Questions — OCR MEI Paper 1 (118 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 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 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 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 Paper 1 2019 June Q13
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
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
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
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
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
2 Express \(\frac { 13 - x } { ( x - 3 ) ( x + 2 ) }\) in partial fractions.
OCR MEI Paper 1 2022 June Q3
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 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 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
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.
OCR MEI Paper 1 2022 June Q7
7 In this question the \(x\) - and \(y\)-directions are horizontal and vertically upwards respectively and the origin is on horizontal ground.
A ball is thrown from a point 5 m above the origin with an initial velocity \(\binom { 14 } { 7 } \mathrm {~ms} ^ { - 1 }\).
  1. Find the position vector of the ball at time \(t \mathrm {~s}\) after it is thrown.
  2. Find the distance between the origin and the point at which the ball lands on the ground.
OCR MEI Paper 1 2022 June Q8
8 A particle moves in the \(x - y\) plane so that its position at time \(t\) s is given by \(x = t ^ { 3 } - 8 t , y = t ^ { 2 }\) for \(- 3.5 < t < 3.5\). The units of distance are metres. The graph shows the path of the particle and the direction of travel at the point \(\mathrm { P } ( 8,4 )\).
\includegraphics[max width=\textwidth, alt={}, center]{9dd6fc6d-b51e-4a73-ace5-d26a7558032c-07_492_924_415_242}
  1. Find \(\frac { \mathrm { dy } } { \mathrm { dx } }\) in terms of \(t\).
  2. Hence show that the value of \(\frac { \mathrm { dy } } { \mathrm { dx } }\) at P is - 1 .
  3. Find the time at which the particle is travelling in the direction opposite to that at P .
  4. Find the cartesian equation of the path, giving \(x ^ { 2 }\) as a function of \(y\).
OCR MEI Paper 1 2022 June Q9
9 In this question, the vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively.
The velocity \(\mathbf { v } \mathrm { ms } ^ { - 1 }\) of a particle at time \(t\) s is given by \(\mathbf { v } = k t ^ { 2 } \mathbf { i } + 6 t\), where \(k\) is a positive constant. The magnitude of the acceleration when \(t = 2\) is \(10 \mathrm {~ms} ^ { - 2 }\).
  1. Calculate the value of \(k\). The particle is at the origin when \(t = 0\).
  2. Determine an expression for the position vector of the particle at time \(t\).
  3. Determine the time when the particle is directly north-east of the origin.
OCR MEI Paper 1 2022 June Q10
10 A triangle ABC is made from two thin rods hinged together at A and a piece of elastic which joins \(B\) and \(C\). \(A B\) is a 30 cm rod and \(A C\) is a 15 cm rod. The angle \(B A C\) is \(\theta\) radians as shown in the diagram. The angle \(\theta\) increases at a rate of 0.1 radians per second.
Determine the rate of change of the length BC when \(\theta = \frac { 1 } { 3 } \pi\).
OCR MEI Paper 1 2022 June Q11
11 Given that \(k\) is a positive constant, show that \(\int _ { k } ^ { 2 k } \frac { 2 } { ( 2 x + k ) ^ { 2 } } d x\) is inversely proportional to \(k\).
OCR MEI Paper 1 2022 June Q12
12 Prove by contradiction that 3 is the only prime number which is 1 less than a square number.
OCR MEI Paper 1 2022 June Q13
13 A toy train consists of an engine of mass 0.5 kg pulling a coach of mass 0.4 kg . The coupling between the engine and the coach is light and inextensible. The train is pulled along with a string attached to the front of the engine. At first, the train is pulled from rest along a horizontal carpet where there is a resistance to motion of 0.8 N on each part of the train. The string is horizontal, and the tension in the string is 5 N .
  1. Determine the velocity of the train after 1.5 s . The train is then pulled up a track inclined at \(20 ^ { \circ }\) to the horizontal. The string is parallel to the track and the tension in the string is \(P \mathrm {~N}\). The resistance on each part of the train along the track is \(R \mathrm {~N}\).
  2. Draw a diagram showing all the forces acting on the train modelled as two connected particles.
  3. Find the equation of motion for the train modelled as a single particle.
  4. The acceleration of the train when \(P = 5.5\) is double the acceleration when \(P = 5\). Calculate the value of \(R\).
OCR MEI Paper 1 2022 June Q14
14 Alex places a hot object into iced water and records the temperature \(\theta ^ { \circ } \mathrm { C }\) of the object every minute. The temperature of an object \(t\) minutes after being placed in iced water is modelled by \(\theta = \theta _ { 0 } \mathrm { e } ^ { - k t }\) where \(\theta _ { 0 }\) and \(k\) are constants whose values depend on the characteristics of the object. The temperature of Alex's object is \(82 ^ { \circ } \mathrm { C }\) when it is placed into the water. After 5 minutes the temperature is \(27 ^ { \circ } \mathrm { C }\).
  1. Find the values of \(\theta _ { 0 }\) and \(k\) that best model the data.
  2. Explain why the model may not be suitable in the long term if Alex does not top up the ice in the water.
  3. Show that the model with the values found in part (a) can be written as \(\ln \theta = \mathrm { a } -\) bt where \(a\) and \(b\) are constants to be determined. Ben places a different object into iced water at the same time as Alex. The model for Ben's object is \(\ln \theta = 3.4 - 0.08 t\).
  4. Determine each of the following:
    • the initial temperature of Ben's object
    • the rate at which Ben's object is cooling initially.
    • According to the models, there is a time at which both objects have the same temperature.
    Find this time and the corresponding temperature.
OCR MEI Paper 1 2023 June Q1
1 A ball is thrown vertically upwards with a speed of \(8 \mathrm {~ms} ^ { - 1 }\).
Find the times at which the ball is 3 m above the point of projection.
OCR MEI Paper 1 2023 June Q2
2 Express \(\frac { 5 x + 1 } { x ^ { 2 } - x - 12 }\) in partial fractions.
OCR MEI Paper 1 2023 June Q3
3 Find \(\int \left( 2 x ^ { 4 } - x \sqrt { x } \right) d x\).
OCR MEI Paper 1 2023 June Q4
4 A ruler PQRS is a uniform rectangular lamina with mass 20 grams. The length of PQ is 30 cm and the length of PS is 4 cm . The ruler is attached at P to a smooth hinge and held with S vertically below P by a horizontal force of magnitude \(F \mathrm {~N}\) as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{8eeff88d-8b05-43c6-86a5-bd82221c0bea-04_303_1495_1363_239}
  1. Calculate the value of \(F\).
  2. Explain what would happen to the lamina if the force at S were removed.
OCR MEI Paper 1 2023 June Q6
6
  1. Show that the equation \(\sin \left( x + \frac { 1 } { 6 } \pi \right) = \cos \left( x - \frac { 1 } { 4 } \pi \right)\) can be written in the form \(\tan x = \frac { \sqrt { 2 } - 1 } { \sqrt { 3 } - \sqrt { 2 } }\).
  2. Hence solve the equation \(\sin \left( x + \frac { 1 } { 6 } \pi \right) = \cos \left( x - \frac { 1 } { 4 } \pi \right)\) for \(0 \leqslant x \leqslant 2 \pi\).
OCR MEI Paper 1 2023 June Q7
7 Determine the exact distance between the two points at which the line through ( 4,5 ) and ( \(6 , - 1\) ) meets the curve \(y = 2 x ^ { 2 } - 7 x + 1\).
OCR MEI Paper 1 2023 June Q8
8 A bus is travelling along a straight road at \(5.4 \mathrm {~ms} ^ { - 1 }\). At \(t = 0\), as the bus passes a boy standing on the pavement, the boy starts running in the same direction as the bus, accelerating at \(1.2 \mathrm {~ms} ^ { - 2 }\) from rest for 5 s . He then runs at constant speed until he catches up with the bus.
  1. The diagram in the Printed Answer Booklet shows the velocity-time graph for the bus. Draw the velocity-time graph for the boy on this diagram.
  2. Determine the time at which the boy is running at the same speed as the bus.
  3. Find the maximum distance between the bus and the boy.
  4. Find the distance the boy has run when he catches up with the bus.