Questions — OCR MEI (4301 questions)

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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 MEI Further Statistics B AS Specimen Q7
Standard +0.3
7 Two flatmates work at the same location. One of them takes the bus to work and the other one cycles. Journey times, measured in minutes, are distributed as follows.
  • By bus: Normally distributed with mean 23 and standard deviation 6
  • By bicycle: Normally distributed with mean 21 and standard deviation 2
You should assume that all journey times are independent.
  1. One morning the two flatmates set out at the same time. Find the probability that the person who takes the bus arrives before the cyclist.
  2. Find the probability that the total time taken for 5 bus journeys is less than 2 hours.
  3. Comment on the assumption that all journey times are independent. \section*{END OF QUESTION PAPER} OCR is committed to seeking permission to reproduce all third-party content that it uses in the assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements booklet. This is produced for each series of examinations and is freely available to download from our public website (\href{http://www.ocr.org.uk}{www.ocr.org.uk}) after the live examination series. If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity.
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OCR MEI Further Pure Core 2019 June Q1
Easy -1.2
1 Find \(\sum _ { r = 1 } ^ { n } \left( 2 r ^ { 2 } - 1 \right)\), expressing your answer in fully factorised form.
OCR MEI Further Pure Core 2019 June Q2
Moderate -0.5
2 The plane \(x + 2 y + c z = 4\) is perpendicular to the plane \(2 x - c y + 6 z = 9\), where \(c\) is a constant. Find the value of \(c\).
OCR MEI Further Pure Core 2019 June Q3
Moderate -0.8
3 Matrices \(\mathbf { A }\) and \(\mathbf { B }\) are defined by \(\mathbf { A } = \left( \begin{array} { l l } 3 & 1 \\ 2 & 1 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { l l } k & 1 \\ 2 & 0 \end{array} \right)\), where \(k\) is a constant.
  1. Verify the result \(( \mathbf { A B } ) ^ { - 1 } = \mathbf { B } ^ { - 1 } \mathbf { A } ^ { - 1 }\) in this case.
  2. Investigate whether \(\mathbf { A }\) and \(\mathbf { B }\) are commutative under matrix multiplication.
OCR MEI Further Pure Core 2019 June Q5
Standard +0.3
5 Using the Maclaurin series for \(\cos 2 x\), show that, for small values of \(x\), \(\sin ^ { 2 } x \approx a x ^ { 2 } + b x ^ { 4 } + c x ^ { 6 }\),
where the values of \(a , b\) and \(c\) are to be given in exact form.
OCR MEI Further Pure Core 2019 June Q6
Standard +0.8
6 In this question you must show detailed reasoning.
Find \(\int _ { 2 } ^ { \infty } \frac { 1 } { 4 + x ^ { 2 } } \mathrm {~d} x\).
OCR MEI Further Pure Core 2019 June Q7
Standard +0.3
7 A curve has cartesian equation \(\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 2 c ^ { 2 } x y\), where \(c\) is a positive constant.
  1. Show that the polar equation of the curve is \(r ^ { 2 } = c ^ { 2 } \sin 2 \theta\).
  2. Sketch the curves \(r = c \sqrt { \sin 2 \theta }\) and \(r = - c \sqrt { \sin 2 \theta }\) for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
  3. Find the area of the region enclosed by one of the loops in part (b). Section B (110 marks)
    Answer all the questions.
OCR MEI Further Pure Core 2019 June Q9
Moderate -0.3
9 Prove by induction that \(5 ^ { n } + 2 \times 11 ^ { n }\) is divisible by 3 for all positive integers \(n\).
OCR MEI Further Pure Core 2019 June Q11
Standard +0.3
11
  1. Specify fully the transformations represented by the following matrices.
    • \(\mathbf { M } _ { 1 } = \left( \begin{array} { r r } \frac { 3 } { 5 } & - \frac { 4 } { 5 } \\ \frac { 4 } { 5 } & \frac { 3 } { 5 } \end{array} \right)\)
    • \(\mathbf { M } _ { 2 } = \left( \begin{array} { r r } 1 & 0 \\ 0 & - 1 \end{array} \right)\)
    • Find the equation of the mirror line of the reflection R represented by the matrix \(\mathbf { M } _ { 3 } = \mathbf { M } _ { 1 } \mathbf { M } _ { 2 }\).
    • It is claimed that the reflection represented by the matrix \(\mathbf { M } _ { 4 } = \mathbf { M } _ { 2 } \mathbf { M } _ { 1 }\) has the same mirror line as R . Explain whether or not this claim is correct.
OCR MEI Further Pure Core 2019 June Q12
Challenging +1.2
12 Three intersecting lines \(L _ { 1 } , L _ { 2 }\) and \(L _ { 3 }\) have equations
\(L _ { 1 } : \frac { x } { 2 } = \frac { y } { 3 } = \frac { z } { 1 } , \quad L _ { 2 } : \frac { x } { 1 } = \frac { y } { 2 } = \frac { z } { - 4 } \quad\) and \(\quad L _ { 3 } : \frac { x - 1 } { 1 } = \frac { y - 2 } { 1 } = \frac { z + 4 } { 5 }\).
Find the area of the triangle enclosed by these lines.
OCR MEI Further Pure Core 2019 June Q13
Standard +0.8
13
  1. Using the logarithmic form of \(\operatorname { arcosh } x\), prove that the derivative of \(\operatorname { arcosh } x\) is \(\frac { 1 } { \sqrt { x ^ { 2 } - 1 } }\).
  2. Hence find \(\int _ { 1 } ^ { 2 } \operatorname { arcosh } x \mathrm {~d} x\), giving your answer in exact logarithmic form.
  3. Ali tries to evaluate \(\int _ { 0 } ^ { 1 } \operatorname { arcosh } x \mathrm {~d} x\) using his calculator, and gets an 'error'. Explain why.
OCR MEI Further Pure Core 2019 June Q14
Challenging +1.2
14 Three planes have equations $$\begin{aligned} - x + a y & = 2 \\ 2 x + 3 y + z & = - 3 \\ x + b y + z & = c \end{aligned}$$ where \(a\), \(b\) and \(c\) are constants.
  1. In the case where the planes do not intersect at a unique point,
    1. find \(b\) in terms of \(a\),
    2. find the value of \(c\) for which the planes form a sheaf.
  2. In the case where \(b = a\) and \(c = 1\), find the coordinates of the point of intersection of the planes in terms of \(a\).
OCR MEI Further Pure Core 2019 June Q16
Challenging +1.2
16
  1. Show that \(\left( 2 - \mathrm { e } ^ { \mathrm { i } \theta } \right) \left( 2 - \mathrm { e } ^ { - \mathrm { i } \theta } \right) = 5 - 4 \cos \theta\). Series \(C\) and \(S\) are defined by
    \(C = \frac { 1 } { 2 } \cos \theta + \frac { 1 } { 4 } \cos 2 \theta + \frac { 1 } { 8 } \cos 3 \theta + \ldots + \frac { 1 } { 2 ^ { n } } \cos n \theta\),
    \(S = \frac { 1 } { 2 } \sin \theta + \frac { 1 } { 4 } \sin 2 \theta + \frac { 1 } { 8 } \sin 3 \theta + \ldots + \frac { 1 } { 2 ^ { n } } \sin n \theta\).
  2. Show that \(C = \frac { 2 ^ { n } ( 2 \cos \theta - 1 ) - 2 \cos ( n + 1 ) \theta + \cos n \theta } { 2 ^ { n } ( 5 - 4 \cos \theta ) }\).
OCR MEI Further Pure Core 2019 June Q17
17 A cyclist accelerates from rest for 5 seconds then brakes for 5 seconds, coming to rest at the end of the 10 seconds. The total mass of the cycle and rider is \(m \mathrm {~kg}\), and at time \(t\) seconds, for \(0 \leqslant t \leqslant 10\), the cyclist's velocity is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A resistance to motion, modelled by a force of magnitude 0.1 mvN , acts on the cyclist during the whole 10 seconds.
  1. Explain why modelling the resistance to motion in this way is likely to be more realistic than assuming this force is constant. During the braking phase of the motion, for \(5 \leqslant t \leqslant 10\), the brakes apply an additional constant resistance force of magnitude \(2 m \mathrm {~N}\) and the cyclist does not provide any driving force.
  2. Show that, for \(5 \leqslant t \leqslant 10 , \frac { \mathrm {~d} v } { \mathrm {~d} t } + 0.1 v = - 2\).
    1. Solve the differential equation in part (b).
    2. Hence find the velocity of the cyclist when \(t = 5\). During the acceleration phase ( \(0 \leqslant t \leqslant 5\) ), the cyclist applies a driving force of magnitude directly proportional to \(t\).
  4. Show that, for \(0 \leqslant t \leqslant 5 , \frac { \mathrm {~d} v } { \mathrm {~d} t } + 0.1 v = \lambda t\), where \(\lambda\) is a positive constant.
    1. Show by integration that, for \(0 \leqslant t \leqslant 5 , v = 10 \lambda \left( t - 10 + 10 \mathrm { e } ^ { - 0.1 t } \right)\).
    2. Hence find \(\lambda\).
  6. Find the total distance, to the nearest metre, travelled by the cyclist during the motion.
OCR MEI Further Pure Core 2022 June Q1
Standard +0.3
1
  1. By considering \(( r + 1 ) ^ { 3 } - r ^ { 3 }\), find \(\sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } \left( 3 \mathrm { r } ^ { 2 } + 3 \mathrm { r } + 1 \right)\).
  2. Use this result to find \(\sum _ { r = 1 } ^ { n } r ( r + 1 )\), expressing your answer in fully factorised form.
OCR MEI Further Pure Core 2022 June Q2
Challenging +1.2
2 In this question you must show detailed reasoning. Find the exact value of \(\int _ { 3 } ^ { \infty } \frac { 1 } { x ^ { 2 } - 4 x + 5 } d x\)
OCR MEI Further Pure Core 2022 June Q3
Standard +0.3
3 In this question you must show detailed reasoning.
Solve the equation \(3 \cosh x = 2 \sinh ^ { 2 } x\), giving your solutions in exact logarithmic form.
OCR MEI Further Pure Core 2022 June Q4
Standard +0.8
4
  1. A transformation with associated matrix \(\left( \begin{array} { r r r } m & 2 & 1 \\ 0 & 1 & - 2 \\ 2 & 0 & 3 \end{array} \right)\), where \(m\) is a constant, maps the vertices of a cube to points that all lie in a plane. Find \(m\).
  2. The transformations S and T of the plane have associated matrices \(\mathbf { M }\) and \(\mathbf { N }\) respectively, where \(\mathbf { M } = \left( \begin{array} { r r } k & 1 \\ - 3 & 4 \end{array} \right)\) and the determinant of \(\mathbf { N }\) is \(3 k + 1\). The transformation \(U\) is equivalent to the combined transformation consisting of S followed by T . Given that U preserves orientation and has an area scale factor 2, find the possible values of \(k\).
OCR MEI Further Pure Core 2022 June Q5
Standard +0.3
5
  1. Sketch the polar curve \(\mathrm { r } = \mathrm { a } ( 1 - \cos \theta ) , 0 \leqslant \theta < 2 \pi\), where \(a\) is a positive constant.
  2. Determine the exact area of the region enclosed by the curve.
OCR MEI Further Pure Core 2022 June Q6
Standard +0.3
6 Prove by mathematical induction that \(\left( \begin{array} { r l } 2 & 0 \\ - 1 & 1 \end{array} \right) ^ { n } = \left( \begin{array} { c c } 2 ^ { n } & 0 \\ 1 - 2 ^ { n } & 1 \end{array} \right)\) for all positive integers \(n\). Answer all the questions.
Section B (107 marks)
OCR MEI Further Pure Core 2022 June Q7
Challenging +1.2
7 In this question you must show detailed reasoning.
Show that \(\int _ { 2 } ^ { 3 } \frac { x + 1 } { ( x - 1 ) \left( x ^ { 2 } + 1 \right) } d x = \frac { 1 } { 2 } \ln 2\).
OCR MEI Further Pure Core 2022 June Q8
Challenging +1.2
8 Two sets of complex numbers are given by \(\left\{ z : \arg ( z - 10 ) = \frac { 3 } { 4 } \pi \right\}\) and \(\{ z : | z - 3 - 6 i | = k \}\), where \(k\) is a positive constant. In an Argand diagram, one of the points of intersection of the two loci representing these sets lies on the imaginary axis.
  1. Sketch the loci on an Argand diagram.
  2. In this question you must show detailed reasoning. Find the complex numbers represented by the points of intersection.
OCR MEI Further Pure Core 2022 June Q9
Challenging +1.2
9 The function \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = \ln ( 1 + \sinh x )\).
  1. Given that \(k\) lies in the domain of this function, explain why \(k\) must be greater than \(\ln ( \sqrt { 2 } - 1 )\).
    1. Find \(\mathrm { f } ^ { \prime } ( x )\).
    2. Show that \(\mathrm { f } ^ { \prime \prime } ( \mathrm { x } ) = \frac { \mathrm { a } \sinh \mathrm { x } + \mathrm { b } } { ( 1 + \sinh \mathrm { x } ) ^ { 2 } }\), where \(a\) and \(b\) are integers to be determined.
  3. Hence find a quadratic approximation to \(\mathrm { f } ( x )\) for small values of \(x\).
  4. Find the percentage error in this approximation when \(x = 0.1\).
OCR MEI Further Pure Core 2022 June Q10
Challenging +1.2
10 The equation
\(4 x ^ { 4 } + 16 x ^ { 3 } + a x ^ { 2 } + b x + 6 = 0\),
where \(a\) and \(b\) are real, has roots \(\alpha , \frac { 2 } { \alpha } , \beta\) and \(3 \beta\).
  1. Given that \(\beta < 0\), determine all 4 roots.
  2. Determine the values of \(a\) and \(b\).
OCR MEI Further Pure Core 2022 June Q11
Standard +0.3
11 An Argand diagram with the point A representing a complex number \(z _ { 1 }\) is shown below.
\includegraphics[max width=\textwidth, alt={}, center]{b57a2590-84e8-4998-9633-902db861f23a-4_716_778_932_239} The complex numbers \(z _ { 2 }\) and \(z _ { 3 }\) are \(z _ { 1 } \mathrm { e } ^ { \frac { 2 } { 3 } \mathrm { i } \pi }\) and \(z _ { 1 } \mathrm { e } ^ { \frac { 4 } { 3 } \mathrm { i } \pi }\) respectively.
    1. On the copy of the Argand diagram in the Printed Answer Booklet, mark the points B and C representing the complex numbers \(z _ { 2 }\) and \(z _ { 3 }\).
    2. Show that \(z _ { 1 } + z _ { 2 } + z _ { 3 } = 0\).
  2. Given now that \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\) are roots of the equation \(z ^ { 3 } = 8 \mathrm { i }\), find these three roots, giving your answers in the form \(\mathrm { a } + \mathrm { ib }\), where \(a\) and \(b\) are real and exact.