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 Pure Core AS 2020 November Q5
6 marks Standard +0.3
5 You are given that \(u _ { 1 } = 5\) and \(u _ { n + 1 } = u _ { n } + 2 n + 4\).
Prove by induction that \(u _ { n } = n ^ { 2 } + 3 n + 1\) for all positive integers \(n\).
OCR MEI Further Pure Core AS 2020 November Q6
8 marks Moderate -0.3
6 The matrices \(\mathbf { M }\) and \(\mathbf { N }\) are \(\left( \begin{array} { l l } \lambda & 2 \\ 2 & \lambda \end{array} \right)\) and \(\left( \begin{array} { c c } \mu & 1 \\ 1 & \mu \end{array} \right)\) respectively, where \(\lambda\) and \(\mu\) are constants.
  1. Investigate whether \(\mathbf { M }\) and \(\mathbf { N }\) are commutative under multiplication.
  2. You are now given that \(\mathbf { M N } = \mathbf { I }\).
    1. Write down a relationship between \(\operatorname { det } \mathbf { M }\) and \(\operatorname { det } \mathbf { N }\).
    2. Given that \(\lambda > 0\), find the exact values of \(\lambda\) and \(\mu\).
    3. Hence verify your answer to part (i).
OCR MEI Further Pure Core AS 2020 November Q7
7 marks Standard +0.8
7 In the quartic equation \(2 x ^ { 4 } - 20 x ^ { 3 } + a x ^ { 2 } + b x + 250 = 0\), the coefficients \(a\) and \(b\) are real. One root of the equation is \(2 + \mathrm { i }\). Find the other roots.
OCR MEI Further Pure Core AS 2020 November Q8
7 marks Moderate -0.3
8
  1. The matrix \(\mathbf { M }\) is \(\left( \begin{array} { r r } 0 & - 1 \\ - 1 & 0 \end{array} \right)\).
    1. Find \(\mathbf { M } ^ { 2 }\).
    2. Write down the transformation represented by \(\mathbf { M }\).
    3. Hence state the geometrical significance of the result of part (i).
  2. The matrix \(\mathbf { N }\) is \(\left( \begin{array} { c c } k + 1 & 0 \\ k & k + 2 \end{array} \right)\), where \(k\) is a constant. Using determinants, investigate whether \(\mathbf { N }\) can represent a reflection.
OCR MEI Further Pure Core AS 2020 November Q9
7 marks Standard +0.3
9 Three planes have equations
\(k x + y - 2 z = 0\)
\(2 x + 3 y - 6 z = - 5\)
\(3 x - 2 y + 5 z = 1\)
where \(k\) is a constant. Investigate the arrangement of the planes for each of the following cases. If in either case the planes meet at a unique point, find the coordinates of that point.
  1. \(k = - 1\)
  2. \(k = \frac { 2 } { 3 }\)
OCR MEI Further Pure Core AS 2020 November Q10
7 marks Challenging +1.2
10 A vector \(\mathbf { v }\) has magnitude 1 unit. The angle between \(\mathbf { v }\) and the positive \(z\)-axis is \(60 ^ { \circ }\), and \(\mathbf { v }\) is parallel to the plane \(x - 2 y = 0\). Given that \(\mathbf { v } = a \mathbf { i } + b \mathbf { j } + c \mathbf { k }\), where \(a , b\) and \(c\) are all positive, find \(\mathbf { v }\). \section*{END OF QUESTION PAPER}
OCR MEI Further Pure Core AS 2021 November Q1
3 marks Moderate -0.8
1 Using standard summation formulae, find \(\sum _ { r = 1 } ^ { n } \left( r ^ { 2 } - 3 r \right)\), giving your answer in fully factorised form.
OCR MEI Further Pure Core AS 2021 November Q2
3 marks Standard +0.3
2 The equation \(3 x ^ { 2 } - 4 x + 2 = 0\) has roots \(\alpha\) and \(\beta\).
Find an equation with integer coefficients whose roots are \(3 - 2 \alpha\) and \(3 - 2 \beta\).
OCR MEI Further Pure Core AS 2021 November Q3
7 marks Standard +0.3
3 Three planes have the following equations. $$\begin{aligned} 2 x - 3 y + z & = - 3 \\ x - 4 y + 2 z & = 1 \\ - 3 x - 2 y + 3 z & = 14 \end{aligned}$$
    1. Write the system of equations in matrix form.
    2. Hence find the point of intersection of the planes.
  2. In this question you must show detailed reasoning. Find the acute angle between the planes \(2 x - 3 y + z = - 3\) and \(x - 4 y + 2 z = 1\).
OCR MEI Further Pure Core AS 2021 November Q4
5 marks Moderate -0.3
4 Anika thinks that, for two square matrices \(\mathbf { A }\) and \(\mathbf { B }\), the inverse of \(\mathbf { A B }\) is \(\mathbf { A } ^ { - 1 } \mathbf { B } ^ { - 1 }\). Her attempted proof of this is as follows. $$\begin{aligned} ( \mathbf { A B } ) \left( \mathbf { A } ^ { - 1 } \mathbf { B } ^ { - 1 } \right) & = \mathbf { A } \left( \mathbf { B A } ^ { - 1 } \right) \mathbf { B } ^ { - 1 } \\ & = \mathbf { A } \left( \mathbf { A } ^ { - 1 } \mathbf { B } \right) \mathbf { B } ^ { - 1 } \\ & = \left( \mathbf { A } \mathbf { A } ^ { - 1 } \right) \left( \mathbf { B B } ^ { - 1 } \right) \\ & = \mathbf { I } \times \mathbf { I } \\ & = \mathbf { I } \\ \text { Hence } ( \mathbf { A B } ) ^ { - 1 } & = \mathbf { A } ^ { - 1 } \mathbf { B } ^ { - 1 } \end{aligned}$$
  1. Explain the error in Anika's working.
  2. State the correct inverse of the matrix \(\mathbf { A B }\) and amend Anika's working to prove this.
OCR MEI Further Pure Core AS 2021 November Q5
5 marks Standard +0.3
5 Prove by induction that \(\sum _ { r = 1 } ^ { n } r \times 2 ^ { r - 1 } = 1 + ( n - 1 ) 2 ^ { n }\) for all positive integers \(n\).
OCR MEI Further Pure Core AS 2021 November Q6
12 marks Standard +0.8
6 A transformation T of the plane has associated matrix \(\mathbf { M } = \left( \begin{array} { c c } 1 & \lambda + 1 \\ \lambda - 1 & - 1 \end{array} \right)\), where \(\lambda\) is a non-zero
constant.
    1. Show that T reverses orientation.
    2. State, in terms of \(\lambda\), the area scale factor of T .
    1. Show that \(\mathbf { M } ^ { 2 } - \lambda ^ { 2 } \mathbf { I } = \mathbf { 0 }\).
    2. Hence specify the transformation equivalent to two applications of T .
  3. In the case where \(\lambda = 1 , \mathrm {~T}\) is equivalent to a transformation S followed by a reflection in the \(x\)-axis.
    1. Determine the matrix associated with S .
    2. Hence describe the transformation S .
OCR MEI Further Pure Core AS 2021 November Q7
9 marks Challenging +1.2
7
    1. Find the modulus and argument of \(z _ { 1 }\), where \(z _ { 1 } = 1 + \mathrm { i }\).
    2. Given that \(\left| z _ { 2 } \right| = 2\) and \(\arg \left( z _ { 2 } \right) = \frac { 1 } { 6 } \pi\), express \(z _ { 2 }\) in a + bi form, where \(a\) and \(b\) are exact real numbers.
  2. Using these results, find the exact value of \(\sin \frac { 5 } { 12 } \pi\), giving the answer in the form \(\frac { \sqrt { m } + \sqrt { n } } { p }\), where \(m , n\) and \(p\) are integers.
OCR MEI Further Pure Core AS 2021 November Q9
9 marks Challenging +1.2
9
  1. On a single Argand diagram, sketch the loci defined by
    • \(\arg ( z - 2 ) = \frac { 3 } { 4 } \pi\),
    • \(\quad | z | = | z + 2 - i |\).
    • In this question you must show detailed reasoning.
    The point of intersection of the two loci in part (a) represents the complex number \(w\). Find \(w\), giving your answer in exact form. \section*{END OF QUESTION PAPER}
OCR MEI Further Pure Core AS Specimen Q1
4 marks Moderate -0.8
1 The complex number \(z _ { 1 }\) is \(1 + \mathrm { i }\) and the complex number \(z _ { 2 }\) has modulus 4 and argument \(\frac { \pi } { 3 }\).
  1. Express \(z _ { 2 }\) in the form \(a + b \mathrm { i }\), giving \(a\) and \(b\) in exact form.
  2. Express \(\frac { z _ { 2 } } { z _ { 1 } }\) in the form \(c + d i\), giving \(c\) and \(d\) in exact form.
  3. Describe fully the transformation represented by the matrix \(\left( \begin{array} { l l } 1 & 2 \\ 0 & 1 \end{array} \right)\).
  4. A triangle of area 5 square units undergoes the transformation represented by the matrix \(\left( \begin{array} { l l } 1 & 2 \\ 0 & 1 \end{array} \right)\). Explaining your reasoning, find the area of the image of the triangle following this transformation.
OCR MEI Further Pure Core AS Specimen Q3
4 marks Standard +0.3
3
  1. Write down, in complex form, the equation of the locus represented by the circle in the Argand diagram shown in Fig. 3. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{7728fdf9-2000-4265-a0cb-f34a6561c2ca-2_917_825_1334_699} \captionsetup{labelformat=empty} \caption{Fig. 3}
    \end{figure}
  2. On the copy of Fig. 3 in the Printed Answer Booklet mark with a cross any point(s) on the circle for which \(\arg ( z - 2 \mathrm { i } ) = \frac { \pi } { 4 }\).
OCR MEI Further Pure Core AS Specimen Q4
6 marks Standard +0.3
4
  1. Find the coordinates of the point where the following three planes intersect. Give your answers in terms of \(a\). $$\begin{aligned} x - 2 y - z & = 6 \\ 3 x + y + 5 z & = - 4 \\ - 4 x + 2 y - 3 z & = a \end{aligned}$$
  2. Determine whether the intersection of the three planes could be on the \(z\)-axis.
OCR MEI Further Pure Core AS Specimen Q5
7 marks Standard +0.8
5 The cubic equation \(x ^ { 3 } - 4 x ^ { 2 } + p x + q = 0\) has roots \(\alpha , \frac { 2 } { \alpha }\) and \(\alpha + \frac { 2 } { \alpha }\). Find
  • the values of the roots of the equation,
  • the value of \(p\).
OCR MEI Further Pure Core AS Specimen Q6
5 marks Standard +0.8
6
  1. Show that, when \(n = 5 , \sum _ { r = n + 1 } ^ { 2 n } r ^ { 2 } = 330\).
  2. Find, in terms of \(n\), a fully factorised expression for \(\sum _ { r = n + 1 } ^ { 2 n } r ^ { 2 }\).
OCR MEI Further Pure Core AS Specimen Q7
7 marks Standard +0.3
7 The plane \(\Pi\) has equation \(3 x - 5 y + z = 9\).
  1. Show that \(\Pi\) contains
    • the point (4,1,2)
      and
    • the vector \(\left( \begin{array} { l } 1 \\ 1 \\ 2 \end{array} \right)\).
    • Determine the equation of a plane which is perpendicular to \(\Pi\) and which passes through ( \(4,1,2\) ).
OCR MEI Further Pure Core AS Specimen Q9
14 marks Standard +0.8
9 You are given that matrix \(\mathbf { M } = \left( \begin{array} { l l } - 3 & 8 \\ - 2 & 5 \end{array} \right)\).
  1. Prove that, for all positive integers \(n , \mathbf { M } ^ { n } = \left( \begin{array} { c c } 1 - 4 n & 8 n \\ - 2 n & 1 + 4 n \end{array} \right)\).
  2. Determine the equation of the line of invariant points of the transformation represented by the matrix \(\mathbf { M }\). It is claimed that the answer to part (ii) is also a line of invariant points of the transformation represented by the matrix \(\mathbf { M } ^ { n }\), for any positive integer \(n\).
  3. Explain geometrically why this claim is true.
  4. Verify algebraically that this claim is true. \section*{END OF QUESTION PAPER} {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. For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
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OCR MEI Further Mechanics A AS 2018 June Q1
6 marks Moderate -0.8
1 Forces of magnitude \(4 \mathrm {~N} , 3 \mathrm {~N} , 5 \mathrm {~N}\) and \(R \mathrm {~N}\) act on a particle in the directions shown in Fig. 1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fa99d9e6-e174-42dd-ac92-7b7d112c08be-2_697_780_443_639} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} The particle is in equilibrium. Find each of the following.
  • The value of \(R\).
  • The value of \(\theta\).
OCR MEI Further Mechanics A AS 2018 June Q2
12 marks Standard +0.3
2 A car of mass 1350 kg travels along a straight horizontal road. Throughout this question the resistance force to the motion of the car is modelled as constant and equal to 920 N .
  1. Calculate the power, in kW , developed by the car when the car is travelling at a constant speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car is now used to tow a caravan of mass 1050 kg along the same road. When the car tows the caravan at a constant speed of \(20 \mathrm {~ms} ^ { - 1 }\) the power developed by the car is 45 kW .
  2. Find the additional resistance force due to the caravan. In the remaining parts of this question the power developed by the car is constant and equal to 68 kW and the resistance force due to the caravan is modelled as constant and equal to the value found in part (ii). When the car and caravan pass a point A on the same straight horizontal road the speed of the car and caravan is \(20 \mathrm {~ms} ^ { - 1 }\).
  3. Find the acceleration of the car and caravan at point A . The car and caravan later pass a point B on the same straight horizontal road with speed \(28 \mathrm {~ms} ^ { - 1 }\). The distance \(A B\) is \(1024 m\).
  4. Find the time taken for the car and caravan to travel from point A to point B .
  5. Suggest one way in which any of the modelling assumptions used in this question could have been improved.
OCR MEI Further Mechanics A AS 2018 June Q3
9 marks Moderate -0.8
3 Jodie is doing an experiment involving a simple pendulum. The pendulum consists of a small object tied to one end of a piece of string. The other end of the string is attached to a fixed point O and the object is allowed to swing between two fixed points A and B and back again, as shown in Fig. 3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fa99d9e6-e174-42dd-ac92-7b7d112c08be-3_328_350_584_886} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} Jodie thinks that \(P\), the time the pendulum takes to swing from A to B and back again, depends on the mass, \(m\), of the small object, the length, \(l\), of the piece of string, and the acceleration due to gravity \(g\). She proposes the formula \(P = k m ^ { \alpha } l ^ { \beta } g ^ { \gamma }\).
  1. What is the significance of \(k\) in Jodie's formula?
  2. Use dimensional analysis to determine the values of \(\alpha , \beta\) and \(\gamma\). Jodie finds that when the mass of the object is 1.5 kg and the length of the string is 80 cm the time taken for the pendulum to swing from A to B and back again is 1.8 seconds.
  3. Use Jodie's formula and your answers to part (ii) to find each of the following.
    (A) The value of \(k\)
    (B) The time taken for the pendulum to swing from A to B and back again when the mass of the object is 0.9 kg and the length of the string is 1.4 m
  4. Comment on the assumption made by Jodie that the formula for the time taken for the pendulum to swing from A to B and back again is dependent on \(m , l\) and \(g\).
OCR MEI Further Mechanics A AS 2018 June Q4
9 marks Standard +0.3
4 A uniform lamina ABDE is in the shape of an equilateral triangle ABC of side 12 cm from which an equilateral triangle of side 6 cm has been removed from corner \(C\). The lamina is situated on coordinate axes as shown in Fig. 4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fa99d9e6-e174-42dd-ac92-7b7d112c08be-4_501_753_406_646} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure}
  1. Explain why angle \(\mathrm { BDA } = 90 ^ { \circ }\).
  2. Find the coordinates of the centre of mass of the lamina ABDE . The lamina ABDE is now freely suspended from D and hangs in equilibrium.
  3. Calculate the angle DE makes with the downward vertical.