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OCR MEI Paper 1 2019 June Q5
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
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
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
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
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
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
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
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
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 \(R\).