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OCR MEI AS Paper 1 2022 June Q2
3 marks Easy -1.2
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 marks Easy -1.2
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
6 marks Moderate -0.8
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
6 marks Moderate -0.3
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
8 marks Moderate -0.8
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
4 marks Moderate -0.3
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
7 marks Standard +0.3
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.
OCR MEI AS Paper 1 2022 June Q9
6 marks Moderate -0.8
9 A tractor of mass 1800 kg uses a towbar to pull a trailer of mass 1000 kg on a level field. The tractor and trailer experience resistances to motion of 1600 N and 800 N respectively. The tractor provides a driving force of 6600 N .
  1. Draw a force diagram showing all the horizontal forces acting on the tractor and trailer.
  2. Find the tension in the towbar.
OCR MEI AS Paper 1 2022 June Q10
9 marks Standard +0.3
10 A triangle has vertices \(A ( 1,4 ) , B ( 7,0 )\) and \(C ( - 4 , - 1 )\).
  1. Show that the equation of the line AC is \(\mathrm { y } = \mathrm { x } + 3\). M is the midpoint of AB . The line AC intersects the \(x\)-axis at D .
  2. Determine the angle DMA.
OCR MEI AS Paper 1 2022 June Q11
9 marks Moderate -0.8
11 A sports car accelerates along a straight road from rest. After 5 s its velocity is \(9 \mathrm {~ms} ^ { - 1 }\). In model A, the acceleration is assumed to be constant.
  1. Calculate the distance travelled by the car in the first 5 seconds according to model A . In model B , the velocity \(v\) in \(\mathrm { ms } ^ { - 1 }\) is given by \(\mathrm { v } = 0.05 \mathrm { t } ^ { 3 } + \mathrm { kt }\), where \(t\) is the time in seconds after the start and \(k\) is a constant.
  2. Find the value of \(k\) which gives the correct value of \(v\) when \(t = 5\).
  3. Using this value of \(k\) in model B , calculate the acceleration of the car when \(t = 5\). The car travels 16 m in the first 5 seconds.
  4. Show that model B, with the value of \(k\) found in part (b), better fits this information than model A does.
OCR MEI AS Paper 1 2022 June Q12
6 marks Moderate -0.5
12 Below is a faulty argument that appears to show that the gradient of the curve \(y = x ^ { 2 }\) at the point \(( 3,9 )\) is 1 . Consider the chord joining \(( 3,9 )\) to the point \(\left( 3 + h , ( 3 + h ) ^ { 2 } \right)\) The gradient is \(\frac { ( 3 + h ) ^ { 2 } - 9 } { h } = \frac { 6 h + h ^ { 2 } } { h }\) When \(h = 0\) the gradient is \(\frac { 0 } { 0 }\) so the gradient of the curve is 1
  1. Identify a fault in the argument.
  2. Write a valid first principles argument leading to the correct value for the gradient at (3, 9).
  3. Find the equation of the normal to the curve at the point ( 3,9 ).
OCR MEI AS Paper 1 2023 June Q1
2 marks Easy -1.2
1 A particle moves along a straight line. Its velocity \(v \mathrm {~ms} ^ { - 1 }\) at time \(t\) s is given by \(\mathbf { v } = 2 \mathbf { t } + 0.6 \mathbf { t } ^ { 2 }\).
Find an expression for the acceleration of the particle at time \(t\).
OCR MEI AS Paper 1 2023 June Q2
3 marks Moderate -0.3
2 The height of the first part of a rollercoaster track is \(h \mathrm {~m}\) at a horizontal distance of \(x \mathrm {~m}\) from the start. A student models this using the equation \(h = 17 + 15 \cos 6 x\), for \(0 \leqslant x \leqslant 40\), using the values of \(h\) given when their calculator is set to work in degrees.
  1. Find the height that the student's model predicts when the horizontal distance from the start is 40 m .
  2. The student argues that the model predicts that the rollercoaster track will achieve a maximum height of 32 m more than once because the cosine function is periodic. Comment on the validity of the student's argument.
OCR MEI AS Paper 1 2023 June Q3
4 marks Easy -1.3
3 The points \(A\) and \(B\) have position vectors \(\binom { 2 } { - 1 }\) and \(\binom { 5 } { 4 }\) respectively. The vector \(\overrightarrow { \mathrm { AC } }\) is \(\binom { - 2 } { 2 }\).
  1. Write down the position vector of C as a column vector.
  2. Show that B is equidistant from A and C .
OCR MEI AS Paper 1 2023 June Q4
5 marks Moderate -0.3
4 In this question you must show detailed reasoning.
Solve the equation \(6 \cos ^ { 2 } x + \sin x = 5\), giving all the roots in the interval \(- 180 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
OCR MEI AS Paper 1 2023 June Q5
5 marks Moderate -0.8
5 The graph shows displacement \(s m\) against time \(t \mathrm {~s}\) for a model of the motion of a bead moving along a straight wire. The points \(( 0,4 ) , ( 2,7 ) , ( 5,7 )\) and \(( 9 , - 7 )\) are the endpoints of the line segments. \includegraphics[max width=\textwidth, alt={}, center]{1d1e41f3-a834-4230-b6e1-4b0be9450d30-4_741_1301_404_239}
  1. Find an expression for the displacement of the bead for \(0 \leqslant t \leqslant 2\).
  2. Sketch the velocity-time graph for this model.
  3. Explain why the model may not be suitable at \(t = 2\) and \(t = 5\).
OCR MEI AS Paper 1 2023 June Q6
5 marks Moderate -0.8
6 Show that the expression \(3 x ^ { 3 } + x ^ { 2 } - 6 x - 5\) can be written in the form \(( x + 2 ) \left( a x ^ { 2 } + b x + c \right) + d\) where \(a\), \(b\), \(c\) and \(d\) are constants to be determined.
OCR MEI AS Paper 1 2023 June Q7
6 marks Moderate -0.8
7 In this question you must show detailed reasoning. \includegraphics[max width=\textwidth, alt={}, center]{1d1e41f3-a834-4230-b6e1-4b0be9450d30-5_643_716_303_242} Find the exact area of the shaded region shown in the diagram, enclosed by the \(x\)-axis and the curve \(y = - 3 x ^ { 2 } + 7 x - 2\).
OCR MEI AS Paper 1 2023 June Q8
9 marks Moderate -0.3
8 In this question you must show detailed reasoning.
  1. Find the centre and radius of the circle with equation \(x ^ { 2 } + y ^ { 2 } - 2 x + 4 y - 20 = 0\).
  2. Find the points of intersection of the circle with the line \(x + 3 y - 10 = 0\).
OCR MEI AS Paper 1 2023 June Q9
9 marks Moderate -0.3
9 The graph shows the function \(\mathrm { y } = \mathrm { e } ^ { 2 \mathrm { x } }\). \includegraphics[max width=\textwidth, alt={}, center]{1d1e41f3-a834-4230-b6e1-4b0be9450d30-6_595_732_322_242}
  1. Describe the transformation of the graph of \(y = e ^ { x }\) that gives the graph of \(y = e ^ { 2 x }\). A second function is defined by \(\mathrm { y } = \mathrm { k } + \mathrm { e } ^ { \mathrm { x } }\).
  2. A copy of the graph of \(\mathrm { y } = \mathrm { e } ^ { 2 \mathrm { x } }\) is given in the Printed Answer Booklet. Add a sketch of the graph of \(\mathrm { y } = \mathrm { k } + \mathrm { e } ^ { \mathrm { x } }\) in a case where \(k\) is a positive constant.
  3. Show that the two graphs do not intersect for values of \(k\) less than \(- \frac { 1 } { 4 }\).
  4. In the case where \(k = 2\), show that the only point of intersection occurs when \(x = \ln 2\).
OCR MEI AS Paper 1 2023 June Q10
7 marks Easy -1.3
10 Layla invests money in the bank and receives compound interest. The amount \(\pounds L\) that she has after \(t\) years is given by the equation \(\mathrm { L } = 2800 \times 1.023 ^ { \mathrm { t } }\).
    1. State the amount she invests.
    2. State the annual rate of interest. Amit invests \(\pounds 3000\) and receives \(2 \%\) compound interest per year. The amount \(\pounds A\) that he has after \(t\) years is given by the equation \(\mathrm { A } = \mathrm { ab } ^ { \mathrm { t } }\).
  2. Determine the values of the constants \(a\) and \(b\).
  3. Layla and Amit invest their money in the bank at the same time. Determine the value of \(t\) for which Layla and Amit have equal amounts in the bank. Give your answer correct to \(\mathbf { 1 }\) decimal place.
OCR MEI AS Paper 1 2023 June Q11
7 marks Moderate -0.8
11 A block of mass 3 kg is at rest on a smooth horizontal table. It is attached to a light inextensible string which passes over a smooth pulley. This part of the string is horizontal. A sphere of mass 1.2 kg is attached to the other end of the string. The sphere hangs with this part of the string vertical as shown in the diagram. A horizontal force of magnitude \(F\) N is applied to the block to prevent motion. \includegraphics[max width=\textwidth, alt={}, center]{1d1e41f3-a834-4230-b6e1-4b0be9450d30-7_268_718_493_244}
  1. Complete the copy of the diagram in the Printed Answer Booklet to show all the forces acting on the block and the sphere.
  2. Find the value of \(F\). The force \(F\) N is removed, and the system begins to move.
  3. The equation of motion of the block is \(\mathrm { T } = 3 \mathrm { a }\), where \(T \mathrm {~N}\) is the tension in the string and \(a \mathrm {~ms} ^ { - 2 }\) is the acceleration of the block. Write down the equation of motion of the sphere.
  4. Find the value of \(T\).
OCR MEI AS Paper 1 2023 June Q12
8 marks Standard +0.3
12 Points A, B and C lie in a straight line in that order on horizontal ground. A box of mass 5 kg is pushed from A to C by a horizontal force of magnitude 8 N . The box is at rest at A and takes 3 seconds to reach B . The ground is smooth between A and B . Between B and C the ground is rough and the resistance to motion is 28 N . The box comes to rest at C . Determine the distance AC.
OCR MEI AS Paper 1 2024 June Q1
2 marks Moderate -0.8
1 The triangle ABC has an obtuse angle at A . The angle at B is \(15 ^ { \circ }\). The length of AC is 10 cm and the length of BC is 13 cm . Calculate the size of the angle at A .
OCR MEI AS Paper 1 2024 June Q2
2 marks Easy -1.2
2 Two forces \(\mathbf { F } _ { 1 } \mathrm {~N}\) and \(\mathbf { F } _ { 2 } \mathrm {~N}\) are given by \(\mathbf { F } _ { 1 } = - 6 \mathbf { i } + 2 \mathbf { j }\) and \(\mathbf { F } _ { 2 } = - 8 \mathbf { i } + \mathbf { j }\).
Show that the magnitude of the resultant of these two forces is \(\sqrt { 205 } \mathrm {~N}\).