Questions — OCR MEI (4333 questions)

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OCR MEI Paper 1 2018 June Q11
7 marks Standard +0.3
11 Fig. 11 shows two blocks at rest, connected by a light inextensible string which passes over a smooth pulley. Block A of mass 4.7 kg rests on a smooth plane inclined at \(60 ^ { \circ }\) to the horizontal. Block B of mass 4 kg rests on a rough plane inclined at \(25 ^ { \circ }\) to the horizontal. On either side of the pulley, the string is parallel to a line of greatest slope of the plane. Block B is on the point of sliding up the plane. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{904025c9-6d68-4344-bd41-8c0fccfcf92f-07_332_931_443_575} \captionsetup{labelformat=empty} \caption{Fig. 11}
\end{figure}
  1. Show that the tension in the string is 39.9 N correct to 3 significant figures.
  2. Find the coefficient of friction between the rough plane and Block B.
OCR MEI Paper 1 2018 June Q12
14 marks Standard +0.8
12 Fig. 12 shows the circle \(( x - 1 ) ^ { 2 } + ( y + 1 ) ^ { 2 } = 25\), the line \(4 y = 3 x - 32\) and the tangent to the circle at the point \(\mathrm { A } ( 5,2 )\). D is the point of intersection of the line \(4 y = 3 x - 32\) and the tangent at A . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{904025c9-6d68-4344-bd41-8c0fccfcf92f-07_750_773_1311_632} \captionsetup{labelformat=empty} \caption{Fig. 12}
\end{figure}
  1. Write down the coordinates of C , the centre of the circle.
  2. (A) Show that the line \(4 y = 3 x - 32\) is a tangent to the circle.
    (B) Find the coordinates of B , the point where the line \(4 y = 3 x - 32\) touches the circle.
  3. Prove that ADBC is a square.
  4. The point E is the lowest point on the circle. Find the area of the sector ECB .
OCR MEI Paper 1 2018 June Q13
12 marks Standard +0.8
13 The function \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = \sqrt [ 3 ] { 27 - 8 x ^ { 3 } }\). Jenny uses her scientific calculator to create a table of values for \(\mathrm { f } ( x )\) and \(\mathrm { f } ^ { \prime } ( x )\).
\(x\)\(f ( x )\)\(f ^ { \prime } ( x )\)
030
0.252.9954- 0.056
0.52.9625- 0.228
0.752.8694- 0.547
12.6684- 1.124
1.252.2490- 1.977
1.50ERROR
  1. Use calculus to find an expression for \(\mathrm { f } ^ { \prime } ( x )\) and hence explain why the calculator gives an error for \(\mathrm { f } ^ { \prime } ( 1.5 )\).
  2. Find the first three terms of the binomial expansion of \(\mathrm { f } ( x )\).
  3. Jenny integrates the first three terms of the binomial expansion of \(\mathrm { f } ( x )\) to estimate the value of \(\int _ { 0 } ^ { 1 } \sqrt [ 3 ] { 27 - 8 x ^ { 3 } } \mathrm {~d} x\). Explain why Jenny's method is valid in this case. (You do not need to evaluate Jenny's approximation.)
  4. Use the trapezium rule with 4 strips to obtain an estimate for \(\int _ { 0 } ^ { 1 } \sqrt [ 3 ] { 27 - 8 x ^ { 3 } } \mathrm {~d} x\). The calculator gives 2.92117438 for \(\int _ { 0 } ^ { 1 } \sqrt [ 3 ] { 27 - 8 x ^ { 3 } } \mathrm {~d} x\). The graph of \(y = \mathrm { f } ( x )\) is shown in Fig. 13. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{904025c9-6d68-4344-bd41-8c0fccfcf92f-08_490_906_1505_568} \captionsetup{labelformat=empty} \caption{Fig. 13}
    \end{figure}
  5. Explain why the trapezium rule gives an underestimate.
OCR MEI Paper 1 2018 June Q14
17 marks Standard +0.3
14 The velocity of a car, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t\) seconds, is being modelled. Initially the car has velocity \(5 \mathrm {~ms} ^ { - 1 }\) and it accelerates to \(11.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in 4 seconds. In model A, the acceleration is assumed to be uniform.
  1. Find an expression for the velocity of the car at time \(t\) using this model.
  2. Explain why this model is not appropriate in the long term. Model A is refined so that the velocity remains constant once the car reaches \(17.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  3. Sketch a velocity-time graph for the motion of the car, making clear the time at which the acceleration changes.
  4. Calculate the displacement of the car in the first 20 seconds according to this refined model. In model B, the velocity of the car is given by $$v = \begin{cases} 5 + 0.6 t ^ { 2 } - 0.05 t ^ { 3 } & \text { for } 0 \leqslant t \leqslant 8 \\ 17.8 & \text { for } 8 < t \leqslant 20 \end{cases}$$
  5. Show that this model gives an appropriate value for \(v\) when \(t = 4\).
  6. Explain why the value of the acceleration immediately before the velocity becomes constant is likely to mean that model B is a better model than model A.
  7. Show that model B gives the same value as model A for the displacement at time 20 s .
OCR MEI Paper 1 2019 June Q2
3 marks Moderate -0.8
2 Show that the line which passes through the points \(( 2 , - 4 )\) and \(( - 1,5 )\) does not intersect the line \(3 x + y = 10\).
OCR MEI Paper 1 2019 June Q3
8 marks Standard +0.3
3 The function \(\mathrm { f } ( x )\) is given by \(\mathrm { f } ( x ) = ( 1 - a x ) ^ { - 3 }\), where \(a\) is a non-zero constant. In the binomial expansion of \(\mathrm { f } ( x )\), the coefficients of \(x\) and \(x ^ { 2 }\) are equal.
  1. Find the value of \(a\).
  2. Using this value for \(a\),
    1. state the set of values of \(x\) for which the binomial expansion is valid,
    2. write down the quadratic function which approximates \(\mathrm { f } ( x )\) when \(x\) is small.
OCR MEI Paper 1 2019 June Q4
3 marks Moderate -0.3
4 Fig. 4 shows a uniform beam of mass 4 kg and length 2.4 m resting on two supports P and Q . P is at one end of the beam and Q is 0.3 m from the other end.
Determine whether a person of mass 50 kg can tip the beam by standing on it. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{59e924e6-8fa9-4035-9173-705fce487bd9-4_195_977_1676_262} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure}
OCR MEI Paper 1 2019 June Q5
4 marks Easy -1.2
5 A car of mass 1200 kg travels from rest along a straight horizontal road. The driving force is 4000 N and the total of all resistances to motion is 800 N .
Calculate the velocity of the car after 9 seconds.
OCR MEI Paper 1 2019 June Q6
7 marks Standard +0.3
6
  1. Prove that \(\frac { \sin \theta } { 1 - \cos \theta } - \frac { 1 } { \sin \theta } = \cot \theta\).
  2. Hence find the exact roots of the equation \(\frac { \sin \theta } { 1 - \cos \theta } - \frac { 1 } { \sin \theta } = 3 \tan \theta\) in the interval \(0 \leqslant \theta \leqslant \pi\). Answer all the questions.
    Section B (75 marks)
OCR MEI Paper 1 2019 June Q7
4 marks Standard +0.3
7 The velocity \(v \mathrm {~ms} ^ { - 1 }\) of a particle at time \(t \mathrm {~s}\) is given by \(v = 0.5 t ( 7 - t )\). Determine whether the speed of the particle is increasing or decreasing when \(t = 8\).
OCR MEI Paper 1 2019 June Q8
7 marks Standard +0.3
8 An arithmetic series has first term 9300 and 10th term 3900.
  1. Show that the 20th term of the series is negative.
  2. The sum of the first \(n\) terms is denoted by \(S\). Find the greatest value of \(S\) as \(n\) varies.
OCR MEI Paper 1 2019 June Q9
7 marks Moderate -0.8
9 A cannonball is fired from a point on horizontal ground at \(100 \mathrm {~ms} ^ { - 1 }\) at an angle of \(25 ^ { \circ }\) above the horizontal. Ignoring air resistance, calculate
  1. the greatest height the cannonball reaches,
  2. the range of the cannonball.
OCR MEI Paper 1 2019 June Q10
7 marks Standard +0.3
10
  1. Express \(7 \cos x - 2 \sin x\) in the form \(R \cos ( x + \alpha )\) where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\), giving the exact value of \(R\) and the value of \(\alpha\) correct to 3 significant figures.
  2. Give details of a sequence of two transformations which maps the curve \(y = \sec x\) onto the curve \(y = \frac { 1 } { 7 \cos x - 2 \sin x }\).
OCR MEI Paper 1 2019 June Q11
5 marks Moderate -0.8
11 In this question, the unit vector \(\mathbf { i }\) is horizontal and the unit vector \(\mathbf { j }\) is vertically upwards. A particle of mass 0.8 kg moves under the action of its weight and two forces given by ( \(k \mathbf { i } + 5 \mathbf { j }\) ) N and \(( 4 \mathbf { i } + 3 \mathbf { j } ) \mathrm { N }\). The acceleration of the particle is vertically upwards.
  1. Write down the value of \(k\). Initially the velocity of the particle is \(( 4 \mathbf { i } + 7 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\).
  2. Find the velocity of the particle 10 seconds later.
OCR MEI Paper 1 2019 June Q12
6 marks Moderate -0.3
12 Fig. 12 shows a curve C with parametric equations \(x = 4 t ^ { 2 } , y = 4 t\). The point P , with parameter \(t\), is a general point on the curve. Q is the point on the line \(x + 4 = 0\) such that PQ is parallel to the \(x\)-axis. R is the point \(( 4,0 )\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{59e924e6-8fa9-4035-9173-705fce487bd9-6_766_584_413_255} \captionsetup{labelformat=empty} \caption{Fig. 12}
\end{figure}
  1. Show algebraically that P is equidistant from Q and R .
  2. Find a cartesian equation of C .
OCR MEI Paper 1 2019 June Q13
5 marks Moderate -0.8
13 A 15 kg box is suspended in the air by a rope which makes an angle of \(30 ^ { \circ }\) with the vertical. The box is held in place by a string which is horizontal.
  1. Draw a diagram showing the forces acting on the box.
  2. Calculate the tension in the rope.
  3. Calculate the tension in the string.
OCR MEI Paper 1 2019 June Q14
8 marks Standard +0.3
14 Fig. 14 shows a circle with centre O and radius \(r \mathrm {~cm}\). The chord AB is such that angle \(\mathrm { AOB } = x\) radians. The area of the shaded segment formed by AB is \(5 \%\) of the area of the circle. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{59e924e6-8fa9-4035-9173-705fce487bd9-7_497_496_356_251} \captionsetup{labelformat=empty} \caption{Fig. 14}
\end{figure}
  1. Show that \(x - \sin x - \frac { 1 } { 10 } \pi = 0\). The Newton-Raphson method is to be used to find \(x\).
  2. Write down the iterative formula to be used for the equation in part (a).
  3. Use three iterations of the Newton-Raphson method with \(x _ { 0 } = 1.2\) to find the value of \(x\) to a suitable degree of accuracy.
OCR MEI Paper 1 2019 June Q15
12 marks Standard +0.3
15 A model for the motion of a small object falling through a thick fluid can be expressed using the differential equation \(\frac { \mathrm { d } v } { \mathrm {~d} t } = 9.8 - k v\),
where \(v \mathrm {~ms} ^ { - 1 }\) is the velocity after \(t \mathrm {~s}\) and \(k\) is a positive constant.
  1. Given that \(v = 0\) when \(t = 0\), solve the differential equation to find \(v\) in terms of \(t\) and \(k\).
  2. Sketch the graph of \(v\) against \(t\). Experiments show that for large values of \(t\), the velocity tends to \(7 \mathrm {~ms} ^ { - 1 }\).
  3. Find the value of \(k\).
  4. Find the value of \(t\) for which \(v = 3.5\).
OCR MEI Paper 1 2019 June Q16
14 marks Standard +0.3
16 A particle of mass 2 kg slides down a plane inclined at \(20 ^ { \circ }\) to the horizontal. The particle has an initial velocity of \(1.4 \mathrm {~ms} ^ { - 1 }\) down the plane. Two models for the particle's motion are proposed. In model A the plane is taken to be smooth.
  1. Calculate the time that model A predicts for the particle to slide the first 0.7 m .
  2. Explain why model A is likely to underestimate the time taken. In model B the plane is taken to be rough, with a constant coefficient of friction between the particle and the plane.
  3. Calculate the acceleration of the particle predicted by model B given that it takes 0.8 s to slide the first 0.7 m .
  4. Find the coefficient of friction predicted by model B , giving your answer correct to 3 significant figures. \section*{END OF QUESTION PAPER}
OCR MEI Paper 1 2022 June Q1
4 marks Easy -1.2
1 A particle moves along a straight line. The displacement \(s \mathrm {~m}\) at time \(t \mathrm {~s}\) is shown in the displacementtime graph below. The graph consists of straight line segments joining the points \(( 0 , - 2 ) , ( 10,5 )\) and \(( 15,1 )\). \includegraphics[max width=\textwidth, alt={}, center]{9dd6fc6d-b51e-4a73-ace5-d26a7558032c-04_641_848_641_242}
  1. Find the distance travelled by the particle in the first 15 s .
  2. Calculate the velocity of the particle between \(t = 10\) and \(t = 15\).
OCR MEI Paper 1 2022 June Q2
3 marks Easy -1.2
2 Express \(\frac { 13 - x } { ( x - 3 ) ( x + 2 ) }\) in partial fractions.
OCR MEI Paper 1 2022 June Q3
8 marks
3
  1. Sketch the graph of \(\mathrm { y } = \arctan \mathrm { x }\) where \(x\) is in radians.
  2. In this question you must show detailed reasoning. Find all points of intersection of the curves \(\mathrm { y } = 3 \sin \mathrm { xcos } \mathrm { x }\) and \(\mathrm { y } = \cos ^ { 2 } \mathrm { x }\) for \(- \pi \leqslant x \leqslant \pi\).
OCR MEI Paper 1 2022 June Q4
4 marks Standard +0.3
4 Using an appropriate expansion show that, for sufficiently small values of \(x\), \(\frac { 1 - x } { ( 2 + x ) ^ { 2 } } \approx \frac { 1 } { 4 } - \frac { 1 } { 2 } x + \frac { 7 } { 16 } x ^ { 2 }\).
OCR MEI Paper 1 2022 June Q5
5 marks Moderate -0.8
5 A sphere of mass 3 kg hangs on a string. A horizontal force of magnitude \(F \mathrm {~N}\) acts on the sphere so that it hangs in equilibrium with the string making an angle of \(25 ^ { \circ }\) to the vertical. The force diagram for the sphere is shown below. \includegraphics[max width=\textwidth, alt={}, center]{9dd6fc6d-b51e-4a73-ace5-d26a7558032c-05_502_513_408_244}
  1. Sketch the triangle of forces for these forces.
  2. Hence or otherwise determine each of the following:
    • the tension in the string
    • the value of \(F\).
    Answer all the questions.
    Section B (76 marks)
OCR MEI Paper 1 2022 June Q6
9 marks Standard +0.3
6 A shelf consists of a horizontal uniform plank AB of length 0.8 m and mass 5 kg with light inextensible vertical strings attached at each end. A stack of bricks each of mass 2.3 kg is placed on the plank as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{9dd6fc6d-b51e-4a73-ace5-d26a7558032c-06_397_734_641_242}
  1. Explain the meaning of each of the following modelling assumptions.
    • The stack of bricks is modelled as a particle.
    • The plank is modelled as uniform.
    Either of the strings will break if the tension exceeds 75 N.
  2. Find the greatest number of bricks that can be placed at the centre of the plank without breaking the strings.
  3. Find an expression for the moment about A of the weight of a stack of \(n\) bricks when the stack is at a distance of \(x \mathrm {~m}\) from A . State the units for your answer.
  4. Calculate the greatest distance from A that the largest stack of bricks can be placed without a string breaking.