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OCR MEI AS Paper 1 2021 November Q5
5 marks Standard +0.3
5 The diagram shows the triangle ABC in which \(\mathrm { AC } = 13 \mathrm {~cm}\) and AB is the shortest side. The perimeter of the triangle is 32 cm . The area is \(24 \mathrm {~cm} ^ { 2 }\) and \(\sin \mathrm { B } = \frac { 4 } { 5 }\). Determine the lengths of AB and BC .
OCR MEI AS Paper 1 2021 November Q6
8 marks Moderate -0.3
6 The displacement of a particle is modelled by the equation \(\mathrm { s } = 7 + 4 \mathrm { t } - \mathrm { t } ^ { 2 }\), where \(s\) metres is the displacement from the origin at time \(t\) seconds. The diagram shows part of the displacement-time graph for the particle. The point \(( 2,11 )\) is the maximum point on the graph. \includegraphics[max width=\textwidth, alt={}, center]{5428eabf-431d-4db1-8c25-1f2b9570d9aa-4_513_1381_422_255}
  1. Kai argues that the point \(( 2,11 )\) is on the graph, so the particle has travelled a distance of 11 metres in the first 2 seconds. Comment on the validity of Kai's argument.
  2. Determine the total distance the particle travels in the first 10 seconds.
  3. Find an expression for the velocity of the particle at time \(t\).
  4. Find the speed of the particle when \(t = 10\).
OCR MEI AS Paper 1 2021 November Q7
6 marks Moderate -0.3
7 The diagram shows part of a curve which passes through the point \(( 1,0 )\). \includegraphics[max width=\textwidth, alt={}, center]{5428eabf-431d-4db1-8c25-1f2b9570d9aa-4_711_704_1722_258} The gradient of the curve is given by \(\frac { d y } { d x } = 6 x + \frac { 8 } { x ^ { 3 } }\).
Determine whether the curve passes through the point \(( 2,12 )\).
OCR MEI AS Paper 1 2021 November Q9
9 marks Moderate -0.8
9
  1. Sketch both of the following on the axes provided in the Printed Answer Booklet.
    1. The curve \(\mathrm { y } = \frac { 12 } { \mathrm { x } }\), stating the coordinates of at least one point on the curve.
    2. The line \(y = 2 x + 8\), stating the coordinates of the points at which the line crosses the axes.
  2. In this question you must show detailed reasoning. Determine the exact coordinates of the points of intersection of the curve and the line.
OCR MEI AS Paper 1 2021 November Q10
10 marks Moderate -0.3
10 A rescue worker is lowered from a helicopter on a rope. She attaches a second rope to herself and to a woman in difficulties on the ground. The helicopter winches both women upwards with the rescued woman vertically below the rescue worker, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{5428eabf-431d-4db1-8c25-1f2b9570d9aa-6_509_460_408_262} The model for this motion uses the following modelling assumptions:
  • each woman can be modelled as a particle;
  • the ropes are both light and inextensible;
  • there is no air resistance to the motion;
  • the motion is in a vertical line.
    1. Explain what it means when the women are each 'modelled as a particle'.
    2. Explain what 'light' means in this context.
The tension in the rope to the helicopter is 1500 N . The rescue worker has a mass of 65 kg and the rescued woman has a mass of 75 kg .
  • Draw a diagram showing the forces on the two women.
  • Write down the equation of motion of the two women considered as a single particle.
  • Calculate the acceleration of the women.
  • Determine the tension in the rope connecting the two women.
    •
    OCR MEI AS Paper 1 2021 November Q11
    10 marks Moderate -0.8
    11 On the day that a new consumer product went on sale (day zero), a call centre received 1 call about it. On the 2nd day after day zero the call centre received 3 calls, and on the 10th day after day zero there were 200 calls. Two models were proposed to model \(N\), the number of calls received \(t\) days after day zero.
    Model 1 is a linear model \(\mathrm { N } = \mathrm { mt } + \mathrm { c }\).
    1. Determine the values of \(m\) and \(c\) which best model the data for 2 days and 10 days after day zero.
    2. State the rate of increase in calls according to model 1.
    3. Explain why this model is not suitable when \(t = 1\). Model 2 is an exponential model \(\mathbf { N } = e ^ { 0.53 t }\).
    4. Verify that this is a good model for the number of calls when \(t = 2\) and \(t = 10\).
    5. Determine the rate of increase in calls when \(t = 10\) according to model 2 .
    OCR MEI AS Paper 1 Specimen Q1
    2 marks Easy -1.2
    1 Simplify \(\frac { \left( 2 x ^ { 2 } y \right) ^ { 3 } \times 4 x ^ { 3 } y ^ { 5 } } { 2 x y ^ { 10 } }\).
    OCR MEI AS Paper 1 Specimen Q2
    3 marks Easy -1.2
    2 Find the coefficient of \(x ^ { 4 }\) in the binomial expansion of \(( x - 3 ) ^ { 5 }\).
    OCR MEI AS Paper 1 Specimen Q3
    3 marks Easy -1.8
    3 Fig. 3 shows a particle of weight 8 N on a rough horizontal table.
    The particle is being pulled by a horizontal force of 10 N .
    It remains at rest in equilibrium. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{970d2349-7705-4b66-9931-83613e5d852f-3_204_454_1311_255} \captionsetup{labelformat=empty} \caption{Fig. 3}
    \end{figure}
    1. What information given in the question, tells you that the forces shown in Fig. 3 cannot be the only forces acting on the particle?
    2. The only other forces acting on the particle are due to the particle being on the table. State the types of these forces and their magnitudes.
    OCR MEI AS Paper 1 Specimen Q4
    3 marks Easy -1.2
    4
    1. Express \(x ^ { 2 } + 4 x + 7\) in the form \(( x + b ) ^ { 2 } + c\).
    2. Explain why the minimum point on the curve \(y = ( x + b ) ^ { 2 } + c\) occurs when \(x = - b\).
    OCR MEI AS Paper 1 Specimen Q5
    5 marks Moderate -0.8
    5 Particle P moves on a straight line that contains the point O .
    At time \(t\) seconds the displacement of P from O is \(s\) metres, where \(s = t ^ { 3 } - 3 t ^ { 2 } + 3\).
    1. Determine the times when the particle has zero velocity.
    2. Find the distances of P from O at the times when it has zero velocity.
    OCR MEI AS Paper 1 Specimen Q6
    4 marks Moderate -0.3
    6 Two points, \(A\) and \(B\), have position vectors \(\mathbf { a } = \mathbf { i } - 3 \mathbf { j }\) and \(\mathbf { b } = 4 \mathbf { i } + 3 \mathbf { j }\).
    The point C lies on the line \(y = 1\). The lengths of the line segments AC and BC are equal. Determine the position vector of \(C\).
    OCR MEI AS Paper 1 Specimen Q7
    4 marks Moderate -0.8
    7 A car is usually driven along the whole of a 5 km stretch of road at a constant speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). On one occasion, during a period of 50 seconds, the speed of the car is as shown by the speed-time graph in Fig. 7.
    The rest of the 5 km is travelled at \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{970d2349-7705-4b66-9931-83613e5d852f-5_510_1016_589_296} \captionsetup{labelformat=empty} \caption{Fig. 7}
    \end{figure} How much more time than usual did the journey take on this occasion?
    Show your working clearly.
    OCR MEI AS Paper 1 Specimen Q8
    11 marks Moderate -0.8
    8 A circle has equation \(( x - 2 ) ^ { 2 } + ( y + 3 ) ^ { 2 } = 25\).
    1. Write down
    OCR MEI AS Paper 1 Specimen Q9
    8 marks Moderate -0.3
    9 A biologist is investigating the growth of bacteria in a piece of bread.
    He believes that the number, \(N\), of bacteria after \(t\) hours may be modelled by the relationship \(N = A \times 2 ^ { k t }\), where \(A\) and \(k\) are constants.
    1. Show that, according to the model, the graph of \(\log _ { 10 } N\) against \(t\) is a straight line. Give, in terms of \(A\) and \(k\),
      The biologist measures the number of bacteria at regular intervals over 22 hours and plots a graph of \(\log _ { 10 } N\) against \(t\). He finds that the graph is approximately a straight line with gradient 0.20 . The line crosses the vertical axis at 2.0 .
    2. Find the values of \(A\) and \(k\).
    3. Use the model to predict the number of bacteria after 24 hours.
    4. Give a reason why the model may not be appropriate for large values of \(t\).
    OCR MEI AS Paper 1 Specimen Q10
    12 marks Standard +0.8
    10
    1. Sketch the graph of \(y = \frac { 1 } { x } + a\), where \(a\) is a positive constant.
    OCR MEI AS Paper 1 Specimen Q11
    6 marks Moderate -0.3
    11 In this question you must show detailed reasoning.
    Determine for what values of \(k\) the graphs \(y = 2 x ^ { 2 } - k x\) and \(y = x ^ { 2 } - k\) intersect.
    OCR MEI AS Paper 1 Specimen Q12
    9 marks Moderate -0.3
    12 A box hangs from a balloon by means of a light inelastic string. The string is always vertical. The mass of the box is 15 kg . Catherine initially models the situation by assuming that there is no air resistance to the motion of the box. Use Catherine's model to calculate the tension in the string if:
    1. the box is held at rest by the tension in the string,
    2. the box is instantaneously at rest and accelerating upwards at \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\),
    3. the box is moving downwards at \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and accelerating upwards at \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Catherine now carries out an experiment to find the magnitude of the air resistance on the box when it is moving.
      At a time when the box is accelerating downwards at \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), she finds that the tension in the string is 140 N .
    4. Calculate the magnitude of the air resistance at that time. Give, with a reason, the direction of motion of the box. \section*{END OF QUESTION PAPER}
    OCR MEI AS Paper 2 2019 June Q1
    3 marks Easy -1.2
    1 Solve the equation \(4 x ^ { - \frac { 1 } { 2 } } = 7\), giving your answer as a fraction in its lowest terms.
    OCR MEI AS Paper 2 2019 June Q2
    2 marks Moderate -0.8
    2 Fig. 2 shows a triangle with one angle of \(117 ^ { \circ }\) given. The lengths are given in centimetres. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{11e5167f-9f95-4494-9b66-b59fdce8b1ef-3_300_791_589_244} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} Calculate the area of the triangle, giving your answer correct to 3 significant figures.
    OCR MEI AS Paper 2 2019 June Q3
    3 marks Moderate -0.8
    3 Without using a calculator, prove that \(3 \sqrt { 2 } > 2 \sqrt { 3 }\).
    OCR MEI AS Paper 2 2019 June Q4
    3 marks Easy -1.2
    4 The equation of a circle is \(x ^ { 2 } + y ^ { 2 } + 8 x - 6 y - 39 = 0\).
    1. Find the coordinates of the centre of the circle.
    2. Find the radius of the circle.
    OCR MEI AS Paper 2 2019 June Q5
    8 marks Easy -1.2
    5 Each day John either cycles to work or goes on the bus.
    • If it is raining when John is ready to set off for work, the probability that he cycles to work is 0.4.
    • If it is not raining when John is ready to set off for work, the probability that he cycles to work is 0.9 .
    • The probability that it is raining when he is ready to set off for work is 0.2 .
    You should assume that days on which it rains occur randomly and independently.
    1. Draw a tree diagram to show the possible outcomes and their associated probabilities.
    2. Calculate the probability that, on a randomly chosen day, John cycles to work. John works 5 days each week.
    3. Calculate the probability that he cycles to work every day in a randomly chosen working week.
    OCR MEI AS Paper 2 2019 June Q6
    13 marks Moderate -0.8
    6 The large data set gives information about life expectancy at birth for males and females in different London boroughs. Fig. 6.1 shows summary statistics for female life expectancy at birth for the years 2012-2014. Fig. 6.2 shows summary statistics for male life expectancy at birth for the years 2012-2014. \section*{Female Life Expectancy at Birth} \begin{table}[h]
    n32
    Mean84.2313
    s1.1563
    \(\sum x\)2695.4
    \(\sum x ^ { 2 }\)227078.36
    Min82.1
    Q183.45
    Median84
    Q384.9
    Max86.7
    \captionsetup{labelformat=empty} \caption{Fig. 6.1}
    \end{table} Male Life Expectancy at Birth \begin{table}[h]
    n32
    Mean80.2844
    s1.4294
    \(\sum x\)2569.1
    \(\sum x ^ { 2 }\)206321.93
    Min77.6
    Q179
    Median80.25
    Q381.15
    Max83.3
    \captionsetup{labelformat=empty} \caption{Fig. 6.2}
    \end{table}
    1. Use the information in Fig. 6.1 and Fig. 6.2 to draw two box plots. Draw one box plot for female life expectancy at birth in London boroughs and one box plot for male life expectancy at birth in London boroughs.
    2. Compare and contrast the distribution of male life expectancy at birth with the distribution of female life expectancy at birth in London boroughs in 2012-2014. Lorraine, who lives in Lancashire, says she wishes her daughter (who was born in 2013) had been born in the London borough of Barnet, because her daughter would have had a higher life expectancy.
    3. Give two reasons why there is no evidence in the large data set to support Lorraine's comment.
    4. Use the mean and standard deviation for the summary statistics given in Fig. 6.1 and Fig. 6.2 to show that there is at least one outlier in each set. The scatter diagram in Fig. 6.3 shows male life expectancy at birth plotted against female life expectancy at birth for London boroughs in 2012-14. The outliers have been removed. Male life expectancy at birth against female life expectancy at birth \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{11e5167f-9f95-4494-9b66-b59fdce8b1ef-5_593_1054_1260_246} \captionsetup{labelformat=empty} \caption{Fig. 6.3}
      \end{figure}
    5. Describe the association between male life expectancy at birth and female life expectancy at birth in London boroughs in 2012-14.
    OCR MEI AS Paper 2 2019 June Q7
    8 marks Moderate -0.8
    7
    1. Find \(\int x ^ { 3 } \left( 15 x + \frac { 11 } { \sqrt [ 3 ] { x } } \right) \mathrm { d } x\).
    2. Show that \(\int _ { 0 } ^ { 8 } x ^ { 3 } \left( 15 x + \frac { 11 } { \sqrt [ 3 ] { x } } \right) \mathrm { d } x = a \times 2 ^ { 11 }\), where \(a\) is a positive integer to be determined.