Questions — OCR MEI (4456 questions)

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OCR MEI Further Pure Core AS 2020 November Q2
4 marks Moderate -0.8
2 Fig. 2 shows two complex numbers \(z _ { 1 }\) and \(z _ { 2 }\) represented on an Argand diagram. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{55a4a9f1-ed86-44bb-8759-dfee0b66f56d-2_985_997_781_239} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure}
  1. On the copy of Fig. 2 in the Printed Answer Booklet, mark points representing each of the following complex numbers.
    In the case where \(z _ { 1 } = 1 + 2 \mathrm { i }\) and \(z _ { 2 } = 3 + \mathrm { i }\), find \(\frac { z _ { 2 } - z _ { 1 } } { z _ { 1 } ^ { * } }\) in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are real numbers.
OCR MEI Further Pure Core AS 2020 November Q3
7 marks Standard +0.3
3 In this question you must show detailed reasoning.
The roots of the equation \(x ^ { 2 } - 2 x + 4 = 0\) are \(\alpha\) and \(\beta\).
  1. Find \(\alpha\) and \(\beta\) in modulus-argument form.
  2. Hence or otherwise show that \(\alpha\) and \(\beta\) are both roots of \(x ^ { 3 } + \lambda = 0\), where \(\lambda\) is a real constant to be determined.
OCR MEI Further Pure Core AS 2020 November Q4
4 marks Moderate -0.3
4 The matrix \(\mathbf { M }\) is \(\left( \begin{array} { r r r } 0 & - 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{array} \right)\).
    1. Calculate \(\operatorname { det } \mathbf { M }\).
    2. State two geometrical consequences of this value for the transformation associated with \(\mathbf { M }\).
  2. Describe fully the transformation associated with \(\mathbf { M }\).
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
    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 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.
OCR MEI Further Mechanics A AS 2018 June Q5
13 marks Moderate -0.3
5 A small ball is held at a height of 160 cm above a horizontal surface. The ball is released from rest and rebounds from the surface. After its first bounce on the surface the ball reaches a height of 122.5 cm .
  1. Find the height reached by the ball after its second bounce on the surface. After \(n\) bounces the height reached by the ball is less than 10 cm .
  2. Find the minimum possible value of \(n\).
  3. State what would happen if the same ball is released from rest from a height of 160 cm above a different horizontal surface and
    (A) the coefficient of restitution between the ball and the new surface is 0 ,
    (B) the coefficient of restitution between the ball and the new surface is 1 .
OCR MEI Further Mechanics A AS 2018 June Q6
11 marks Standard +0.3
6 A uniform rod AB has length \(2 a\) and weight \(W\). The rod is in equilibrium in a horizontal position. The end A rests on a smooth plane which is inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The force exerted on AB by the plane is \(R\). The end B is attached to a light inextensible string inclined at an angle of \(\theta\) to AB as shown in Fig. 6. The rod and string are in the same vertical plane, which also contains the line of greatest slope of the plane on which A lies. The tension in the string is \(T\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fa99d9e6-e174-42dd-ac92-7b7d112c08be-5_474_862_479_616} \captionsetup{labelformat=empty} \caption{Fig. 6}
\end{figure}
  1. Add the forces \(R\) and \(T\) to the copy of Fig. 6 in the Printed Answer Booklet.
  2. By taking moments about B , find an expression for \(R\) in terms of \(W\).
  3. By resolving horizontally, show that \(6 T \cos \theta = W \sqrt { 3 }\).
  4. By finding a second equation connecting \(T\) and \(\theta\), determine
OCR MEI Further Mechanics A AS 2019 June Q1
3 marks Moderate -0.8
1 A child is pulling a toy block in a straight line along a horizontal floor.
The block is moving with a constant speed of \(2 \mathrm {~ms} ^ { - 1 }\) by means of a constant force of magnitude 20 N acting at an angle of \(\theta ^ { \circ }\) above the horizontal. The work done by the force in 10 s is 350 J . Calculate the value of \(\theta\).
OCR MEI Further Mechanics A AS 2019 June Q2
12 marks Moderate -0.3
2 The surface tension of a liquid allows a metal needle to be at rest on the surface of the liquid.
The greatest mass \(m\) of a needle of length \(l\) which can be supported in this way by a liquid of surface tension \(S\) is given by the formula \(m = \frac { 2 S l } { g }\) where \(g\) is the acceleration due to gravity.
  1. Determine the dimensions of surface tension. Surface tension also allows liquids to rise up capillary tubes. Molly is experimenting with liquids in capillary tubes and she arrives at the formula \(h = \frac { 2 S } { \rho g r }\), where \(h\) is the height to which a liquid of surface tension \(S\) rises, \(\rho\) is the density of the liquid, and \(r\) is the radius of the capillary tube.
  2. Show that the equation for \(h\) is dimensionally consistent. In SI units, the surface tension of mercury is \(0.475 \mathrm {~kg} \mathrm {~s} ^ { - 2 }\) and its density is \(13500 \mathrm {~kg} \mathrm {~m} ^ { - 3 }\).
  3. Find the diameter of a capillary tube in which mercury will rise to a height of 10 cm . In another experiment, Molly finds that when liquid of surface tension \(S\) is poured onto a horizontal surface, puddles of depth \(d\) are formed. For this experiment she finds that \(d = k S ^ { \alpha } \rho ^ { \beta } g ^ { \gamma }\) where \(k\) is a dimensionless constant.
  4. Determine the values of \(\alpha , \beta\) and \(\gamma\).