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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.
OCR MEI Paper 1 2023 June Q9
9 The gradient of a curve is given by \(\frac { d y } { d x } = e ^ { x } - 4 e ^ { - x }\).
  1. Show that the \(x\)-coordinate of any point on the curve at which the gradient is 3 satisfies the equation \(\left( e ^ { x } \right) ^ { 2 } - 3 e ^ { x } - 4 = 0\).
  2. Hence show that there is only one point on the curve at which the gradient is 3 , stating the exact value of its \(x\)-coordinate.
  3. The curve passes through the point \(( 0,0 )\). Show that when \(x = 1\) the curve is below the \(x\)-axis.
OCR MEI Paper 1 2023 June Q10
10 The diagram shows the graph of \(\mathrm { y } = 1.5 + \sin ^ { 2 } \mathrm { x }\) for \(0 \leqslant x \leqslant 2 \pi\).
\includegraphics[max width=\textwidth, alt={}, center]{8eeff88d-8b05-43c6-86a5-bd82221c0bea-07_512_1278_322_242}
  1. Show that the equation of the graph can be written in the form \(\mathrm { y } = \mathrm { a } - \mathrm { b } \cos 2 \mathrm { x }\) where \(a\) and \(b\) are constants to be determined.
  2. Write down the period of the function \(1.5 + \sin ^ { 2 } x\).
  3. Determine the \(x\)-coordinates of the points of intersection of the graph of \(y = 1.5 + \sin ^ { 2 } x\) with the graph of \(\mathrm { y } = 1 + \cos 2 \mathrm { x }\) in the interval \(0 \leqslant x \leqslant 2 \pi\).
OCR MEI Paper 1 2023 June Q11
11 The height \(h \mathrm {~cm}\) of a sunflower plant \(t\) days after planting the seed is modelled by \(\mathrm { h } = \mathrm { a } + \mathrm { b }\) Int for \(t \geqslant 9\), where \(a\) and \(b\) are constants. The sunflower is 10 cm tall 10 days after planting and 200 cm tall 85 days after planting.
    1. Show that the value of \(b\) which best models these values is 88.8 correct to \(\mathbf { 3 }\) significant figures.
    2. Find the corresponding value of \(a\).
    1. Explain why the model is not suitable for small positive values of \(t\).
    2. Explain why the model is not suitable for very large positive values of \(t\).
  3. Show that the model indicates that the sunflower grows to 1 m in height in less than half the time it takes to grow to 2 m .
  4. Find the value of \(t\) for which the rate of growth is 3 cm per day.
OCR MEI Paper 1 2023 June Q12
12 In this question the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal and vertically upwards respectively. A particle has mass 2 kg .
  1. Write down its weight as a vector. A horizontal force of 3 N in the \(\mathbf { i }\) direction and a force \(\mathbf { F } = ( - 4 \mathbf { i } + 12 \mathbf { j } ) \mathrm { N }\) act on the particle.
  2. Determine the acceleration of the particle.
  3. The initial velocity of the particle is \(5 \mathbf { i } \mathrm {~ms} ^ { - 1 }\). Find the velocity of the particle after 4 s .
  4. Find the extra force that must be applied to the particle for it to move at constant velocity.
OCR MEI Paper 1 2023 June Q13
13 A block of mass 8 kg is placed on a rough plane inclined at \(15 ^ { \circ }\) to the horizontal. The coefficient of friction between the block and the plane is 0.3 . One end of a light rope is attached to the block. The rope passes over a smooth pulley fixed at the top of the plane, and a sphere of mass 5 kg , attached to the other end of the rope, hangs vertically below the pulley. The part of the rope between the block and the pulley is parallel to the plane. The system is released from rest, and as the sphere falls the block moves directly up the plane with acceleration \(a \mathrm {~ms} ^ { - 2 }\).
\includegraphics[max width=\textwidth, alt={}, center]{8eeff88d-8b05-43c6-86a5-bd82221c0bea-08_252_803_1560_246}
  1. On the diagram in the Printed Answer Booklet, show all the forces acting on the block and on the sphere.
  2. Write down the equation of motion for the sphere.
  3. Determine the value of \(a\).
OCR MEI Paper 1 2023 June Q14
14
  1. Use the laws of logarithms to show that \(\log _ { 10 } 200 - \log _ { 10 } 20\) is equal to 1 . The first three terms of a sequence are \(\log _ { 10 } 20 , \log _ { 10 } 200 , \log _ { 10 } 2000\).
  2. Show that the sequence is arithmetic.
  3. Find the exact value of the sum of the first 50 terms of this sequence.
OCR MEI Paper 1 2023 June Q15
15 A projectile is launched from a point on level ground with an initial velocity \(u\) at an angle \(\theta\) above the horizontal.
  1. Show that the range of the projectile is given by \(\frac { 2 u ^ { 2 } \sin \theta \cos \theta } { g }\).
  2. Determine the set of values of \(\theta\) for which the maximum height of the projectile is greater than the range, where \(\theta\) is an acute angle. Give your answer in degrees.
OCR MEI Paper 1 2024 June Q1
1 A student states that \(1 + x ^ { 2 } < ( 1 + x ) ^ { 2 }\) for all values of \(x\).
Using a counter example, show that the student is wrong.
OCR MEI Paper 1 2024 June Q2
2 A car of mass 1400 kg pulls a trailer of mass 400 kg along a straight horizontal road. The engine of the car produces a driving force of 6000 N . A resistance of 800 N acts on the car. A resistance of 300 N acts on the trailer. The tow-bar between the car and the trailer is light and horizontal.
  1. Draw a force diagram showing all the horizontal forces on the car and the trailer.
  2. Calculate the acceleration of the car and trailer.
OCR MEI Paper 1 2024 June Q3
3 A particle hangs at the end of a string. A horizontal force of magnitude \(F \mathrm {~N}\) acting on the particle holds it in equilibrium so that the string makes an angle of \(20 ^ { \circ }\) with the vertical, as shown in the diagram. The tension in the string is 12 N .
\includegraphics[max width=\textwidth, alt={}, center]{1d0ca3d5-6529-435f-a0b8-50ea4859adde-04_357_374_1409_239}
  1. Find the value of \(F\).
  2. Find the mass of the particle.
OCR MEI Paper 1 2024 June Q4
4 The vectors \(\mathbf { v } _ { 1 }\) and \(\mathbf { v } _ { 2 }\) are defined by \(\mathbf { v } _ { 1 } = 2 \mathrm { a } \mathbf { i } + \mathrm { bj }\) and \(\mathbf { v } _ { 2 } = b \mathbf { i } - 3 \mathbf { j }\) where \(a\) and \(b\) are constants. Given that \(3 \mathbf { v } _ { 1 } + \mathbf { v } _ { 2 } = 22 \mathbf { i } - 9 \mathbf { j }\), find the values of \(a\) and \(b\).
OCR MEI Paper 1 2024 June Q5
5
  1. Make \(y\) the subject of the formula \(\log _ { 10 } ( y - k ) = x \log _ { 10 } 2\), where \(k\) is a positive constant.
  2. Sketch the graph of \(y\) against \(x\).
OCR MEI Paper 1 2024 June Q6
6 Given that \(\mathrm { f } ( x ) = 2 x ^ { 2 } + 3\), show from first principles that \(\mathrm { f } ^ { \prime } ( x ) = 4 x\).
OCR MEI Paper 1 2024 June Q7
7 A rectangular book ABCD rests on a smooth horizontal table. The length of AB is 28 cm and the length of AD is 18 cm . The following five forces act on the book, as shown in the diagram.
  • 4 N at A in the direction AD
  • 5 N at B in the direction BC
  • 3 N at B in the direction BA
  • 9 N at D in the direction DA
  • 3 N at D in the direction DC
    \includegraphics[max width=\textwidth, alt={}, center]{1d0ca3d5-6529-435f-a0b8-50ea4859adde-06_663_830_774_242}
    1. Show that the resultant of the forces acting on the book has zero magnitude.
    2. Find the total moment of the forces about the centre of the book. Give your answer in Nm .
    3. Describe how the book will move under the action of these forces.
OCR MEI Paper 1 2024 June Q8
8 The equation of a curve is \(\mathrm { y } = \sqrt { \sin 4 \mathrm { x } } + 2 \cos 2 \mathrm { x }\), where \(x\) is in radians.
  1. Show that, for small values of \(x , y \approx 2 \sqrt { x } + 2 - 4 x ^ { 2 }\). The diagram shows the region bounded by the curve \(\mathrm { y } = \sqrt { \sin 4 \mathrm { x } } + 2 \cos 2 \mathrm { x }\), the axes and the line \(x = 0.1\).
    \includegraphics[max width=\textwidth, alt={}, center]{1d0ca3d5-6529-435f-a0b8-50ea4859adde-07_499_881_589_223}
  2. In this question you must show detailed reasoning. Use the approximation in part (a) to estimate the area of this region.
OCR MEI Paper 1 2024 June Q9
9 A child throws a pebble of mass 40 g vertically downwards with a speed of \(6 \mathrm {~ms} ^ { - 1 }\) from a point 0.8 m above a sandy beach.
  1. Calculate the speed at which the pebble hits the beach. The pebble travels 3 cm through the sand before coming to rest.
  2. Find the magnitude of the resistance force of the sand on the pebble, assuming it is constant. Give your answer correct to \(\mathbf { 3 }\) significant figures.
OCR MEI Paper 1 2024 June Q10
10 Zac is measuring the growth of a culture of bacteria in a laboratory. The initial area of the culture is \(8 \mathrm {~cm} ^ { 2 }\). The area one day later is \(8.8 \mathrm {~cm} ^ { 2 }\). At first, Zac uses a model of the form \(\mathrm { A } = \mathrm { a } + \mathrm { bt }\), where \(A \mathrm {~cm} ^ { 2 }\) is the area \(t\) days after he begins measuring and \(a\) and \(b\) are constants.
  1. Find the values of \(a\) and \(b\) that best model the initial area and the area one day later.
  2. Calculate the value of \(t\) for which the model predicts an area of \(15 \mathrm {~cm} ^ { 2 }\).
  3. Zac notices the area covered by the culture increases by \(10 \%\) each day. Explain why this model may not be suitable after the first day. Zac decides to use a different model for \(A\). His new model is \(\mathrm { A } = \mathrm { Pe } ^ { \mathrm { kt } }\), where \(P\) and \(k\) are constants.
  4. Find the values of \(P\) and \(k\) that best model the initial area and the area one day later.
  5. Calculate the value of \(t\) for which the area reaches \(15 \mathrm {~cm} ^ { 2 }\) according to this model.
  6. Explain why this model may not be suitable for large values of \(t\).