OCR MEI Paper 1 (Paper 1) 2022 June

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 \(( 15,1 )\).
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1. Find the distance travelled by the particle in the first 15 s .
2. Calculate the velocity of the particle between \(t = 10\) and \(t = 15\).

2 Express \(\frac { 13 - x } { ( x - 3 ) ( x + 2 ) }\) in partial fractions.

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 \(- \pi \leqslant x \leqslant \pi\).

4 Using an appropriate expansion show that, for sufficiently small values of \(x\), \(\frac { 1 - x } { ( 2 + x ) ^ { 2 } } \approx \frac { 1 } { 4 } - \frac { 1 } { 2 } x + \frac { 7 } { 16 } x ^ { 2 }\).

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.
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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)

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.
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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.

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.

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 \(\mathrm { P } ( 8,4 )\).
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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 \(y\).

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.

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 \(\theta = \frac { 1 } { 3 } \pi\).

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 \(k\).

12 Prove by contradiction that 3 is the only prime number which is 1 less than a square number.

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 \(R\).

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.