Questions — OCR MEI (4456 questions)

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OCR MEI AS Paper 2 Specimen Q8
7 marks Moderate -0.8
8 In an experiment, the temperature of a hot liquid is measured every minute.
The difference between the temperature of the hot liquid and room temperature is \(D ^ { \circ } \mathrm { C }\) at time \(t\) minutes. Fig. 8 shows the experimental data. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{05376a51-e768-4b45-9c18-c98255a4bd70-07_1144_1541_497_276} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure} It is thought that the model \(D = 70 \mathrm { e } ^ { - 0.03 t }\) might fit the data.
  1. Write down the derivative of \(\mathrm { e } ^ { - 0.03 t }\).
  2. Explain how you know that \(70 \mathrm { e } ^ { - 0.03 t }\) is a decreasing function of \(t\).
  3. Calculate the value of \(70 \mathrm { e } ^ { - 0.03 t }\) when
    1. \(\quad t = 0\),
    2. \(t = 20\).
  4. Using your answers to parts (b) and (c), discuss how well the model \(D = 70 \mathrm { e } ^ { - 0.03 t }\) fits the data.
OCR MEI AS Paper 2 Specimen Q9
7 marks Easy -1.3
9 Fig. 9.1 shows box and whisker diagrams which summarise the birth rates per 1000 people for all the countries in three of the regions as given in the pre-release data set.
The diagrams were drawn as part of an investigation comparing birth rates in different regions of the world. Africa (Sub-Saharan) \includegraphics[max width=\textwidth, alt={}, center]{05376a51-e768-4b45-9c18-c98255a4bd70-08_104_991_557_730} East and South East Asia \includegraphics[max width=\textwidth, alt={}, center]{05376a51-e768-4b45-9c18-c98255a4bd70-08_109_757_744_671} Caribbean \includegraphics[max width=\textwidth, alt={}, center]{05376a51-e768-4b45-9c18-c98255a4bd70-08_99_369_982_730} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{05376a51-e768-4b45-9c18-c98255a4bd70-08_202_1595_1153_299} \captionsetup{labelformat=empty} \caption{Fig. 9.1}
\end{figure}
  1. Discuss the distributions of birth rates in these regions of the world. Make three different statements. You should refer to both information from the box and whisker diagrams and your knowledge of the large data set.
  2. The birth rates for all the countries in Australasia are shown below.
    CountryBirth rate per 1000
    Australia12.19
    New Zealand13.4
    Papua New Guinea24.89
    1. Explain why the calculation below is not a correct method for finding the birth rate per 1000 for Australasia as a whole. $$\frac { 12.19 + 13.4 + 24.89 } { 3 } \approx 16.83$$
    2. Without doing any calculations, explain whether the birth rate per 1000 for Australasia as a whole is higher or lower than 16.83 . The scatter diagram in Fig. 9.2 shows birth rate per 1000 and physicians/ 1000 population for all the countries in the pre-release data set. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{05376a51-e768-4b45-9c18-c98255a4bd70-09_898_1698_386_274} \captionsetup{labelformat=empty} \caption{Fig. 9.2}
      \end{figure}
  3. Describe the correlation in the scatter diagram.
  4. Discuss briefly whether the scatter diagram shows that high birth rates would be reduced by increasing the number of physicians in a country.
OCR MEI AS Paper 2 Specimen Q10
9 marks Moderate -0.3
10 A company operates trains. The company claims that \(92 \%\) of its trains arrive on time. You should assume that in a random sample of trains, they arrive on time independently of each other.
  1. Assuming that \(92 \%\) of the company's trains arrive on time, find the probability that in a random sample of 30 trains operated by this company
    1. exactly 28 trains arrive on time,
    2. more than 27 trains arrive on time. A journalist believes that the percentage of trains operated by this company which arrive on time is lower than \(92 \%\).
  2. To investigate the journalist's belief a hypothesis test will be carried out at the \(1 \%\) significance level. A random sample of 18 trains is selected. For this hypothesis test,
OCR MEI AS Paper 2 Specimen Q12
3 marks Standard +0.3
12 Given that \(\arcsin x = \arccos y\), prove that \(x ^ { 2 } + y ^ { 2 } = 1\). [Hint: Let \(\arcsin x = \theta\) ] \section*{END OF QUESTION PAPER}
OCR MEI Paper 1 2018 June Q1
3 marks Easy -1.8
1 Show that ( \(x - 2\) ) is a factor of \(3 x ^ { 3 } - 8 x ^ { 2 } + 3 x + 2\).
OCR MEI Paper 1 2018 June Q2
2 marks Moderate -0.8
2 By considering a change of sign, show that the equation \(\mathrm { e } ^ { x } - 5 x ^ { 3 } = 0\) has a root between 0 and 1 .
OCR MEI Paper 1 2018 June Q3
4 marks Standard +0.3
3 In this question you must show detailed reasoning.
Solve the equation \(\sec ^ { 2 } \theta + 2 \tan \theta = 4\) for \(0 ^ { \circ } \leqslant \theta < 360 ^ { \circ }\).
OCR MEI Paper 1 2018 June Q4
4 marks Easy -1.2
4 Rory pushes a box of mass 2.8 kg across a rough horizontal floor against a resistance of 19 N . Rory applies a constant horizontal force. The box accelerates from rest to \(1.2 \mathrm {~ms} ^ { - 1 }\) as it travels 1.8 m .
  1. Calculate the acceleration of the box.
  2. Find the magnitude of the force that Rory applies.
OCR MEI Paper 1 2018 June Q5
4 marks Moderate -0.3
5 The position vector \(\mathbf { r }\) metres of a particle at time \(t\) seconds is given by $$\mathbf { r } = \left( 1 + 12 t - 2 t ^ { 2 } \right) \mathbf { i } + \left( t ^ { 2 } - 6 t \right) \mathbf { j }$$
  1. Find an expression for the velocity of the particle at time \(t\).
  2. Determine whether the particle is ever stationary.
OCR MEI Paper 1 2018 June Q6
6 marks Moderate -0.8
6 Aleela and Baraka are saving to buy a car. Aleela saves \(\pounds 50\) in the first month. She increases the amount she saves by \(\pounds 20\) each month.
  1. Calculate how much she saves in two years. Baraka also saves \(\pounds 50\) in the first month. The amount he saves each month is \(12 \%\) more than the amount he saved in the previous month.
  2. Explain why the amounts Baraka saves each month form a geometric sequence.
  3. Determine whether Baraka saves more in two years than Aleela. Answer all the questions
    Section B (77 marks)
OCR MEI Paper 1 2018 June Q7
3 marks Moderate -0.3
7 A rod of length 2 m hangs vertically in equilibrium. Parallel horizontal forces of 30 N and 50 N are applied to the top and bottom and the rod is held in place by a horizontal force \(F \mathrm {~N}\) applied \(x \mathrm {~m}\) below the top of the rod as shown in Fig. 7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{904025c9-6d68-4344-bd41-8c0fccfcf92f-05_445_390_609_824} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure}
  1. Find the value of \(F\).
  2. Find the value of \(x\).
OCR MEI Paper 1 2018 June Q8
6 marks Standard +0.3
8
  1. Show that \(8 \sin ^ { 2 } x \cos ^ { 2 } x\) can be written as \(1 - \cos 4 x\).
  2. Hence find \(\int \sin ^ { 2 } x \cos ^ { 2 } x \mathrm {~d} x\).
OCR MEI Paper 1 2018 June Q9
10 marks Standard +0.3
9 A pebble is thrown horizontally at \(14 \mathrm {~ms} ^ { - 1 }\) from a window which is 5 m above horizontal ground. The pebble goes over a fence 2 m high \(d \mathrm {~m}\) away from the window as shown in Fig. 9. The origin is on the ground directly below the window with the \(x\)-axis horizontal in the direction in which the pebble is thrown and the \(y\)-axis vertically upwards. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{904025c9-6d68-4344-bd41-8c0fccfcf92f-06_538_1082_452_488} \captionsetup{labelformat=empty} \caption{Fig. 9}
\end{figure}
  1. Find the time the pebble takes to reach the ground.
  2. Find the cartesian equation of the trajectory of the pebble.
  3. Find the range of possible values for \(d\).
OCR MEI Paper 1 2018 June Q10
8 marks Challenging +1.2
10 Fig. 10 shows the graph of \(y = ( k - x ) \ln x\) where \(k\) is a constant ( \(k > 1\) ). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{904025c9-6d68-4344-bd41-8c0fccfcf92f-06_454_1266_1564_395} \captionsetup{labelformat=empty} \caption{Fig. 10}
\end{figure} Find, in terms of \(k\), the area of the finite region between the curve and the \(x\)-axis.
OCR MEI Paper 1 2018 June Q11
7 marks Standard +0.3
11 Fig. 11 shows two blocks at rest, connected by a light inextensible string which passes over a smooth pulley. Block A of mass 4.7 kg rests on a smooth plane inclined at \(60 ^ { \circ }\) to the horizontal. Block B of mass 4 kg rests on a rough plane inclined at \(25 ^ { \circ }\) to the horizontal. On either side of the pulley, the string is parallel to a line of greatest slope of the plane. Block B is on the point of sliding up the plane. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{904025c9-6d68-4344-bd41-8c0fccfcf92f-07_332_931_443_575} \captionsetup{labelformat=empty} \caption{Fig. 11}
\end{figure}
  1. Show that the tension in the string is 39.9 N correct to 3 significant figures.
  2. Find the coefficient of friction between the rough plane and Block B.
OCR MEI Paper 1 2018 June Q12
14 marks Standard +0.8
12 Fig. 12 shows the circle \(( x - 1 ) ^ { 2 } + ( y + 1 ) ^ { 2 } = 25\), the line \(4 y = 3 x - 32\) and the tangent to the circle at the point \(\mathrm { A } ( 5,2 )\). D is the point of intersection of the line \(4 y = 3 x - 32\) and the tangent at A . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{904025c9-6d68-4344-bd41-8c0fccfcf92f-07_750_773_1311_632} \captionsetup{labelformat=empty} \caption{Fig. 12}
\end{figure}
  1. Write down the coordinates of C , the centre of the circle.
  2. (A) Show that the line \(4 y = 3 x - 32\) is a tangent to the circle.
    (B) Find the coordinates of B , the point where the line \(4 y = 3 x - 32\) touches the circle.
  3. Prove that ADBC is a square.
  4. The point E is the lowest point on the circle. Find the area of the sector ECB .
OCR MEI Paper 1 2018 June Q13
12 marks Standard +0.8
13 The function \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = \sqrt [ 3 ] { 27 - 8 x ^ { 3 } }\). Jenny uses her scientific calculator to create a table of values for \(\mathrm { f } ( x )\) and \(\mathrm { f } ^ { \prime } ( x )\).
\(x\)\(f ( x )\)\(f ^ { \prime } ( x )\)
030
0.252.9954- 0.056
0.52.9625- 0.228
0.752.8694- 0.547
12.6684- 1.124
1.252.2490- 1.977
1.50ERROR
  1. Use calculus to find an expression for \(\mathrm { f } ^ { \prime } ( x )\) and hence explain why the calculator gives an error for \(\mathrm { f } ^ { \prime } ( 1.5 )\).
  2. Find the first three terms of the binomial expansion of \(\mathrm { f } ( x )\).
  3. Jenny integrates the first three terms of the binomial expansion of \(\mathrm { f } ( x )\) to estimate the value of \(\int _ { 0 } ^ { 1 } \sqrt [ 3 ] { 27 - 8 x ^ { 3 } } \mathrm {~d} x\). Explain why Jenny's method is valid in this case. (You do not need to evaluate Jenny's approximation.)
  4. Use the trapezium rule with 4 strips to obtain an estimate for \(\int _ { 0 } ^ { 1 } \sqrt [ 3 ] { 27 - 8 x ^ { 3 } } \mathrm {~d} x\). The calculator gives 2.92117438 for \(\int _ { 0 } ^ { 1 } \sqrt [ 3 ] { 27 - 8 x ^ { 3 } } \mathrm {~d} x\). The graph of \(y = \mathrm { f } ( x )\) is shown in Fig. 13. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{904025c9-6d68-4344-bd41-8c0fccfcf92f-08_490_906_1505_568} \captionsetup{labelformat=empty} \caption{Fig. 13}
    \end{figure}
  5. Explain why the trapezium rule gives an underestimate.
OCR MEI Paper 1 2018 June Q14
17 marks Standard +0.3
14 The velocity of a car, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t\) seconds, is being modelled. Initially the car has velocity \(5 \mathrm {~ms} ^ { - 1 }\) and it accelerates to \(11.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in 4 seconds. In model A, the acceleration is assumed to be uniform.
  1. Find an expression for the velocity of the car at time \(t\) using this model.
  2. Explain why this model is not appropriate in the long term. Model A is refined so that the velocity remains constant once the car reaches \(17.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  3. Sketch a velocity-time graph for the motion of the car, making clear the time at which the acceleration changes.
  4. Calculate the displacement of the car in the first 20 seconds according to this refined model. In model B, the velocity of the car is given by $$v = \begin{cases} 5 + 0.6 t ^ { 2 } - 0.05 t ^ { 3 } & \text { for } 0 \leqslant t \leqslant 8 \\ 17.8 & \text { for } 8 < t \leqslant 20 \end{cases}$$
  5. Show that this model gives an appropriate value for \(v\) when \(t = 4\).
  6. Explain why the value of the acceleration immediately before the velocity becomes constant is likely to mean that model B is a better model than model A.
  7. Show that model B gives the same value as model A for the displacement at time 20 s .
OCR MEI Paper 1 2019 June Q2
3 marks Moderate -0.8
2 Show that the line which passes through the points \(( 2 , - 4 )\) and \(( - 1,5 )\) does not intersect the line \(3 x + y = 10\).
OCR MEI Paper 1 2019 June Q3
8 marks Standard +0.3
3 The function \(\mathrm { f } ( x )\) is given by \(\mathrm { f } ( x ) = ( 1 - a x ) ^ { - 3 }\), where \(a\) is a non-zero constant. In the binomial expansion of \(\mathrm { f } ( x )\), the coefficients of \(x\) and \(x ^ { 2 }\) are equal.
  1. Find the value of \(a\).
  2. Using this value for \(a\),
    1. state the set of values of \(x\) for which the binomial expansion is valid,
    2. write down the quadratic function which approximates \(\mathrm { f } ( x )\) when \(x\) is small.
OCR MEI Paper 1 2019 June Q4
3 marks Moderate -0.3
4 Fig. 4 shows a uniform beam of mass 4 kg and length 2.4 m resting on two supports P and Q . P is at one end of the beam and Q is 0.3 m from the other end.
Determine whether a person of mass 50 kg can tip the beam by standing on it. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{59e924e6-8fa9-4035-9173-705fce487bd9-4_195_977_1676_262} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure}
OCR MEI Paper 1 2019 June Q5
4 marks Easy -1.2
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
7 marks Standard +0.3
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
4 marks Standard +0.3
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
7 marks Standard +0.3
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.