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 Q7
7. The second and third terms of a geometric series are \(\log _ { 3 } 4\) and \(\log _ { 3 } 16\) respectively.
  1. Find the common ratio of the series.
  2. Show that the first term of the series is \(\log _ { 3 } 2\).
  3. Find, to 3 significant figures, the sum of the first six terms of the series.
OCR C2 Q8
8. (i) Find, to 2 decimal places, the values of \(x\) in the interval \(0 \leq x < \pi\) for which $$\tan 2 x = 3$$ (ii) Find, in terms of \(\pi\), the values of \(y\) in the interval \(0 \leq y < 2 \pi\) for which $$2 \sin y = \tan y$$
OCR C2 Q9
9.
\includegraphics[max width=\textwidth, alt={}]{0744b3cf-2941-45cb-b6df-2aaf44588e5c-3_592_771_683_541}
The diagram shows the curve \(C\) with equation \(y = 3 x - 4 \sqrt { x } + 2\) and the tangent to \(C\) at the point \(A\). Given that \(A\) has \(x\)-coordinate 4,
  1. show that the tangent to \(C\) at \(A\) has the equation \(y = 2 x - 2\). The shaded region is bounded by \(C\), the tangent to \(C\) at \(A\) and the \(y\)-axis.
  2. Find the area of the shaded region.
OCR C2 Q1
  1. (i) Sketch on the same diagram the graphs of \(y = \sin 2 x\) and \(y = \tan \frac { x } { 2 }\) for \(x\) in the interval \(0 \leq x \leq 360 ^ { \circ }\).
    (ii) Hence state how many solutions exist to the equation
$$\sin 2 x = \tan \frac { x } { 2 } ,$$ for \(x\) in the interval \(0 \leq x \leq 360 ^ { \circ }\) and give a reason for your answer.
OCR C2 Q2
2.
\includegraphics[max width=\textwidth, alt={}, center]{e5d62032-84ad-4e0b-9b72-ccfd8f4dbac8-1_588_513_813_593} The diagram shows a circle of radius \(r\) and centre \(O\) in which \(A D\) is a diameter.
The points \(B\) and \(C\) lie on the circle such that \(O B\) and \(O C\) are arcs of circles of radius \(r\) with centres \(A\) and \(D\) respectively. Show that the area of the shaded region \(O B C\) is \(\frac { 1 } { 6 } r ^ { 2 } ( 3 \sqrt { 3 } - \pi )\).
OCR C2 Q3
3. The sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { n + 1 } = \left( u _ { n } \right) ^ { 2 } - 1 , \quad n \geq 1 .$$ Given that \(u _ { 1 } = k\), where \(k\) is a constant,
  1. find expressions for \(u _ { 2 }\) and \(u _ { 3 }\) in terms of \(k\). Given also that \(u _ { 2 } + u _ { 3 } = 11\),
  2. find the possible values of \(k\).
OCR C2 Q4
4.
\includegraphics[max width=\textwidth, alt={}, center]{e5d62032-84ad-4e0b-9b72-ccfd8f4dbac8-2_465_844_246_516} The diagram shows the curve with equation \(y = \frac { 1 } { x ^ { 2 } + 1 }\).
The shaded region \(R\) is bounded by the curve, the coordinate axes and the line \(x = 2\).
  1. Use the trapezium rule with four strips of equal width to estimate the area of \(R\). The cross-section of a support for a bookshelf is modelled by \(R\) with 1 unit on each axis representing 8 cm . Given that the support is 2 cm thick,
  2. find an estimate for the volume of the support.
OCR C2 Q5
5. (i) Find the value of \(a\) such that $$\log _ { a } 27 = 3 + \log _ { a } 8$$ (ii) Solve the equation $$2 ^ { x + 3 } = 6 ^ { x - 1 }$$ giving your answer to 3 significant figures.
OCR C2 Q6
6. (i) Evaluate $$\int _ { 2 } ^ { 4 } \left( 2 - \frac { 1 } { x ^ { 2 } } \right) \mathrm { d } x$$ (ii) Given that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 x ^ { 3 } + 1$$ and that \(y = 3\) when \(x = 0\), find the value of \(y\) when \(x = 2\).
OCR C2 Q7
7.
\includegraphics[max width=\textwidth, alt={}, center]{e5d62032-84ad-4e0b-9b72-ccfd8f4dbac8-3_499_721_248_552} The diagram shows part of the curve \(y = \mathrm { f } ( x )\) where \(\mathrm { f } ( x ) = \frac { 1 - 8 x ^ { 3 } } { x ^ { 2 } } , x \neq 0\).
  1. Solve the equation \(\mathrm { f } ( x ) = 0\).
  2. Find \(\int \mathrm { f } ( x ) \mathrm { d } x\).
  3. Find the area of the shaded region bounded by the curve \(y = \mathrm { f } ( x )\), the \(x\)-axis and the line \(x = 2\).
OCR C2 Q8
8. A store begins to stock a new range of DVD players and achieves sales of \(\pounds 1500\) of these products during the first month. In a model it is assumed that sales will decrease by \(\pounds x\) in each subsequent month, forming an arithmetic sequence. Given that sales total \(\pounds 8100\) during the first six months, use the model to
  1. find the value of \(x\),
  2. find the expected value of sales in the eighth month,
  3. show that the expected total of sales in pounds during the first \(n\) months is given by \(k n ( 51 - n )\), where \(k\) is an integer to be found.
  4. Explain why this model cannot be valid over a long period of time.
OCR C2 Q9
9. \(f ( x ) = 2 x ^ { 3 } - 5 x ^ { 2 } + x + 2\).
  1. Show that \(( x - 2 )\) is a factor of \(\mathrm { f } ( x )\).
  2. Fully factorise \(\mathrm { f } ( x )\).
  3. Solve the equation \(\mathrm { f } ( x ) = 0\).
  4. Find, in terms of \(\pi\), the values of \(\theta\) in the interval \(0 \leq \theta \leq 2 \pi\) for which $$2 \sin ^ { 3 } \theta - 5 \sin ^ { 2 } \theta + \sin \theta + 2 = 0$$
OCR C2 2009 January Q1
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
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
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
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
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
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
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
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
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
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
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
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
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.