Questions — OCR MEI Paper 1 (118 questions)

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OCR MEI Paper 1 2023 June Q9
9 The gradient of a curve is given by \(\frac { d y } { d x } = e ^ { x } - 4 e ^ { - x }\).
  1. Show that the \(x\)-coordinate of any point on the curve at which the gradient is 3 satisfies the equation \(\left( e ^ { x } \right) ^ { 2 } - 3 e ^ { x } - 4 = 0\).
  2. Hence show that there is only one point on the curve at which the gradient is 3 , stating the exact value of its \(x\)-coordinate.
  3. The curve passes through the point \(( 0,0 )\). Show that when \(x = 1\) the curve is below the \(x\)-axis.
OCR MEI Paper 1 2023 June Q10
10 The diagram shows the graph of \(\mathrm { y } = 1.5 + \sin ^ { 2 } \mathrm { x }\) for \(0 \leqslant x \leqslant 2 \pi\).
\includegraphics[max width=\textwidth, alt={}, center]{8eeff88d-8b05-43c6-86a5-bd82221c0bea-07_512_1278_322_242}
  1. Show that the equation of the graph can be written in the form \(\mathrm { y } = \mathrm { a } - \mathrm { b } \cos 2 \mathrm { x }\) where \(a\) and \(b\) are constants to be determined.
  2. Write down the period of the function \(1.5 + \sin ^ { 2 } x\).
  3. Determine the \(x\)-coordinates of the points of intersection of the graph of \(y = 1.5 + \sin ^ { 2 } x\) with the graph of \(\mathrm { y } = 1 + \cos 2 \mathrm { x }\) in the interval \(0 \leqslant x \leqslant 2 \pi\).
OCR MEI Paper 1 2023 June Q11
11 The height \(h \mathrm {~cm}\) of a sunflower plant \(t\) days after planting the seed is modelled by \(\mathrm { h } = \mathrm { a } + \mathrm { b }\) Int for \(t \geqslant 9\), where \(a\) and \(b\) are constants. The sunflower is 10 cm tall 10 days after planting and 200 cm tall 85 days after planting.
    1. Show that the value of \(b\) which best models these values is 88.8 correct to \(\mathbf { 3 }\) significant figures.
    2. Find the corresponding value of \(a\).
    1. Explain why the model is not suitable for small positive values of \(t\).
    2. Explain why the model is not suitable for very large positive values of \(t\).
  3. Show that the model indicates that the sunflower grows to 1 m in height in less than half the time it takes to grow to 2 m .
  4. Find the value of \(t\) for which the rate of growth is 3 cm per day.
OCR MEI Paper 1 2023 June Q12
12 In this question the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal and vertically upwards respectively. A particle has mass 2 kg .
  1. Write down its weight as a vector. A horizontal force of 3 N in the \(\mathbf { i }\) direction and a force \(\mathbf { F } = ( - 4 \mathbf { i } + 12 \mathbf { j } ) \mathrm { N }\) act on the particle.
  2. Determine the acceleration of the particle.
  3. The initial velocity of the particle is \(5 \mathbf { i } \mathrm {~ms} ^ { - 1 }\). Find the velocity of the particle after 4 s .
  4. Find the extra force that must be applied to the particle for it to move at constant velocity.
OCR MEI Paper 1 2023 June Q13
13 A block of mass 8 kg is placed on a rough plane inclined at \(15 ^ { \circ }\) to the horizontal. The coefficient of friction between the block and the plane is 0.3 . One end of a light rope is attached to the block. The rope passes over a smooth pulley fixed at the top of the plane, and a sphere of mass 5 kg , attached to the other end of the rope, hangs vertically below the pulley. The part of the rope between the block and the pulley is parallel to the plane. The system is released from rest, and as the sphere falls the block moves directly up the plane with acceleration \(a \mathrm {~ms} ^ { - 2 }\).
\includegraphics[max width=\textwidth, alt={}, center]{8eeff88d-8b05-43c6-86a5-bd82221c0bea-08_252_803_1560_246}
  1. On the diagram in the Printed Answer Booklet, show all the forces acting on the block and on the sphere.
  2. Write down the equation of motion for the sphere.
  3. Determine the value of \(a\).
OCR MEI Paper 1 2023 June Q14
14
  1. Use the laws of logarithms to show that \(\log _ { 10 } 200 - \log _ { 10 } 20\) is equal to 1 . The first three terms of a sequence are \(\log _ { 10 } 20 , \log _ { 10 } 200 , \log _ { 10 } 2000\).
  2. Show that the sequence is arithmetic.
  3. Find the exact value of the sum of the first 50 terms of this sequence.
OCR MEI Paper 1 2023 June Q15
15 A projectile is launched from a point on level ground with an initial velocity \(u\) at an angle \(\theta\) above the horizontal.
  1. Show that the range of the projectile is given by \(\frac { 2 u ^ { 2 } \sin \theta \cos \theta } { g }\).
  2. Determine the set of values of \(\theta\) for which the maximum height of the projectile is greater than the range, where \(\theta\) is an acute angle. Give your answer in degrees.
OCR MEI Paper 1 2024 June Q1
1 A student states that \(1 + x ^ { 2 } < ( 1 + x ) ^ { 2 }\) for all values of \(x\).
Using a counter example, show that the student is wrong.
OCR MEI Paper 1 2024 June Q2
2 A car of mass 1400 kg pulls a trailer of mass 400 kg along a straight horizontal road. The engine of the car produces a driving force of 6000 N . A resistance of 800 N acts on the car. A resistance of 300 N acts on the trailer. The tow-bar between the car and the trailer is light and horizontal.
  1. Draw a force diagram showing all the horizontal forces on the car and the trailer.
  2. Calculate the acceleration of the car and trailer.
OCR MEI Paper 1 2024 June Q3
3 A particle hangs at the end of a string. A horizontal force of magnitude \(F \mathrm {~N}\) acting on the particle holds it in equilibrium so that the string makes an angle of \(20 ^ { \circ }\) with the vertical, as shown in the diagram. The tension in the string is 12 N .
\includegraphics[max width=\textwidth, alt={}, center]{1d0ca3d5-6529-435f-a0b8-50ea4859adde-04_357_374_1409_239}
  1. Find the value of \(F\).
  2. Find the mass of the particle.
OCR MEI Paper 1 2024 June Q4
4 The vectors \(\mathbf { v } _ { 1 }\) and \(\mathbf { v } _ { 2 }\) are defined by \(\mathbf { v } _ { 1 } = 2 \mathrm { a } \mathbf { i } + \mathrm { bj }\) and \(\mathbf { v } _ { 2 } = b \mathbf { i } - 3 \mathbf { j }\) where \(a\) and \(b\) are constants. Given that \(3 \mathbf { v } _ { 1 } + \mathbf { v } _ { 2 } = 22 \mathbf { i } - 9 \mathbf { j }\), find the values of \(a\) and \(b\).
OCR MEI Paper 1 2024 June Q5
5
  1. Make \(y\) the subject of the formula \(\log _ { 10 } ( y - k ) = x \log _ { 10 } 2\), where \(k\) is a positive constant.
  2. Sketch the graph of \(y\) against \(x\).
OCR MEI Paper 1 2024 June Q6
6 Given that \(\mathrm { f } ( x ) = 2 x ^ { 2 } + 3\), show from first principles that \(\mathrm { f } ^ { \prime } ( x ) = 4 x\).
OCR MEI Paper 1 2024 June Q7
7 A rectangular book ABCD rests on a smooth horizontal table. The length of AB is 28 cm and the length of AD is 18 cm . The following five forces act on the book, as shown in the diagram.
  • 4 N at A in the direction AD
  • 5 N at B in the direction BC
  • 3 N at B in the direction BA
  • 9 N at D in the direction DA
  • 3 N at D in the direction DC
    \includegraphics[max width=\textwidth, alt={}, center]{1d0ca3d5-6529-435f-a0b8-50ea4859adde-06_663_830_774_242}
    1. Show that the resultant of the forces acting on the book has zero magnitude.
    2. Find the total moment of the forces about the centre of the book. Give your answer in Nm .
    3. Describe how the book will move under the action of these forces.
OCR MEI Paper 1 2024 June Q8
8 The equation of a curve is \(\mathrm { y } = \sqrt { \sin 4 \mathrm { x } } + 2 \cos 2 \mathrm { x }\), where \(x\) is in radians.
  1. Show that, for small values of \(x , y \approx 2 \sqrt { x } + 2 - 4 x ^ { 2 }\). The diagram shows the region bounded by the curve \(\mathrm { y } = \sqrt { \sin 4 \mathrm { x } } + 2 \cos 2 \mathrm { x }\), the axes and the line \(x = 0.1\).
    \includegraphics[max width=\textwidth, alt={}, center]{1d0ca3d5-6529-435f-a0b8-50ea4859adde-07_499_881_589_223}
  2. In this question you must show detailed reasoning. Use the approximation in part (a) to estimate the area of this region.
OCR MEI Paper 1 2024 June Q9
9 A child throws a pebble of mass 40 g vertically downwards with a speed of \(6 \mathrm {~ms} ^ { - 1 }\) from a point 0.8 m above a sandy beach.
  1. Calculate the speed at which the pebble hits the beach. The pebble travels 3 cm through the sand before coming to rest.
  2. Find the magnitude of the resistance force of the sand on the pebble, assuming it is constant. Give your answer correct to \(\mathbf { 3 }\) significant figures.
OCR MEI Paper 1 2024 June Q10
10 Zac is measuring the growth of a culture of bacteria in a laboratory. The initial area of the culture is \(8 \mathrm {~cm} ^ { 2 }\). The area one day later is \(8.8 \mathrm {~cm} ^ { 2 }\). At first, Zac uses a model of the form \(\mathrm { A } = \mathrm { a } + \mathrm { bt }\), where \(A \mathrm {~cm} ^ { 2 }\) is the area \(t\) days after he begins measuring and \(a\) and \(b\) are constants.
  1. Find the values of \(a\) and \(b\) that best model the initial area and the area one day later.
  2. Calculate the value of \(t\) for which the model predicts an area of \(15 \mathrm {~cm} ^ { 2 }\).
  3. Zac notices the area covered by the culture increases by \(10 \%\) each day. Explain why this model may not be suitable after the first day. Zac decides to use a different model for \(A\). His new model is \(\mathrm { A } = \mathrm { Pe } ^ { \mathrm { kt } }\), where \(P\) and \(k\) are constants.
  4. Find the values of \(P\) and \(k\) that best model the initial area and the area one day later.
  5. Calculate the value of \(t\) for which the area reaches \(15 \mathrm {~cm} ^ { 2 }\) according to this model.
  6. Explain why this model may not be suitable for large values of \(t\).
OCR MEI Paper 1 2024 June Q11
11 The first three terms of a geometric sequence are \(5 k - 2,3 k - 6 , k + 2\), where \(k\) is a constant.
  1. Show that \(k\) satisfies the equation \(k ^ { 2 } - 11 k + 10 = 0\).
  2. When \(k\) takes the smaller of the two possible values, find the sum of the first 20 terms of the sequence.
  3. When \(k\) takes the larger of the two possible values, find the sum to infinity of the sequence.
OCR MEI Paper 1 2024 June Q12
12 In this question the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in the \(x\) - and \(y\)-directions respectively.
The velocity \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\) of a particle is given by \(\mathbf { v } = 3 \mathbf { i } + \left( 6 t ^ { 2 } - 5 \right) \mathbf { j }\). The initial position of the particle is \(7 \mathbf { j } \mathrm {~m}\).
  1. Find an expression for the position vector of the particle at time \(t \mathrm {~s}\).
  2. Find the Cartesian equation of the path of the particle.
OCR MEI Paper 1 2024 June Q13
13 The curve with equation \(\mathrm { y } = \mathrm { px } + \frac { 8 } { \mathrm { x } ^ { 2 } } + \mathrm { q }\), where \(p\) and \(q\) are constants, has a stationary point at \(( 2,7 )\).
  1. Determine the values of \(p\) and \(q\).
  2. Find \(\frac { d ^ { 2 } y } { d x ^ { 2 } }\).
  3. Hence determine the nature of the stationary point at (2, 7).
OCR MEI Paper 1 2024 June Q14
14 A man runs at a constant speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along a straight horizontal road. A woman is standing on a bridge that spans the road. At the instant that the man passes directly below the woman she throws a ball with initial speed \(u \mathrm {~ms} ^ { - 1 }\) at \(\alpha ^ { \circ }\) above the horizontal. The path of the ball is directly above the road. The man catches the ball 2.4 s after it is thrown. At the instant the man catches it, the ball is 3.6 m below the level of the point of projection.
  1. Explain what it means that the ball is modelled as a particle.
  2. Find the vertical component of the ball's initial velocity.
  3. Find each of the following.
    • The value of \(u\)
    • The value of \(\alpha\)
OCR MEI Paper 1 2024 June Q15
15 The circle \(x ^ { 2 } + y ^ { 2 } + 2 x - 14 y + 25 = 0\) has its centre at the point \(C\). The line \(7 y = x + 25\) intersects the circle at points A and B . Prove that triangle ABC is a right-angled triangle.
OCR MEI Paper 1 2024 June Q16
16 A block of mass \(m\) kg rests on rough horizontal ground. The coefficient of friction between the block and the ground is \(\mu\). A force of magnitude \(T \mathrm {~N}\) is applied at an angle \(\theta\) radians above the horizontal as shown in the diagram and the block slides without tilting or lifting.
\includegraphics[max width=\textwidth, alt={}, center]{1d0ca3d5-6529-435f-a0b8-50ea4859adde-10_291_707_388_239}
  1. Show that the acceleration of the block is given by \(\frac { T } { m } \cos \theta - \mu g + \frac { T } { m } \mu \sin \theta\). For a fixed value of \(T\), the acceleration of the block depends on the value of \(\theta\). The acceleration has its greatest value when \(\theta = \alpha\).
  2. Find an expression for \(\alpha\) in terms of \(\mu\).
OCR MEI Paper 1 2020 November Q1
2 marks
1 Simplify \(\left( \frac { 27 } { x ^ { 9 } } \right) ^ { \frac { 2 } { 3 } } \times \left( \frac { x ^ { 4 } } { 9 } \right)\).
[0pt] [2]
OCR MEI Paper 1 2020 November Q2
2 Express \(\frac { a + \sqrt { 2 } } { 3 - \sqrt { 2 } }\) in the form \(\mathrm { p } + \mathrm { q } \sqrt { 2 }\), giving \(p\) and \(q\) in terms of \(a\).