Questions Paper 1 (373 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 2021 November Q7
7 In this question you must show detailed reasoning.
The points \(\mathrm { A } ( - 1,4 )\) and \(\mathrm { B } ( 7 , - 2 )\) are at opposite ends of a diameter of a circle.
  1. Find the equation of the circle.
  2. Find the coordinates of the points of intersection of the circle and the line \(y = 2 x + 5\).
  3. Q is the point of intersection with the larger \(y\)-coordinate. Calculate the area of the triangle ABQ .
OCR MEI Paper 1 2021 November Q8
8 Kareem wants to solve the equation \(\sin 4 x + \mathrm { e } ^ { - x } + 0.75 = 0\). He uses his calculator to create the following table of values for \(\mathrm { f } ( x ) = \sin 4 x + \mathrm { e } ^ { - x } + 0.75\).
\(x\)0123456
\(\mathrm { f } ( x )\)1.7500.3611.8750.2630.4801.670- 0.153
He argues that because \(\mathrm { f } ( 6 )\) is the first negative value in the table, there is a root of the equation between 5 and 6 .
  1. Comment on the validity of his argument. The diagram shows the graph of \(y = \sin 4 x + e ^ { - x } + 0.75\).
    \includegraphics[max width=\textwidth, alt={}, center]{4fac72cb-85cb-48d9-8817-899ef3f80a0f-07_538_1260_920_244}
  2. Explain why Kareem failed to find other roots between 0 and 6 . Kareem decides to use the Newton-Raphson method to find the root close to 3 .
    1. Determine the iterative formula he should use for this equation.
    2. Use the Newton-Raphson method with \(x _ { 0 } = 3\) to find a root of the equation \(\mathrm { f } ( x ) = 0\). Show three iterations and give your answer to a suitable degree of accuracy. Kareem uses the Newton-Raphson method with \(x _ { 0 } = 5\) and also with \(x _ { 0 } = 6\) to try to find the root which lies between 5 and 6 . He produces the following tables.
      \(x _ { 0 }\)5
      \(x _ { 1 }\)3.97288
      \(x _ { 2 }\)4.12125
      \(x _ { 0 }\)6
      \(x _ { 1 }\)6.09036
      \(x _ { 2 }\)6.07110
    1. For the iteration beginning with \(x _ { 0 } = 5\), represent the process on the graph in the Printed Answer Booklet.
    2. Explain why the method has failed to find the root which lies between 5 and 6 .
    3. Explain how Kareem can adapt his method to find the root between 5 and 6 .
OCR MEI Paper 1 2021 November Q9
9 The diagram shows a toy caterpillar consisting of a head and three body sections each connected by a light inextensible ribbon. The head has a mass of 120 g and the body sections each have a mass of 90 g . The toy is pulled on level ground using a horizontal string attached to the head. The tension in the string is 12 N . There are resistances to motion of 2.5 N for the head and each section of the body.
\includegraphics[max width=\textwidth, alt={}, center]{4fac72cb-85cb-48d9-8817-899ef3f80a0f-08_134_794_536_244}
    1. State the equation of motion for the toy caterpillar modelled as a single particle.
    2. Calculate the acceleration of the toy caterpillar.
  2. Draw a diagram showing all the forces acting on the head of the toy caterpillar.
  3. Calculate the tension in the ribbon that joins the head to the body.
OCR MEI Paper 1 2021 November Q10
10 A ball is thrown upwards with a velocity of \(29.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Show that the ball reaches its maximum height after 3 s .
  2. Sketch a velocity-time graph for the first 5 s of motion.
  3. Calculate the speed of the ball 5 s after it is thrown. A second ball is thrown at \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(\alpha ^ { \circ }\) above the horizontal. It reaches the same maximum height as the first ball.
  4. Use this information to write down
    • the vertical component of the second ball's initial velocity,
    • the time taken for the second ball to reach its greatest height.
    This second ball reaches its greatest height at a point which is 48 m horizontally from the point of projection.
  5. Calculate the values of \(u\) and \(\alpha\).
OCR MEI Paper 1 2021 November Q11
11 A balloon is being inflated. The balloon is modelled as a sphere with radius \(x \mathrm {~cm}\) at time \(t \mathrm {~s}\). The volume \(V \mathrm {~cm} ^ { 3 }\) is given by \(\mathrm { V } = \frac { 4 } { 3 } \pi \mathrm { x } ^ { 3 }\). The rate of increase of volume is inversely proportional to the radius of the balloon. Initially, when \(t = 0\), the radius of the balloon is 5 cm and the volume of the balloon is increasing at a rate of \(21 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\).
  1. Show that \(x\) satisfies the differential equation \(\frac { \mathrm { dx } } { \mathrm { dt } } = \frac { 105 } { 4 \pi \mathrm { x } ^ { 3 } }\).
  2. Find the radius of the balloon after two minutes.
  3. Explain why the model may not be suitable for very large values of \(t\).
OCR MEI Paper 1 2021 November Q12
12 A box of mass \(m \mathrm {~kg}\) slides down a rough slope inclined at \(15 ^ { \circ }\) to the horizontal. The coefficient of friction between the box and the slope is 0.4 . The box has an initial velocity of \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) down the slope. Calculate the distance the box travels before coming to rest.
OCR MEI Paper 1 2021 November Q13
13 In this question \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors in the \(x\) - and \(y\)-directions respectively.
The velocity of a particle at time \(t \mathrm {~s}\) is given by \(\left( 3 t ^ { 2 } \mathbf { i } + 7 \mathbf { j } \right) \mathrm { m } \mathrm { s } ^ { - 1 }\). At time \(t = 0\) the position of the particle with respect to the origin is \(( - \mathbf { i } + 2 \mathbf { j } ) \mathrm { m }\).
  1. Determine the distance of the particle from the origin when \(t = 2\).
  2. Show that the cartesian equation of the path of the particle is \(x = \left( \frac { y - 2 } { 7 } \right) ^ { 3 } - 1\).
  3. At time \(t = 2\), the magnitude of the resultant force acting on the particle is 48 N . Find the mass of the particle.
OCR MEI Paper 1 Specimen Q1
1 Fig. 1 shows a sector of a circle of radius 7 cm . The area of the sector is \(5 \mathrm {~cm} ^ { 2 }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ff44367e-c992-4e79-b255-5a04e0b8e21e-04_222_199_621_306} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} Find the angle \(\theta\) in radians.
OCR MEI Paper 1 Specimen Q2
2 A geometric series has first term 3. The sum to infinity of the series is 8 .
Find the common ratio.
OCR MEI Paper 1 Specimen Q3
3 Solve the inequality \(| 2 x - 1 | \geq 4\).
OCR MEI Paper 1 Specimen Q4
4 Differentiate the following.
  1. \(\sqrt { 1 - 3 x ^ { 2 } }\)
  2. \(\frac { x ^ { 2 } } { 3 x + 2 }\)
OCR MEI Paper 1 Specimen Q5
5 A woman is pulling a loaded sledge along horizontal ground. The only resistance to motion of the sledge is due to friction between it and the ground. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ff44367e-c992-4e79-b255-5a04e0b8e21e-05_314_1024_486_356} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure} At first, she pulls with a force of 100 N inclined at \(32 ^ { \circ }\) to the horizontal, as shown in Fig.5, but the sledge does not move.
  1. Determine the frictional force between the ground and the sledge. Give your answer correct to 3 significant figures.
  2. Next she pulls with a force of 100 N inclined at a smaller angle to the horizontal. The sledge still does not move. Compare the frictional force in this new situation with that in part (a), justifying your answer.
OCR MEI Paper 1 Specimen Q6
6 Fig. 6 shows a partially completed spreadsheet.
This spreadsheet uses the trapezium rule with four strips to estimate \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sqrt { 1 + \sin x } \mathrm {~d} x\). \begin{table}[h]
ABCDE
1\(x\)\(\sin x\)\(y\)
200.00000.00001.00000.5000
30.1250.39270.38271.17591.1759
40.250.78540.70711.30661.3066
50.3751.17810.92391.38701.3870
60.51.57081.00001.41420.7071
75.0766
8
\captionsetup{labelformat=empty} \caption{Fig. 6}
\end{table}
  1. Show how the value in cell B3 is calculated.
  2. Show how the values in cells D2 to D6 are used to calculate the value in cell E7.
  3. Complete the calculation to estimate \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sqrt { 1 + \sin x } \mathrm {~d} x\). Give your answer to 3 significant figures.
OCR MEI Paper 1 Specimen Q7
7 In this question take \(\boldsymbol { g } = \mathbf { 1 0 }\).
A small stone is projected from a point O with a speed of \(26 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\theta\) above the horizontal. The initial velocity and part of the path of the stone are shown in Fig. 7.
You are given that \(\sin \theta = \frac { 12 } { 13 }\).
After \(t\) seconds the horizontal displacement of the stone from O is \(x\) metres and the vertical displacement is \(y\) metres. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ff44367e-c992-4e79-b255-5a04e0b8e21e-07_419_479_904_248} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure}
  1. Using the standard model for projectile motion,
    • show that \(y = 24 t - 5 t ^ { 2 }\),
    • find an expression for \(x\) in terms of \(t\).
    The stone passes through a point A . Point A is 16 m above the level of O .
  2. Find the two possible horizontal distances of A from O . A toy balloon is projected from O with the same initial velocity as the small stone.
  3. Suggest two ways in which the standard model could be adapted.
OCR MEI Paper 1 Specimen Q8
8 Find \(\int x ^ { 2 } \mathrm { e } ^ { 2 x } \mathrm {~d} x\).
OCR MEI Paper 1 Specimen Q9
9 In an experiment, a small box is hit across a floor. After it has been hit, the box slides without rotation. The box passes a point A. The distance the box travels after passing A before coming to rest is \(S\) metres and the time this takes is \(T\) seconds. The only resistance to the box's motion is friction due to the floor. The mass of the box is \(m \mathrm {~kg}\) and the frictional force is a constant \(F\).
    1. Find the equation of motion for the box while it is sliding.
    2. Show that \(S = k T ^ { 2 }\) where \(k = \frac { F } { 2 m }\).
  2. Given that \(k = 1.4\), find the value of the coefficient of friction between the box and the floor.
OCR MEI Paper 1 Specimen Q10
10 In a certain region, the populations of grey squirrels, \(P _ { \mathrm { G } }\) and red squirrels \(P _ { \mathrm { R } }\), at time \(t\) years are modelled by the equations:
\(P _ { \mathrm { G } } = 10000 \left( 1 - \mathrm { e } ^ { - k t } \right)\)
\(P _ { \mathrm { R } } = 20000 \mathrm { e } ^ { - k t }\)
where \(t \geq 0\) and \(k\) is a positive constant.
    1. On the axes in your Printed Answer Book, sketch the graphs of \(P _ { \mathrm { G } }\) and \(P _ { \mathrm { R } }\) on the same axes.
    2. Give the equations of any asymptotes.
  2. What does the model predict about the long term population of
    • grey squirrels
    • red squirrels?
    Grey squirrels and red squirrels compete for food and space. Grey squirrels are larger and more successful than red squirrels.
  3. Comment on the validity of the model given by the equations, giving a reason for your answer.
  4. Show that, according to the model, the rate of decrease of the population of red squirrels is always double the rate of increase of the population of grey squirrels.
  5. When \(t = 3\), the numbers of grey and red squirrels are equal. Find the value of \(k\).
OCR MEI Paper 1 Specimen Q11
11 Fig. 11 shows the curve with parametric equations $$x = 2 \cos \theta , y = \sin \theta , 0 \leq \theta \leq 2 \pi .$$ The point P has parameter \(\frac { 1 } { 4 } \pi\). The tangent at P to the curve meets the axes at A and B . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ff44367e-c992-4e79-b255-5a04e0b8e21e-10_668_1075_543_255} \captionsetup{labelformat=empty} \caption{Fig. 11}
\end{figure}
  1. Show that the equation of the line AB is \(x + 2 y = 2 \sqrt { 2 }\).
  2. Determine the area of the triangle AOB .
OCR MEI Paper 1 Specimen Q12
12 A model boat has velocity \(\mathbf { v } = ( ( 2 t - 2 ) \mathbf { i } + ( 2 t + 2 ) \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) for \(t \geq 0\), where \(t\) is the time in seconds. \(\mathbf { i }\) is the unit vector east and \(\mathbf { j }\) is the unit vector north.
When \(t = 3\), the position vector of the boat is \(( 3 \mathbf { i } + 14 \mathbf { j } ) \mathrm { m }\).
  1. Show that the boat is never instantaneously at rest.
  2. Determine any times at which the boat is moving directly northwards.
  3. Determine any times at which the boat is north-east of the origin.
OCR MEI Paper 1 Specimen Q14
14 Blocks A and B are connected by a light rigid horizontal bar and are sliding on a rough horizontal surface. A light horizontal string exerts a force of 40 N on B .
This situation is shown in Fig. 14, which also shows the direction of motion, the mass of each of the blocks and the resistances to their motion. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ff44367e-c992-4e79-b255-5a04e0b8e21e-11_266_1283_664_255} \captionsetup{labelformat=empty} \caption{Fig. 14}
\end{figure}
  1. Calculate the tension in the bar. The string breaks while the blocks are sliding. The resistances to motion are unchanged.
  2. Determine
    • the magnitude of the new force in the bar,
    • whether the bar is in tension or in compression.
OCR MEI Paper 1 Specimen Q15
15 Fig. 15 shows a uniform shelf AB of weight \(W \mathrm {~N}\).
The shelf is 180 cm long and rests on supports at points C and D . Point C is 30 cm from A and point D is 60 cm from B .
side view \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ff44367e-c992-4e79-b255-5a04e0b8e21e-11_284_1169_1987_383} \captionsetup{labelformat=empty} \caption{Fig. 15}
\end{figure} Determine the range of positions a point load of \(3 W\) could be placed on the shelf without the shelf tipping. \section*{Copyright Information:} 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. For queries or further information please contact the Copyright Team, The Triangle Building, Shaftesbury Road, Cambridge CB2 8EA.
OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.
OCR MEI Paper 1 2019 June Q1
1 In this question you must show detailed reasoning. Show that \(\int _ { 4 } ^ { 9 } ( 2 x + \sqrt { x } ) \mathrm { d } x = \frac { 233 } { 3 }\).
OCR MEI Paper 1 2023 June Q5
5 In this question you must show detailed reasoning.
  1. Find the coordinates of the two stationary points on the graph of \(y = 15 - x ^ { 2 } - \frac { 16 } { x ^ { 2 } }\).
  2. Show that both these stationary points are maximum points.
OCR MEI Paper 1 2020 November Q7
7 In this question you must show detailed reasoning. The function \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = x ^ { 3 } + x ^ { 2 } - 8 x - 12\) for all values of \(x\).
  1. Use the factor theorem to show that \(( x + 2 )\) is a factor of \(\mathrm { f } ( x )\).
  2. Solve the equation \(\mathrm { f } ( x ) = 0\).
OCR MEI Paper 1 2020 November Q10
10 In this question you must show detailed reasoning. Fig. 10 shows the curve given parametrically by the equations \(\mathrm { x } = \frac { 1 } { \mathrm { t } ^ { 2 } } , \mathrm { y } = \frac { 1 } { \mathrm { t } ^ { 3 } } - \frac { 1 } { \mathrm { t } }\), for \(t > 0\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7de77679-59c0-4431-a9cb-6ab11d2f9062-07_611_595_708_260} \captionsetup{labelformat=empty} \caption{Fig. 10}
\end{figure}
  1. Show that \(\frac { d y } { d x } = \frac { 3 - t ^ { 2 } } { 2 t }\).
  2. Find the coordinates of the point on the curve at which the tangent to the curve is parallel to the line \(4 \mathrm { y } + \mathrm { x } = 1\).
  3. Find the cartesian equation of the curve. Give your answer in factorised form.