Questions — OCR C2 (296 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 2011 June Q1
1 The diagram shows triangle \(A B C\), with \(A B = 9 \mathrm {~cm} , A C = 17 \mathrm {~cm}\) and angle \(B A C = 40 ^ { \circ }\).
  1. Find the length of \(B C\).
  2. Find the area of triangle \(A B C\).
  3. \(D\) is the point on \(A C\) such that angle \(B D A = 63 ^ { \circ }\). Find the length of \(B D\).
OCR C2 2011 June Q2
2
  1. Find \(\int \left( 6 x ^ { \frac { 1 } { 2 } } - 1 \right) \mathrm { d } x\).
  2. Hence find the equation of the curve for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x ^ { \frac { 1 } { 2 } } - 1\) and which passes through the point \(( 4,17 )\).
OCR C2 2011 June Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{4d03f4e3-ae6c-4a0d-ae4d-d89258d2919a-2_515_501_1439_822} The diagram shows a sector \(A O B\) of a circle, centre \(O\) and radius 8 cm . The perimeter of the sector is 23.2 cm .
  1. Find angle \(A O B\) in radians.
  2. Find the area of the sector.
OCR C2 2011 June Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{4d03f4e3-ae6c-4a0d-ae4d-d89258d2919a-3_588_1136_255_502} The diagram shows the curve \(y = - 1 + \sqrt { x + 4 }\) and the line \(y = 3\).
  1. Show that \(y = - 1 + \sqrt { x + 4 }\) can be rearranged as \(x = y ^ { 2 } + 2 y - 3\).
  2. Hence find by integration the exact area of the shaded region enclosed between the curve, the \(y\)-axis and the line \(y = 3\).
OCR C2 2011 June Q5
5 The first four terms in the binomial expansion of \(( 3 + k x ) ^ { 5 }\), in ascending powers of \(x\), can be written as \(a + b x + c x ^ { 2 } + d x ^ { 3 }\).
  1. State the value of \(a\).
  2. Given that \(b = c\), find the value of \(k\).
  3. Hence find the value of \(d\).
OCR C2 2011 June Q6
6 The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = x ^ { 3 } + x ^ { 2 } - 11 x + 10\).
  1. Use the factor theorem to find a factor of \(\mathrm { f } ( x )\).
  2. Hence solve the equation \(\mathrm { f } ( x ) = 0\), giving each root in an exact form.
OCR C2 2011 June Q7
7
  1. The first term of a geometric progression is 7 and the common ratio is - 2 .
    1. Find the ninth term.
    2. Find the sum of the first 15 terms.
  2. The first term of an arithmetic progression is 7 and the common difference is - 2 . The sum of the first \(N\) terms is - 2900 . Find the value of \(N\).
OCR C2 2011 June Q8
8
\includegraphics[max width=\textwidth, alt={}, center]{4d03f4e3-ae6c-4a0d-ae4d-d89258d2919a-4_417_931_255_607} The diagram shows the curve \(y = 2 ^ { x } - 3\).
  1. Describe the geometrical transformation that transforms the curve \(y = 2 ^ { x }\) to the curve \(y = 2 ^ { x } - 3\).
  2. State the \(y\)-coordinate of the point where the curve \(y = 2 ^ { x } - 3\) crosses the \(y\)-axis.
  3. Find the \(x\)-coordinate of the point where the curve \(y = 2 ^ { x } - 3\) crosses the \(x\)-axis, giving your answer in the form \(\log _ { a } b\).
  4. The curve \(y = 2 ^ { x } - 3\) passes through the point ( \(p , 62\) ). Use logarithms to find the value of \(p\), correct to 3 significant figures.
  5. Use the trapezium rule, with 2 strips each of width 0.5 , to find an estimate for \(\int _ { 3 } ^ { 4 } \left( 2 ^ { x } - 3 \right) \mathrm { d } x\). Give your answer correct to 3 significant figures.
OCR C2 2011 June Q9
9

  1. \includegraphics[max width=\textwidth, alt={}, center]{4d03f4e3-ae6c-4a0d-ae4d-d89258d2919a-4_362_979_1505_625} The diagram shows part of the curve \(y = \cos 2 x\), where \(x\) is in radians. The point \(A\) is the minimum point of this part of the curve.
    1. State the period of \(y = \cos 2 x\).
    2. State the coordinates of \(A\).
    3. Solve the inequality \(\cos 2 x \leqslant 0.5\) for \(0 \leqslant x \leqslant \pi\), giving your answers exactly.
  2. Solve the equation \(\cos 2 x = \sqrt { 3 } \sin 2 x\) for \(0 \leqslant x \leqslant \pi\), giving your answers exactly.
OCR C2 2012 June Q1
1
  1. Find the binomial expansion of \(( 3 + 2 x ) ^ { 5 }\), simplifying the terms.
  2. Hence find the binomial expansion of \(( 3 + 2 x ) ^ { 5 } + ( 3 - 2 x ) ^ { 5 }\).
OCR C2 2012 June Q2
2
  1. Find \(\int \left( x ^ { 2 } - 2 x + 5 \right) \mathrm { d } x\).
  2. Hence find the equation of the curve for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { 2 } - 2 x + 5\) and which passes through the point \(( 3,11 )\).
OCR C2 2012 June Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{f25e2580-ba0b-42ce-bf86-63f2c2075223-2_436_526_762_767} The diagram shows a sector \(A O B\) of a circle, centre \(O\) and radius \(r \mathrm {~cm}\). Angle \(A O B\) is \(72 ^ { \circ }\).
  1. Express \(72 ^ { \circ }\) exactly in radians, simplifying your answer. The area of the sector \(A O B\) is \(45 \pi \mathrm {~cm} ^ { 2 }\).
  2. Find the value of \(r\).
  3. Find the area of the segment bounded by the arc \(A B\) and the chord \(A B\), giving your answer correct to 3 significant figures.
OCR C2 2012 June Q4
4 Solve the equation $$4 \cos ^ { 2 } x + 7 \sin x - 7 = 0$$ giving all values of \(x\) between \(0 ^ { \circ }\) and \(360 ^ { \circ }\).
OCR C2 2012 June Q5
5
  1. A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { 1 } = 4 \quad \text { and } \quad u _ { n + 1 } = \frac { 2 } { u _ { n } } \quad \text { for } n \geqslant 1 .$$
    1. Write down the values of \(u _ { 2 }\) and \(u _ { 3 }\).
    2. Describe the behaviour of the sequence.
  2. In an arithmetic progression the ninth term is 18 and the sum of the first nine terms is 72. Find the first term and the common difference.
OCR C2 2012 June Q6
6
  1. Use the trapezium rule, with 2 strips each of width 4 , to show that an approximate value of \(\int _ { 1 } ^ { 9 } 4 \sqrt { x } \mathrm {~d} x\) is \(32 + 16 \sqrt { 5 }\).
  2. Use a sketch graph to explain why the actual value of \(\int _ { 1 } ^ { 9 } 4 \sqrt { x } \mathrm {~d} x\) is greater than \(32 + 16 \sqrt { 5 }\).
  3. Use integration to find the exact value of \(\int _ { 1 } ^ { 9 } 4 \sqrt { x } \mathrm {~d} x\).
OCR C2 2012 June Q7
7
    1. Given that \(\alpha\) is the acute angle such that \(\tan \alpha = \frac { 2 } { 5 }\), find the exact value of \(\cos \alpha\).
    2. Given that \(\beta\) is the obtuse angle such that \(\sin \beta = \frac { 3 } { 7 }\), find the exact value of \(\cos \beta\).

  2. \includegraphics[max width=\textwidth, alt={}, center]{f25e2580-ba0b-42ce-bf86-63f2c2075223-3_316_662_955_700} The diagram shows a triangle \(A B C\) with \(A C = 6 \mathrm {~cm} , B C = 8 \mathrm {~cm}\), angle \(B A C = 60 ^ { \circ }\) and angle \(A B C = \gamma\). Find the exact value of \(\sin \gamma\), simplifying your answer.
OCR C2 2012 June Q8
8 Two cubic polynomials are defined by $$\mathrm { f } ( x ) = x ^ { 3 } + ( a - 3 ) x + 2 b , \quad \mathrm {~g} ( x ) = 3 x ^ { 3 } + x ^ { 2 } + 5 a x + 4 b$$ where \(a\) and \(b\) are constants.
  1. Given that \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) have a common factor of ( \(x - 2\) ), show that \(a = - 4\) and find the value of \(b\).
  2. Using these values of \(a\) and \(b\), factorise \(\mathrm { f } ( x )\) fully. Hence show that \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) have two common factors.
OCR C2 2012 June Q9
9
  1. An arithmetic progression has first term \(\log _ { 2 } 27\) and common difference \(\log _ { 2 } x\).
    1. Show that the fourth term can be written as \(\log _ { 2 } \left( 27 x ^ { 3 } \right)\).
    2. Given that the fourth term is 6, find the exact value of \(x\).
  2. A geometric progression has first term \(\log _ { 2 } 27\) and common ratio \(\log _ { 2 } y\).
    1. Find the set of values of \(y\) for which the geometric progression has a sum to infinity.
    2. Find the exact value of \(y\) for which the sum to infinity of the geometric progression is 3 . \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.}
OCR C2 2013 June Q1
1 Use the trapezium rule, with 3 strips each of width 2 , to estimate the value of $$\int _ { 5 } ^ { 11 } \frac { 8 } { x } \mathrm {~d} x .$$
OCR C2 2013 June Q2
2 Solve each of the following equations, for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
  1. \(\sin \frac { 1 } { 2 } x = 0.8\)
  2. \(\sin x = 3 \cos x\)
OCR C2 2013 June Q3
3
  1. Find and simplify the first three terms in the expansion of \(( 2 + 5 x ) ^ { 6 }\) in ascending powers of \(x\).
  2. In the expansion of \(( 3 + c x ) ^ { 2 } ( 2 + 5 x ) ^ { 6 }\), the coefficient of \(x\) is 4416. Find the value of \(c\).
OCR C2 2013 June Q4
4
  1. Find \(\int \left( 5 x ^ { 3 } - 6 x + 1 \right) \mathrm { d } x\).
    1. Find \(\int 24 x ^ { - 3 } \mathrm {~d} x\).
    2. Given that \(\int _ { a } ^ { \infty } 24 x ^ { - 3 } \mathrm {~d} x = 3\), find the value of the positive constant \(a\).
OCR C2 2013 June Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{b2c1188d-881e-4fb5-bece-5a51006543c7-2_405_688_1535_685} The diagram shows a sector \(B A C\) of a circle with centre \(A\) and radius 16 cm . The angle \(B A C\) is 0.8 radians. The length \(A D\) is 7 cm .
  1. Find the area of the region \(B D C\).
  2. Find the perimeter of the region \(B D C\).
OCR C2 2013 June Q6
6 Sarah is carrying out a series of experiments which involve using increasing amounts of a chemical. In the first experiment she uses 6 g of the chemical and in the second experiment she uses 7.8 g of the chemical.
  1. Given that the amounts of the chemical used form an arithmetic progression, find the total amount of chemical used in the first 30 experiments.
  2. Instead it is given that the amounts of the chemical used form a geometric progression. Sarah has a total of 1800 g of the chemical available. Show that \(N\), the greatest number of experiments possible, satisfies the inequality $$1.3 ^ { N } \leqslant 91 ,$$ and use logarithms to calculate the value of \(N\).
OCR C2 2013 June Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{b2c1188d-881e-4fb5-bece-5a51006543c7-3_519_611_1087_712} The diagram shows the curve \(y = x ^ { \frac { 3 } { 2 } } - 1\), which crosses the \(x\)-axis at \(( 1,0 )\), and the tangent to the curve at the point \(( 4,7 )\).
  1. Show that \(\int _ { 1 } ^ { 4 } \left( x ^ { \frac { 3 } { 2 } } - 1 \right) \mathrm { d } x = 9 \frac { 2 } { 5 }\).
  2. Hence find the exact area of the shaded region enclosed by the curve, the tangent and the \(x\)-axis.