Questions — OCR MEI AS Paper 1 (93 questions)

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OCR MEI AS Paper 1 2018 June Q1
1 Write \(\frac { 8 } { 3 - \sqrt { 5 } }\) in the form \(a + b \sqrt { 5 }\), where \(a\) and \(b\) are integers to be found.
OCR MEI AS Paper 1 2018 June Q2
2 Find the binomial expansion of \(( 3 - 2 x ) ^ { 3 }\).
OCR MEI AS Paper 1 2018 June Q3
3 A particle is in equilibrium under the action of three forces in newtons given by $$\mathbf { F } _ { 1 } = \binom { 8 } { 0 } , \quad \mathbf { F } _ { 2 } = \binom { 2 a } { - 3 a } \quad \text { and } \quad \mathbf { F } _ { 3 } = \binom { 0 } { b } .$$ Find the values of the constants \(a\) and \(b\).
OCR MEI AS Paper 1 2018 June Q4
4 Fig. 4 shows a block of mass \(4 m \mathrm {~kg}\) and a particle of mass \(m \mathrm {~kg}\) connected by a light inextensible string passing over a smooth pulley. The block is on a horizontal table, and the particle hangs freely. The part of the string between the pulley and the block is horizontal. The block slides towards the pulley and the particle descends. In this motion, the friction force between the table and the block is \(\frac { 1 } { 2 } m g \mathrm {~N}\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1513048a-d53b-4b85-82f4-c86e0d81f8f8-3_204_741_1151_662} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} Find expressions for
  • the acceleration of the system,
  • the tension in the string.
OCR MEI AS Paper 1 2018 June Q5
5
  1. Sketch the graphs of \(y = 4 \cos x\) and \(y = 2 \sin x\) for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\) on the same axes.
  2. Find the exact coordinates of the point of intersection of these graphs, giving your answer in the form (arctan \(a , k \sqrt { b }\) ), where \(a\) and \(b\) are integers and \(k\) is rational.
  3. A student argues that without the condition \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\) all the points of intersection of the graphs would occur at intervals of \(360 ^ { \circ }\) because both \(\sin x\) and \(\cos x\) are periodic functions with this period. Comment on the validity of the student's argument.
OCR MEI AS Paper 1 2018 June Q6
6 In this question you must show detailed reasoning.
You are given that \(\mathrm { f } ( x ) = 4 x ^ { 3 } - 3 x + 1\).
  1. Use the factor theorem to show that \(( x + 1 )\) is a factor of \(\mathrm { f } ( x )\).
  2. Solve the equation \(\mathrm { f } ( x ) = 0\).
OCR MEI AS Paper 1 2018 June Q7
7 A toy boat of mass 1.5 kg is pushed across a pond, starting from rest, for 2.5 seconds. During this time, the boat has an acceleration of \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Subsequently, when the only horizontal force acting on the boat is a constant resistance to motion, the boat travels 10 m before coming to rest. Calculate the magnitude of the resistance to motion.
OCR MEI AS Paper 1 2018 June Q9
9 The curve \(y = ( x - 1 ) ^ { 2 }\) maps onto the curve \(\mathrm { C } _ { 1 }\) following a stretch scale factor \(\frac { 1 } { 2 }\) in the \(x\)-direction.
  1. Show that the equation of \(\mathrm { C } _ { 1 }\) can be written as \(y = 4 x ^ { 2 } - 4 x + 1\). The curve \(\mathrm { C } _ { 2 }\) is a translation of \(y = 4.25 x - x ^ { 2 }\) by \(\binom { 0 } { - 3 }\).
  2. Show that the normal to the curve \(\mathrm { C } _ { 1 }\) at the point \(( 0,1 )\) is a tangent to the curve \(\mathrm { C } _ { 2 }\).
OCR MEI AS Paper 1 2018 June Q10
10 Rory runs a distance of 45 m in 12.5 s . He starts from rest and accelerates to a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). He runs the remaining distance at \(4 \mathrm {~ms} ^ { - 1 }\). Rory proposes a model in which the acceleration is constant until time \(T\) seconds.
  1. Sketch the velocity-time graph for Rory's run using this model.
  2. Calculate \(T\).
  3. Find an expression for Rory's displacement at time \(t \mathrm {~s}\) for \(0 \leqslant t \leqslant T\).
  4. Use this model to find the time taken for Rory to run the first 4 m . Rory proposes a refined model in which the velocity during the acceleration phase is a quadratic function of \(t\). The graph of Rory's quadratic goes through \(( 0,0 )\) and has its maximum point at \(( S , 4 )\). In this model the acceleration phase lasts until time \(S\) seconds, after which the velocity is constant.
  5. Sketch a velocity-time graph that represents Rory's run using this refined model.
  6. State with a reason whether \(S\) is greater than \(T\) or less than \(T\). (You are not required to calculate the value of \(S\).)
OCR MEI AS Paper 1 2018 June Q11
11 The intensity of the sun's radiation, \(y\) watts per square metre, and the average distance from the sun, \(x\) astronomical units, are shown in Fig. 11 for the planets Mercury and Jupiter. \begin{table}[h]
\(x\)\(y\)
Mercury0.307514400
Jupiter4.95055.8
\captionsetup{labelformat=empty} \caption{Fig. 11}
\end{table} The intensity \(y\) is proportional to a power of the distance \(x\).
  1. Write down an equation for \(y\) in terms of \(x\) and two constants.
  2. Show that the equation can be written in the form \(\ln y = a + b \ln x\).
  3. In the Printed Answer Booklet, complete the table for \(\ln x\) and \(\ln y\) correct to 4 significant figures.
  4. Use the values from part (iii) to find \(a\) and \(b\).
  5. Hence rewrite your equation from part (i) for \(y\) in terms of \(x\), using suitable numerical values for the constants.
  6. Sketch a graph of the equation found in part (v).
  7. Earth is 1 astronomical unit from the sun. Find the intensity of the sun's radiation for Earth.
OCR MEI AS Paper 1 2019 June Q3
3 Given that \(k\) is an integer, express \(\frac { 3 \sqrt { 2 } - k } { \sqrt { 8 } + 1 }\) in the form \(a + b \sqrt { 2 }\) where \(a\) and \(b\) are rational expressions in terms of \(k\).
OCR MEI AS Paper 1 2019 June Q4
4 A triangle ABC has sides \(\mathrm { AB } = 5 \mathrm {~cm} , \mathrm { AC } = 9 \mathrm {~cm}\) and \(\mathrm { BC } = 10 \mathrm {~cm}\).
  1. Find the cosine of angle BAC, giving your answer as a fraction in its lowest terms.
  2. Find the exact area of the triangle.
OCR MEI AS Paper 1 2019 June Q5
5 In this question, the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal and vertically upwards respectively. A particle has mass 2.5 kg .
  1. Write the weight of the particle as a vector. The particle moves under the action of its weight and two external forces ( \(3 \mathbf { i } - 2 \mathbf { j }\) ) N and \(( - \mathbf { i } + 18 \mathbf { j } ) N\).
  2. Find the acceleration of the particle, giving your answer in vector form.
OCR MEI AS Paper 1 2019 June Q6
6 Fig. 6 shows a train consisting of an engine of mass 80 tonnes pulling two trucks each of mass 25 tonnes. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0b1c272a-f0f4-4931-be89-5d045804a7af-4_189_1262_938_246} \captionsetup{labelformat=empty} \caption{Fig. 6}
\end{figure} The engine exerts a driving force of \(D \mathrm {~N}\) and experiences a resistance to motion of 2000 N . Each truck experiences a resistance of 600 N . The train travels in a straight line on a level track with an acceleration of \(0.1 \mathrm {~ms} ^ { - 2 }\).
  1. Complete the force diagram in the Printed Answer Booklet to show all the forces acting on the engine and each of the trucks.
  2. Calculate the value of \(D\).
  3. The tension in the coupling between the engine and truck A is larger than that in the coupling between the trucks. Determine how much larger.
OCR MEI AS Paper 1 2019 June Q7
7 In this question you must show detailed reasoning.
  1. Nigel is asked to determine whether \(( x + 7 )\) is a factor of \(x ^ { 3 } - 37 x + 84\). He substitutes \(x = 7\) and calculates \(7 ^ { 3 } - 37 \times 7 + 84\). This comes to 168 , so Nigel concludes that ( \(x + 7\) ) is not a factor. Nigel's conclusion is wrong.
    • Explain why Nigel's argument is not valid.
    • Show that \(( x + 7 )\) is a factor of \(x ^ { 3 } - 37 x + 84\).
    • Sketch the graph of \(y = x ^ { 3 } - 37 x + 84\), indicating the coordinates of the points at which the curve crosses the coordinate axes.
    • The graph in part (b) is translated by \(\binom { 1 } { 0 }\). Find the equation of the translated graph, giving your answer in the form \(y = x ^ { 3 } + a x ^ { 2 } + b x + c\) where \(a , b\) and \(c\) are integers.
OCR MEI AS Paper 1 2019 June Q10
10 In this question you must show detailed reasoning.
  1. Sketch the gradient function for the curve \(y = 24 x - 3 x ^ { 2 } - x ^ { 3 }\).
  2. Determine the set of values of \(x\) for which \(24 x - 3 x ^ { 2 } - x ^ { 3 }\) is decreasing.
OCR MEI AS Paper 1 2019 June Q11
11 David puts a block of ice into a cool-box. He wishes to model the mass \(m \mathrm {~kg}\) of the remaining block of ice at time \(t\) hours later. He finds that when \(t = 5 , m = 2.1\), and when \(t = 50 , m = 0.21\).
  1. David at first guesses that the mass may be inversely proportional to time. Show that this model fits his measurements.
  2. Explain why this model
    1. is not suitable for small values of \(t\),
    2. cannot be used to find the time for the block to melt completely. David instead proposes a linear model \(m = a t + b\), where \(a\) and \(b\) are constants.
  3. Find the values of the constants for which the model fits the mass of the block when \(t = 5\) and \(t = 50\).
  4. Interpret these values of \(a\) and \(b\).
  5. Find the time according to this model for the block of ice to melt completely.
OCR MEI AS Paper 1 2022 June Q1
1 Rationalise the denominator of the fraction \(\frac { 2 + \sqrt { n } } { 3 + \sqrt { n } }\), where \(n\) is a positive integer.
OCR MEI AS Paper 1 2022 June Q2
2
  1. Determine the value of \(\frac { 100 ! } { 98 ! }\).
  2. Find the coefficient of \(x ^ { 98 }\) in the expansion of \(( 1 + x ) ^ { 100 }\).
OCR MEI AS Paper 1 2022 June Q3
3 The velocity-time graph for the motion of a particle is shown below. The velocity \(v \mathrm {~ms} ^ { - 1 }\) at time \(t \mathrm {~s}\) is given by \(\mathrm { v } = - \mathrm { t } ^ { 2 } + 6 \mathrm { t } - 6\) where \(0 \leqslant t \leqslant 5\).
\includegraphics[max width=\textwidth, alt={}, center]{7af62e61-c67f-4d05-b6b9-c1a110345812-3_860_979_1082_239}
  1. Find the times at which the velocity is \(2 \mathrm {~ms} ^ { - 1 }\).
  2. Write down the greatest speed of the particle.
OCR MEI AS Paper 1 2022 June Q4
4 The quadratic function \(\mathrm { f } ( x )\) is given by \(\mathrm { f } ( x ) = x ^ { 2 } - 3 x + 2\).
  1. Write \(\mathrm { f } ( x )\) in the form \(( \mathrm { x } + \mathrm { a } ) ^ { 2 } + \mathrm { b }\), where \(a\) and \(b\) are constants.
  2. Write down the coordinates of the minimum point on the graph of \(y = f ( x )\).
  3. Describe fully the transformation that maps the graph of \(y = f ( x )\) onto the graph of \(y = ( x + 1 ) ^ { 2 } - \frac { 1 } { 4 }\).
OCR MEI AS Paper 1 2022 June Q5
5 Part of the graph of \(y = f ( x )\) is shown below. The graph is the image of \(y = \tan x ^ { \circ }\) after a stretch in the \(x\)-direction.
\includegraphics[max width=\textwidth, alt={}, center]{7af62e61-c67f-4d05-b6b9-c1a110345812-4_791_1022_1014_244}
  1. Find the equation of the graph.
  2. Write down the period of the function \(\mathrm { f } ( x )\).
  3. In this question you must show detailed reasoning. Find all the roots of the equation \(\mathrm { f } ( x ) = 1\) for \(0 ^ { \circ } \leqslant x ^ { \circ } \leqslant 360 ^ { \circ }\).
OCR MEI AS Paper 1 2022 June Q6
6 The gradient of a curve is given by the equation \(\frac { d y } { d x } = 6 x ^ { 2 } - 20 x + 6\). The curve passes through the point \(( 2,6 )\).
  1. Find the equation of the curve.
  2. Verify that the equation of the curve can be written as \(y = 2 ( x + 1 ) ( x - 3 ) ^ { 2 }\).
  3. Sketch the curve, indicating the points where the curve meets the axes.
OCR MEI AS Paper 1 2022 June Q7
7 In this question the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively. A canal narrowboat of mass 9 tonnes is pulled by two ropes. The tensions in the ropes are \(( 450 \mathbf { i } + 20 \mathbf { j } ) \mathbf { N }\) and \(( 420 \mathbf { i } - 20 \mathbf { j } ) \mathbf { N }\). The boat experiences a resistance to motion \(\mathbf { R }\) of magnitude 300 N .
  1. Explain what it means to model the boat as a particle. The boat is travelling in a straight line due east.
  2. Find the equation of motion of the boat.
  3. Find the acceleration of the boat giving your answer as a vector.
OCR MEI AS Paper 1 2022 June Q8
8 A team of volunteers donates cakes for sale at a charity stall. The number of cakes that can be sold depends on the price. A model for this is \(\mathrm { y } = 190 - 70 \mathrm { x }\), where \(y\) cakes can be sold when the price of a cake is \(\pounds\) x.
  1. Find how many cakes could be given away for free according to this model. The number of volunteers who are willing to donate cakes goes up as the price goes up. If the cakes sell for \(\pounds 1.20\) they will donate 50 cakes, but if they sell for \(\pounds 2.40\) they will donate 140 cakes. They use the linear model \(\mathrm { y } = \mathrm { mx } + \mathrm { c }\) to relate the number of cakes donated, \(y\), to the price of a cake, \(\pounds x\).
  2. Find the values of the constants \(m\) and \(c\) for which this linear model fits the two data points.
  3. Explain why the model is not suitable for very low prices.
  4. The team would like to sell all the cakes that they donate. Find the set of possible prices that the cakes could have to achieve this.