Questions AS Paper 1 (363 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 2020 November Q7
7 In this question you must show detailed reasoning.
A curve has equation \(y = 4 x ^ { 3 } - 6 x ^ { 2 } - 9 x + 4\).
  1. Sketch the gradient function for this curve, clearly indicating the points where the gradient is zero.
  2. Find the set of values of \(x\) for which the gradient function is decreasing. Give your answer using set notation.
OCR MEI AS Paper 1 2020 November Q8
8 The point A has coordinates \(( - 1 , - 2 )\) and the point B has coordinates (7,4). The perpendicular bisector of \(A B\) intersects the line \(y + 2 x = k\) at \(P\). Determine the coordinates of P in terms of \(k\).
OCR MEI AS Paper 1 2020 November Q9
9 A car travelling in a straight line accelerates uniformly from rest to \(V \mathrm {~ms} ^ { - 1 }\) in \(T \mathrm {~s}\). It then slows down uniformly, coming to rest after a further \(2 T\) s.
  1. Sketch a velocity-time graph for the car. The acceleration in the first stage of the motion is \(2.5 \mathrm {~ms} ^ { - 2 }\) and the total distance travelled is 240 m .
  2. Calculate the values of \(V\) and \(T\).
OCR MEI AS Paper 1 2020 November Q10
10 An astronaut on the surface of the moon drops a ball from a point 2 m above the surface.
  1. Without any calculations, explain why a standard model using \(g = 9.8 \mathrm {~ms} ^ { - 2 }\) will not be appropriate to model the fall of the ball. The ball takes 1.6s to hit the surface.
  2. Find the acceleration of the ball which best models its motion. Give your answer correct to 2 significant figures.
  3. Use this value to predict the maximum height of the ball above the point of projection when thrown vertically upwards with an initial velocity of \(15 \mathrm {~ms} ^ { - 1 }\).
OCR MEI AS Paper 1 2020 November Q11
11 In this question you must show detailed reasoning.
  1. A student is asked to solve the inequality \(x ^ { \frac { 1 } { 2 } } < 4\). The student argues that \(x ^ { \frac { 1 } { 2 } } < 4 \Leftrightarrow x < 16\), so that the solution is \(\{ x : x < 16 \}\).
    Comment on the validity of the student's argument.
  2. Solve the inequality \(\left( \frac { 1 } { 2 } \right) ^ { x } < 4\).
  3. Show that the equation \(2 \log _ { 2 } ( x + 8 ) - \log _ { 2 } ( x + 6 ) = 3\) has only one root.
OCR MEI AS Paper 1 2021 November Q1
1 Find the coordinates of the point of intersection of the lines \(y = 3 x - 2\) and \(x + 2 y = 10\).
OCR MEI AS Paper 1 2021 November Q2
2 An unmanned craft lands on the planet Mars. A small bolt falls from the craft onto the surface of the planet. It falls 1.5 m from rest in 0.9 s . Calculate the acceleration due to gravity on Mars.
OCR MEI AS Paper 1 2021 November Q3
3 Forces \(\mathbf { F } _ { 1 } = ( 2 \mathbf { i } + 9 \mathbf { j } ) \mathbf { N }\) and \(\mathbf { F } _ { 2 } = ( - \mathbf { i } + \mathbf { j } ) \mathbf { N }\) act on a particle. A third force \(\mathbf { F } _ { 3 }\) acts so that the particle is in equilibrium under the action of the three forces. Find the force \(\mathbf { F } _ { 3 }\).
OCR MEI AS Paper 1 2021 November Q4
4
  1. Show that \(4 ! < 4 ^ { 4 }\).
  2. Nina believes that the statement \(n ! < n ^ { n }\) is true for all positive integers \(n\). Prove that Nina is not correct.
OCR MEI AS Paper 1 2021 November Q5
5 The diagram shows the triangle ABC in which \(\mathrm { AC } = 13 \mathrm {~cm}\) and AB is the shortest side. The perimeter of the triangle is 32 cm . The area is \(24 \mathrm {~cm} ^ { 2 }\) and \(\sin \mathrm { B } = \frac { 4 } { 5 }\). Determine the lengths of AB and BC .
OCR MEI AS Paper 1 2021 November Q6
6 The displacement of a particle is modelled by the equation \(\mathrm { s } = 7 + 4 \mathrm { t } - \mathrm { t } ^ { 2 }\), where \(s\) metres is the displacement from the origin at time \(t\) seconds. The diagram shows part of the displacement-time graph for the particle. The point \(( 2,11 )\) is the maximum point on the graph.
\includegraphics[max width=\textwidth, alt={}, center]{5428eabf-431d-4db1-8c25-1f2b9570d9aa-4_513_1381_422_255}
  1. Kai argues that the point \(( 2,11 )\) is on the graph, so the particle has travelled a distance of 11 metres in the first 2 seconds. Comment on the validity of Kai’s argument.
  2. Determine the total distance the particle travels in the first 10 seconds.
  3. Find an expression for the velocity of the particle at time \(t\).
  4. Find the speed of the particle when \(t = 10\).
OCR MEI AS Paper 1 2021 November Q7
7 The diagram shows part of a curve which passes through the point \(( 1,0 )\).
\includegraphics[max width=\textwidth, alt={}, center]{5428eabf-431d-4db1-8c25-1f2b9570d9aa-4_711_704_1722_258} The gradient of the curve is given by \(\frac { d y } { d x } = 6 x + \frac { 8 } { x ^ { 3 } }\).
Determine whether the curve passes through the point \(( 2,12 )\).
OCR MEI AS Paper 1 2021 November Q9
9
  1. Sketch both of the following on the axes provided in the Printed Answer Booklet.
    1. The curve \(\mathrm { y } = \frac { 12 } { \mathrm { x } }\), stating the coordinates of at least one point on the curve.
    2. The line \(y = 2 x + 8\), stating the coordinates of the points at which the line crosses the axes.
  2. In this question you must show detailed reasoning. Determine the exact coordinates of the points of intersection of the curve and the line.
OCR MEI AS Paper 1 2021 November Q10
10 A rescue worker is lowered from a helicopter on a rope. She attaches a second rope to herself and to a woman in difficulties on the ground. The helicopter winches both women upwards with the rescued woman vertically below the rescue worker, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{5428eabf-431d-4db1-8c25-1f2b9570d9aa-6_509_460_408_262} The model for this motion uses the following modelling assumptions:
  • each woman can be modelled as a particle;
  • the ropes are both light and inextensible;
  • there is no air resistance to the motion;
  • the motion is in a vertical line.
    1. Explain what it means when the women are each 'modelled as a particle'.
    2. Explain what 'light' means in this context.
The tension in the rope to the helicopter is 1500 N . The rescue worker has a mass of 65 kg and the rescued woman has a mass of 75 kg .
  • Draw a diagram showing the forces on the two women.
  • Write down the equation of motion of the two women considered as a single particle.
  • Calculate the acceleration of the women.
  • Determine the tension in the rope connecting the two women.
    •
    OCR MEI AS Paper 1 2021 November Q11
    11 On the day that a new consumer product went on sale (day zero), a call centre received 1 call about it. On the 2nd day after day zero the call centre received 3 calls, and on the 10th day after day zero there were 200 calls. Two models were proposed to model \(N\), the number of calls received \(t\) days after day zero.
    Model 1 is a linear model \(\mathrm { N } = \mathrm { mt } + \mathrm { c }\).
    1. Determine the values of \(m\) and \(c\) which best model the data for 2 days and 10 days after day zero.
    2. State the rate of increase in calls according to model 1.
    3. Explain why this model is not suitable when \(t = 1\). Model 2 is an exponential model \(\mathbf { N } = e ^ { 0.53 t }\).
    4. Verify that this is a good model for the number of calls when \(t = 2\) and \(t = 10\).
    5. Determine the rate of increase in calls when \(t = 10\) according to model 2 .
    OCR MEI AS Paper 1 Specimen Q1
    1 Simplify \(\frac { \left( 2 x ^ { 2 } y \right) ^ { 3 } \times 4 x ^ { 3 } y ^ { 5 } } { 2 x y ^ { 10 } }\).
    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.