Questions — OCR MEI AS Paper 1 (93 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
OCR MEI AS Paper 1 2022 June Q9
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
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
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
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
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
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
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
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 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
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
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
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 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
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
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
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
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 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 \(\sqrt { 205 } \mathrm {~N}\).
OCR MEI AS Paper 1 2024 June Q3
3 Prove that, when \(n\) is an even number, \(n ^ { 3 } + 4\) is a multiple of 4 but not a multiple of 8 .
OCR MEI AS Paper 1 2024 June Q4
4 The perpendicular lines AC and BD intersect at E as shown in the diagram. The point E is the midpoint of AC . The angles BAC and BDC are each equal to \(\chi ^ { \circ }\). The lengths of AB and CD are 4 cm and 7 cm respectively.
\includegraphics[max width=\textwidth, alt={}, center]{b5c47a93-ce43-4aa1-ba7f-fbb650523373-3_606_529_1370_244} Determine the value of \(x\).
OCR MEI AS Paper 1 2024 June Q5
5 In this question you must show detailed reasoning.
  1. Show that the gradient of the curve \(\mathrm { y } = \sqrt { \mathrm { x } } \left( \frac { 1 } { \mathrm { x } ^ { 2 } } - 2 \mathrm { x } \right)\) at the point \(\left( \frac { 1 } { 4 } , \frac { 31 } { 4 } \right)\) is \(- \frac { 99 } { 2 }\).
  2. Find the equation of the tangent to the curve at \(\left( \frac { 1 } { 4 } , \frac { 31 } { 4 } \right)\) giving your answer in the form \(\mathrm { ax } + \mathrm { by } + \mathrm { c } = 0\), where \(a , b\) and \(c\) are integers.
OCR MEI AS Paper 1 2024 June Q6
6 The polynomial \(x ^ { 3 } - 4 x ^ { 2 } + 10 x - 21\) is denoted by \(\mathrm { f } ( x )\).
  1. Use the factor theorem to show that \(( x - 3 )\) is a factor of \(\mathrm { f } ( x )\).
  2. The polynomial \(\mathrm { f } ( x )\) can be written as \(( \mathrm { x } - 3 ) \left( \mathrm { x } ^ { 2 } + \mathrm { bx } + \mathrm { c } \right)\) where \(b\) and \(c\) are constants. Find the values of \(b\) and \(c\).
  3. Show that \(x = 3\) is the only real root of the equation \(\mathrm { f } ( x ) = 0\).
OCR MEI AS Paper 1 2024 June Q7
7 The velocity of a particle moving in a straight line is modelled by \(\mathbf { v } = 0.6 \mathbf { t } ^ { 2 } - 2.1 \mathbf { t } + 1.5\) where \(v\) is the velocity in metres per second and \(t\) is the time in seconds.
  1. Determine the times at which the particle is stationary.
  2. Find the acceleration of the particle at the first of the times at which it is stationary.
  3. Find the distance travelled by the particle between the times at which it is stationary.
OCR MEI AS Paper 1 2024 June Q8
8 A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 6 x - 16 y + 48 = 0\).
  1. Find the coordinates of C . A line has equation \(\mathrm { y } = \mathrm { x } - 2\) and intersects the circle at the points A and B . The midpoints of AC and BC are \(\mathrm { A } ^ { \prime }\) and \(\mathrm { B } ^ { \prime }\) respectively.
  2. Determine the exact distance \(\mathrm { A } ^ { \prime } \mathrm { B } ^ { \prime }\).
OCR MEI AS Paper 1 2024 June Q9
9 Two trains are travelling in the same direction on parallel straight tracks and train A overtakes train B . At time \(t\) seconds after the front of train A overtakes the front of train B the velocities of trains A and B are \(v _ { \mathrm { A } } \mathrm { m } \mathrm { s } ^ { - 1 }\) and \(v _ { \mathrm { B } } \mathrm { ms } ^ { - 1 }\) respectively. The velocity of train A is modelled by \(\mathrm { v } _ { \mathrm { A } } = 25 - 0.6 \mathrm { t }\). The velocity-time graph of train A is shown below.
\includegraphics[max width=\textwidth, alt={}, center]{b5c47a93-ce43-4aa1-ba7f-fbb650523373-5_664_1399_550_242}
  1. A student argues that the speed of train A changes by \(18 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in 30 seconds so its acceleration is \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Comment on the validity of the student's argument.
  2. When the front of train A overtakes the front of train B , train B has a velocity of \(10 \mathrm {~ms} ^ { - 1 }\). The acceleration of train \(B\) is constant and is modelled as \(0.15 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Write down the equation for \(v _ { \mathrm { B } }\) in terms of \(t\) that models the velocity of train B .
  3. Draw the velocity-time graph of train B on the copy of the diagram in the Printed Answer Booklet.
  4. Determine the distance between the fronts of the trains at the time when the trains are travelling at the same velocity.
  5. Explain why the model for train A would not be valid for large values of \(t\).