapezium rule, with 4 strips each of width 0.5 , to find an approximate value for the area of the region bounded by the curve, the \(x\)-axis and the line \(x = 5\). Give your answer correct to 3 significant figures.
  2. State, with a reason, whether this approximation is an underestimate or an overestimate.
OCR C2 2011 January Q4
8 marks Moderate -0.8
4
  1. Use logarithms to solve the equation \(5 ^ { x - 1 } = 120\), giving your answer correct to 3 significant figures.
  2. Solve the equation \(\log _ { 2 } x + 2 \log _ { 2 } 3 = \log _ { 2 } ( x + 5 )\).
OCR C2 2011 January Q5
8 marks Moderate -0.3
5 In a geometric progression, the sum to infinity is four times the first term.
  1. Show that the common ratio is \(\frac { 3 } { 4 }\).
  2. Given that the third term is 9 , find the first term.
  3. Find the sum of the first twenty terms.
OCR C2 2011 January Q6
8 marks Moderate -0.3
6
  1. Find \(\int \frac { x ^ { 3 } + 3 x ^ { \frac { 1 } { 2 } } } { x } \mathrm {~d} x\).
    1. Find, in terms of \(a\), the value of \(\int _ { 2 } ^ { a } 6 x ^ { - 4 } \mathrm {~d} x\), where \(a\) is a constant greater than 2 .
    2. Deduce the value of \(\int _ { 2 } ^ { \infty } 6 x ^ { - 4 } \mathrm {~d} x\).
OCR C2 2011 January Q7
8 marks Moderate -0.3
7 Solve each of the following equations for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
  1. \(3 \tan 2 x = 1\)
  2. \(3 \cos ^ { 2 } x + 2 \sin x - 3 = 0\)