Questions — OCR (4628 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 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 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 FP2 2013 June Q1
5 marks Standard +0.8
1 By using the substitution \(t = \tan \frac { 1 } { 2 } \theta\), find \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \frac { 1 } { 1 + \cos \theta } \mathrm { d } \theta\).
OCR FP2 2013 June Q2
8 marks Standard +0.3
2
  1. Using the definitions for \(\cosh x\) and \(\sinh x\) in terms of \(\mathrm { e } ^ { x }\) and \(\mathrm { e } ^ { - x }\), show that \(\cosh ^ { 2 } x - \sinh ^ { 2 } x \equiv 1\).
  2. Hence solve the equation \(\sinh ^ { 2 } x = 5 \cosh x - 7\), giving your answers in logarithmic form.
OCR FP2 2013 June Q3
10 marks Challenging +1.2
3 It is given that \(\mathrm { f } ( x ) = \tanh ^ { - 1 } \left( \frac { 1 - x } { 3 + x } \right)\) for \(x > - 1\).
  1. Show that \(\mathrm { f } ^ { \prime \prime } ( x ) = \frac { 1 } { 2 ( x + 1 ) ^ { 2 } }\).
  2. Hence find the Maclaurin series for \(\mathrm { f } ( x )\) up to and including the term in \(x ^ { 2 }\).
OCR FP2 2013 June Q4
8 marks Challenging +1.2
4 It is given that \(I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \cos ^ { n } x \mathrm {~d} x\) for \(n \geqslant 0\).
  1. Show that \(I _ { n } = \frac { n - 1 } { n } I _ { n - 2 }\) for \(n \geqslant 2\).
  2. Hence find \(I _ { 11 }\) as an exact fraction.
OCR FP2 2013 June Q5
8 marks Standard +0.3
5 You are given that the equation \(x ^ { 3 } + 4 x ^ { 2 } + x - 1 = 0\) has a root, \(\alpha\), where \(- 1 < \alpha < 0\).
  1. Show that the Newton-Raphson iterative formula for this equation can be written in the form $$x _ { n + 1 } = \frac { 2 x _ { n } ^ { 3 } + 4 x _ { n } ^ { 2 } + 1 } { 3 x _ { n } ^ { 2 } + 8 x _ { n } + 1 } .$$
  2. Using the initial value \(x _ { 1 } = - 0.7\), find \(x _ { 2 }\) and \(x _ { 3 }\) and find \(\alpha\) correct to 5 decimal places.
  3. The diagram shows a sketch of the curve \(y = x ^ { 3 } + 4 x ^ { 2 } + x - 1\) for \(- 1.5 \leqslant x \leqslant 1\). \includegraphics[max width=\textwidth, alt={}, center]{a80eb21f-c273-4b65-8617-16cdee783305-3_602_926_749_566} Using the copy of the diagram in your answer book, explain why the initial value \(x _ { 1 } = 0\) will fail to find \(\alpha\).
OCR FP2 2013 June Q6
8 marks Challenging +1.2
6 \includegraphics[max width=\textwidth, alt={}, center]{a80eb21f-c273-4b65-8617-16cdee783305-4_656_1017_251_525} The diagram shows part of the curve \(y = \ln ( \ln ( x ) )\). The region between the curve and the \(x\)-axis for \(3 \leqslant x \leqslant 6\) is shaded.
  1. By considering \(n\) rectangles of equal width, show that a lower bound, \(L\), for the area of the shaded region is \(\frac { 3 } { n } \sum _ { r = 0 } ^ { n - 1 } \ln \left( \ln \left( 3 + \frac { 3 r } { n } \right) \right)\).
  2. By considering another set of \(n\) rectangles of equal width, find a similar expression for an upper bound, \(U\), for the area of the shaded region.
  3. Find the least value of \(n\) for which \(U - L < 0.001\).
OCR FP2 2013 June Q7
14 marks Challenging +1.2
7 The equation of a curve is \(y = \frac { x ^ { 2 } + 1 } { ( x + 1 ) ( x - 7 ) }\).
  1. Write down the equations of the asymptotes.
  2. Find the coordinates of the stationary points on the curve.
  3. Find the coordinates of the point where the curve meets one of its asymptotes.
  4. Sketch the curve.
OCR FP2 2013 June Q8
11 marks Challenging +1.2
8 The equation of a curve is \(x ^ { 2 } + y ^ { 2 } - x = \sqrt { x ^ { 2 } + y ^ { 2 } }\).
  1. Find the polar equation of this curve in the form \(r = \mathrm { f } ( \theta )\).
  2. Sketch the curve.
  3. The line \(x + 2 y = 2\) divides the region enclosed by the curve into two parts. Find the ratio of the two areas.
OCR FP2 2014 June Q1
3 marks Standard +0.3
1 Find \(\int _ { 0 } ^ { 2 } \frac { 1 } { \sqrt { 4 + x ^ { 2 } } } \mathrm {~d} x\), giving your answer exactly in logarithmic form.
OCR FP2 2014 June Q2
5 marks Moderate -0.3
2 It is given that \(\mathrm { f } ( x ) = \ln \left( 1 + x ^ { 2 } \right)\).
  1. Using the standard Maclaurin expansion for \(\ln ( 1 + x )\), write down the first four terms in the expansion of \(\mathrm { f } ( x )\), stating the set of values of \(x\) for which the expansion is valid.
  2. Hence find the exact value of $$1 - \frac { 1 } { 2 } \left( \frac { 1 } { 2 } \right) ^ { 2 } + \frac { 1 } { 3 } \left( \frac { 1 } { 2 } \right) ^ { 4 } - \frac { 1 } { 4 } \left( \frac { 1 } { 2 } \right) ^ { 6 } + \ldots .$$
OCR FP2 2014 June Q3
7 marks Challenging +1.2
3 The diagram shows the curve \(y = \frac { 1 } { x ^ { 3 } }\) for \(1 \leqslant x \leqslant n\) where \(n\) is an integer. A set of ( \(n - 1\) ) rectangles of unit width is drawn under the curve. \includegraphics[max width=\textwidth, alt={}, center]{736932f1-4007-4a04-a08b-2551db0b136c-2_611_947_1103_557}
  1. Write down the sum of the areas of the rectangles.
  2. Hence show that \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { r ^ { 3 } } < \frac { 3 } { 2 }\).
OCR FP2 2014 June Q4
6 marks Standard +0.8
4 The curves \(y = \cos ^ { - 1 } x\) and \(y = \tan ^ { - 1 } ( \sqrt { 2 } x )\) intersect at a point \(A\).
  1. Verify that the coordinates of \(A\) are \(\left( \frac { 1 } { \sqrt { 2 } } , \frac { 1 } { 4 } \pi \right)\).
  2. Determine whether the tangents to the curves at \(A\) are perpendicular.
OCR FP2 2014 June Q5
9 marks Standard +0.8
5 A curve has equation \(y = \frac { x ^ { 2 } - 8 } { x - 3 }\).
  1. Find the equations of the asymptotes of the curve.
  2. Prove that there are no points on the curve for which \(4 < y < 8\).
  3. Sketch the curve. Indicate the asymptotes in your sketch.
OCR FP2 2014 June Q6
9 marks Standard +0.3
6
  1. Given that \(y = \cosh ^ { - 1 } x\), show that \(y = \ln \left( x + \sqrt { x ^ { 2 } - 1 } \right)\).
  2. Show that \(\frac { \mathrm { d } } { \mathrm { d } x } \left( \cosh ^ { - 1 } x \right) = \frac { 1 } { \sqrt { x ^ { 2 } - 1 } }\).
  3. Solve the equation \(\cosh x = 3\), giving your answers in logarithmic form.
OCR FP2 2014 June Q7
11 marks Challenging +1.8
7 It is given that, for non-negative integers \(n , I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sin ^ { n } x \mathrm {~d} x\).
  1. Show that \(I _ { n } = \frac { n - 1 } { n } I _ { n - 2 }\) for \(n \geqslant 2\).
  2. Explain why \(I _ { 2 n + 1 } < I _ { 2 n - 1 }\).
  3. It is given that \(I _ { 2 n + 1 } < I _ { 2 n } < I _ { 2 n - 1 }\). Take \(n = 5\) to find an interval within which the value of \(\pi\) lies.
OCR FP2 2014 June Q8
10 marks Standard +0.8
8 A curve has polar equation \(r = a ( 1 + \cos \theta )\), where \(a\) is a positive constant and \(0 \leqslant \theta < 2 \pi\).
  1. Find the equation of the tangent at the pole.
  2. Sketch the curve.
  3. Find the area enclosed by the curve.
OCR FP2 2014 June Q9
12 marks Standard +0.3
9 The equation \(10 x - 8 \ln x = 28\) has a root \(\alpha\) in the interval [3,4]. The iteration \(x _ { n + 1 } = \mathrm { g } \left( x _ { n } \right)\), where \(\mathrm { g } ( x ) = 2.8 + 0.8 \ln x\) and \(x _ { 1 } = 3.8\), is to be used to find \(\alpha\).
  1. Find the value of \(\alpha\) correct to 5 decimal places. You should show the result of each step of the iteration to 6 decimal places.
  2. Illustrate this iteration by means of a sketch.
  3. The difference, \(\delta _ { r }\), between successive approximations is given by \(\delta _ { r } = x _ { r + 1 } - x _ { r }\). Find \(\delta _ { 3 }\).
  4. Given that \(\delta _ { n + 1 } \approx \mathrm {~g} ^ { \prime } ( \alpha ) \delta _ { n }\), for all positive integers \(n\), estimate the smallest value of \(n\) such that \(\delta _ { n } < 10 ^ { - 6 } \delta _ { 1 }\). \section*{OCR}
OCR FP2 2015 June Q1
3 marks Standard +0.8
1 By first expressing \(\tanh y\) in terms of exponentials, prove that \(\tanh ^ { - 1 } x = \frac { 1 } { 2 } \ln \left( \frac { 1 + x } { 1 - x } \right)\).
OCR FP2 2015 June Q2
4 marks Standard +0.3
2 It is given that \(\mathrm { f } ( x ) = \ln ( 1 + \sin x )\). Using standard series, find the Maclaurin series for \(\mathrm { f } ( x )\) up to and including the term in \(x ^ { 3 }\).
OCR FP2 2015 June Q3
5 marks Standard +0.3
3 By first completing the square, find the exact value of \(\int _ { \frac { 1 } { 2 } } ^ { 1 } \frac { 1 } { \sqrt { 2 x - x ^ { 2 } } } \mathrm {~d} x\).
OCR FP2 2015 June Q4
9 marks Challenging +1.2
4 It is given that \(I _ { n } = \int _ { 0 } ^ { 1 } x ^ { n } \mathrm { e } ^ { - x } \mathrm {~d} x\) for \(n \geqslant 0\).
  1. Show that \(I _ { n } = n I _ { n - 1 } + k\) for \(n \geqslant 1\), where \(k\) is a constant to be determined.
  2. Find the exact value of \(I _ { 3 }\).
  3. Find the exact value of \(990 I _ { 8 } - I _ { 11 }\).
OCR FP2 2015 June Q5
9 marks Standard +0.8
5 It is given that \(y = \sin ^ { - 1 } 2 x\).
  1. Using the derivative of \(\sin ^ { - 1 } x\) given in the List of Formulae (MF1), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Show that \(\left( 1 - 4 x ^ { 2 } \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 4 x \frac { \mathrm {~d} y } { \mathrm {~d} x }\).
  3. Hence show that \(\left( 1 - 4 x ^ { 2 } \right) \frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } - 12 x \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } = 0\).
  4. Using your results from parts (i), (ii) and (iii), find the Maclaurin series for \(\sin ^ { - 1 } 2 x\) up to and including the term in \(x ^ { 3 }\).
OCR FP2 2015 June Q6
12 marks Challenging +1.8
6 It is given that the equation \(3 x ^ { 3 } + 5 x ^ { 2 } - x - 1 = 0\) has three roots, one of which is positive.
  1. Show that the Newton-Raphson iterative formula for finding this root can be written $$x _ { n + 1 } = \frac { 6 x _ { n } ^ { 3 } + 5 x _ { n } ^ { 2 } + 1 } { 9 x _ { n } ^ { 2 } + 10 x _ { n } - 1 } .$$
  2. A sequence of iterates \(x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots\) which will find the positive root is such that the magnitude of the error in \(x _ { 2 }\) is greater than the magnitude of the error in \(x _ { 1 }\). On the graph given in the Printed Answer Book, mark a possible position for \(x _ { 1 }\).
  3. Apply the iterative formula in part (i) when the initial value is \(x _ { 1 } = - 1\). Describe the behaviour of the iterative sequence, illustrating your answer on the graph given in the Printed Answer Book.
  4. A sequence of approximations to the positive root is given by \(x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots\). Successive differences \(x _ { r } - x _ { r - 1 } = d _ { r }\), where \(r \geqslant 2\), are such that \(d _ { r } \approx k \left( d _ { r - 1 } \right) ^ { 2 }\) where \(k\) is a constant. Show that \(d _ { 4 } \approx \frac { d _ { 3 } ^ { 3 } } { d _ { 2 } ^ { 2 } }\) and demonstrate this numerically when \(x _ { 1 } = 1\).
  5. Find the value of the positive root correct to 5 decimal places.
OCR FP2 2015 June Q7
10 marks Challenging +1.2
7 It is given that \(\mathrm { f } ( x ) = \frac { x ^ { 2 } - 25 } { ( x - 1 ) ( x + 2 ) }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Write down the equations of the asymptotes of the curve \(y = \mathrm { f } ( x )\).
  3. Find the value of \(x\) where the graph of \(y = \mathrm { f } ( x )\) cuts the horizontal asymptote.
  4. Sketch the graph of \(y ^ { 2 } = \mathrm { f } ( x )\).
OCR FP2 2015 June Q8
9 marks Standard +0.3
8 It is given that \(\mathrm { f } ( x ) = 2 \sinh x + 3 \cosh x\).
  1. Show that the curve \(y = \mathrm { f } ( x )\) has a stationary point at \(x = - \frac { 1 } { 2 } \ln 5\) and find the value of \(y\) at this point.
  2. Solve the equation \(\mathrm { f } ( x ) = 5\), giving your answers exactly. \section*{Question 9 begins on page 4.}