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 Specimen Q2
2 Find the coefficient of \(x ^ { 4 }\) in the binomial expansion of \(( x - 3 ) ^ { 5 }\).
OCR MEI AS Paper 1 Specimen Q3
3 Fig. 3 shows a particle of weight 8 N on a rough horizontal table.
The particle is being pulled by a horizontal force of 10 N .
It remains at rest in equilibrium. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{970d2349-7705-4b66-9931-83613e5d852f-3_204_454_1311_255} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure}
  1. What information given in the question, tells you that the forces shown in Fig. 3 cannot be the only forces acting on the particle?
  2. The only other forces acting on the particle are due to the particle being on the table. State the types of these forces and their magnitudes.
OCR MEI AS Paper 1 Specimen Q4
4
  1. Express \(x ^ { 2 } + 4 x + 7\) in the form \(( x + b ) ^ { 2 } + c\).
  2. Explain why the minimum point on the curve \(y = ( x + b ) ^ { 2 } + c\) occurs when \(x = - b\).
OCR MEI AS Paper 1 Specimen Q5
5 Particle P moves on a straight line that contains the point O .
At time \(t\) seconds the displacement of P from O is \(s\) metres, where \(s = t ^ { 3 } - 3 t ^ { 2 } + 3\).
  1. Determine the times when the particle has zero velocity.
  2. Find the distances of P from O at the times when it has zero velocity.
OCR MEI AS Paper 1 Specimen Q6
6 Two points, \(A\) and \(B\), have position vectors \(\mathbf { a } = \mathbf { i } - 3 \mathbf { j }\) and \(\mathbf { b } = 4 \mathbf { i } + 3 \mathbf { j }\).
The point C lies on the line \(y = 1\). The lengths of the line segments AC and BC are equal. Determine the position vector of \(C\).
OCR MEI AS Paper 1 Specimen Q7
7 A car is usually driven along the whole of a 5 km stretch of road at a constant speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). On one occasion, during a period of 50 seconds, the speed of the car is as shown by the speed-time graph in Fig. 7.
The rest of the 5 km is travelled at \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{970d2349-7705-4b66-9931-83613e5d852f-5_510_1016_589_296} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure} How much more time than usual did the journey take on this occasion?
Show your working clearly.
OCR MEI AS Paper 1 Specimen Q8
8 A circle has equation \(( x - 2 ) ^ { 2 } + ( y + 3 ) ^ { 2 } = 25\).
  1. Write down
    • the radius of the circle,
    • the coordinates of the centre of the circle.
    • Find, in exact form, the coordinates of the points of intersection of the circle with the \(y\)-axis.
    • Show that the point \(( 1,2 )\) lies outside the circle.
    • The point \(\mathrm { P } ( - 1,1 )\) lies on the circle. Find the equation of the tangent to the circle at P .
OCR MEI AS Paper 1 Specimen Q9
9 A biologist is investigating the growth of bacteria in a piece of bread.
He believes that the number, \(N\), of bacteria after \(t\) hours may be modelled by the relationship \(N = A \times 2 ^ { k t }\), where \(A\) and \(k\) are constants.
  1. Show that, according to the model, the graph of \(\log _ { 10 } N\) against \(t\) is a straight line. Give, in terms of \(A\) and \(k\),
    • the gradient of the line
    • the intercept on the vertical axis.
    The biologist measures the number of bacteria at regular intervals over 22 hours and plots a graph of \(\log _ { 10 } N\) against \(t\). He finds that the graph is approximately a straight line with gradient 0.20 . The line crosses the vertical axis at 2.0 .
  2. Find the values of \(A\) and \(k\).
  3. Use the model to predict the number of bacteria after 24 hours.
  4. Give a reason why the model may not be appropriate for large values of \(t\).
OCR MEI AS Paper 1 Specimen Q10
10
  1. Sketch the graph of \(y = \frac { 1 } { x } + a\), where \(a\) is a positive constant.
    • State the equations of the horizontal and vertical asymptotes.
    • Give the coordinates of any points where the graph crosses the axes.
    • Find the equation of the normal to the curve \(y = \frac { 1 } { x } + 2\) at the point where \(x = 2\).
    • Find the coordinates of the point where this normal meets the curve again.
OCR MEI AS Paper 1 Specimen Q11
11 In this question you must show detailed reasoning.
Determine for what values of \(k\) the graphs \(y = 2 x ^ { 2 } - k x\) and \(y = x ^ { 2 } - k\) intersect.
OCR MEI AS Paper 1 Specimen Q12
12 A box hangs from a balloon by means of a light inelastic string. The string is always vertical. The mass of the box is 15 kg . Catherine initially models the situation by assuming that there is no air resistance to the motion of the box. Use Catherine's model to calculate the tension in the string if:
  1. the box is held at rest by the tension in the string,
  2. the box is instantaneously at rest and accelerating upwards at \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\),
  3. the box is moving downwards at \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and accelerating upwards at \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Catherine now carries out an experiment to find the magnitude of the air resistance on the box when it is moving.
    At a time when the box is accelerating downwards at \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), she finds that the tension in the string is 140 N .
  4. Calculate the magnitude of the air resistance at that time. Give, with a reason, the direction of motion of the box. \section*{END OF QUESTION PAPER}
OCR MEI AS Paper 1 2019 June Q8
  1. The model gives the correct velocity of \(25.6 \mathrm {~ms} ^ { - 1 }\) at time 8 s . Show that \(k = 0.1\). A second model for the motion uses constant acceleration.
  2. Find the value of the acceleration which gives the correct velocity of \(25.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time 8 s .
  3. Show that these two models give the same value for the displacement in the first 8 s .
OCR MEI AS Paper 1 2018 June Q8
8 In this question you must show detailed reasoning. Fig. 8 shows the graph of a quadratic function. The graph crosses the axes at the points \(( - 1,0 ) , ( 0 , - 4 )\) and \(( 2,0 )\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1513048a-d53b-4b85-82f4-c86e0d81f8f8-4_689_606_1114_731} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure} Find the area of the finite region bounded by the curve and the \(x\)-axis.
OCR MEI AS Paper 1 2019 June Q1
1 In this question you must show detailed reasoning. Show that the equation \(x = 7 + 2 x ^ { 2 }\) has no real roots.
OCR MEI AS Paper 1 2019 June Q2
2 In this question you must show detailed reasoning. Fig. 2 shows the graphs of \(y = 4 \sin x ^ { \circ }\) and \(y = 3 \cos x ^ { \circ }\) for \(0 \leqslant x \leqslant 360\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0b1c272a-f0f4-4931-be89-5d045804a7af-3_549_768_813_258} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} Find the \(x\)-coordinates of the two points of intersection, giving your answers correct to 1 decimal place.
OCR MEI AS Paper 1 2019 June Q9
9 In this question you must show detailed reasoning. A car accelerates from rest along a straight level road. The velocity of the car after 8 s is \(25.6 \mathrm {~ms} ^ { - 1 }\).
In one model for the motion, the velocity \(v \mathrm {~ms} ^ { - 1 }\) at time \(t\) seconds is given by \(v = 1.2 t ^ { 2 } - k t ^ { 3 }\), where \(k\) is a constant and \(0 \leqslant t \leqslant 8\).
  1. The model gives the correct velocity of \(25.6 \mathrm {~ms} ^ { - 1 }\) at time 8 s . Show that \(k = 0.1\). A second model for the motion uses constant acceleration.
  2. Find the value of the acceleration which gives the correct velocity of \(25.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time 8 s .
  3. Show that these two models give the same value for the displacement in the first 8 s .
OCR MEI AS Paper 1 2020 November Q12
12 In this question you must show detailed reasoning. Fig. 12 shows part of the graph of \(y = x ^ { 2 } + \frac { 1 } { x ^ { 2 } }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a1b6c827-7d74-4527-9b60-58872e3d5ef7-7_574_574_402_233} \captionsetup{labelformat=empty} \caption{Fig. 12}
\end{figure} The tangent to the curve \(\mathrm { y } = \mathrm { x } ^ { 2 } + \frac { 1 } { \mathrm { x } ^ { 2 } }\) at the point \(\left( 2 , \frac { 17 } { 4 } \right)\) meets the \(x\)-axis at A and meets the \(y\)-axis at B . O is the origin.
  1. Find the exact area of the triangle OAB .
  2. Use calculus to prove that the complete curve has two minimum points and no maximum point. \section*{END OF QUESTION PAPER}
OCR MEI AS Paper 1 2021 November Q8
8 In this question you must show detailed reasoning.
  1. Use differentiation to find the coordinates of the stationary point on the curve with equation \(y = 2 x ^ { 2 } - 3 x - 2\).
  2. Use the second derivative to determine the nature of the stationary point.
  3. Show by shading on a sketch the region defined by the inequality \(y \geqslant 2 x ^ { 2 } - 3 x - 2\), indicating clearly whether the boundary is included or not.
  4. Solve the inequality \(2 x ^ { 2 } - 3 x - 2 > 0\) using set notation for your answer.