Questions AS Paper 1 (378 questions)

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OCR MEI AS Paper 1 Specimen Q7
4 marks Moderate -0.8
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
11 marks Moderate -0.8
8 A circle has equation \(( x - 2 ) ^ { 2 } + ( y + 3 ) ^ { 2 } = 25\).
  1. Write down
OCR MEI AS Paper 1 Specimen Q9
8 marks Moderate -0.3
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 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
12 marks Standard +0.8
10
  1. Sketch the graph of \(y = \frac { 1 } { x } + a\), where \(a\) is a positive constant.
OCR MEI AS Paper 1 Specimen Q11
6 marks Moderate -0.3
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
9 marks Moderate -0.3
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
7 marks Moderate -0.8
  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 2018 June Q8
8 marks Moderate -0.8
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
3 marks Easy -1.2
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
3 marks Standard +0.3
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 marks Moderate -0.3
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 marks Standard +0.3
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
12 marks Moderate -0.8
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.
AQA AS Paper 1 2018 June Q1
1 marks Easy -1.8
Three of the following points lie on the same straight line. Which point does not lie on this line? Tick one box. [1 mark] \((-2, 14)\) \((-1, 8)\) \((1, -1)\) \((2, -6)\)
AQA AS Paper 1 2018 June Q2
1 marks Easy -1.2
A circle has equation \((x - 2)^2 + (y + 3)^2 = 13\) Find the gradient of the tangent to this circle at the origin. Circle your answer. [1 mark] \(-\frac{3}{2}\) \quad \(-\frac{2}{3}\) \quad \(\frac{2}{3}\) \quad \(\frac{3}{2}\)
AQA AS Paper 1 2018 June Q3
2 marks Easy -1.2
State the interval for which \(\sin x\) is a decreasing function for \(0° \leq x \leq 360°\) [2 marks]
AQA AS Paper 1 2018 June Q4
5 marks Moderate -0.8
  1. Find the first three terms in the expansion of \((1 - 3x)^4\) in ascending powers of \(x\). [3 marks]
  2. Using your expansion, approximate \((0.994)^4\) to six decimal places. [2 marks]
AQA AS Paper 1 2018 June Q5
5 marks Standard +0.3
Point \(C\) has coordinates \((c, 2)\) and point \(D\) has coordinates \((6, d)\). The line \(y + 4x = 11\) is the perpendicular bisector of \(CD\). Find \(c\) and \(d\). [5 marks]
AQA AS Paper 1 2018 June Q6
7 marks Standard +0.8
\(ABC\) is a right-angled triangle. \includegraphics{figure_6} \(D\) is the point on hypotenuse \(AC\) such that \(AD = AB\). The area of \(\triangle ABD\) is equal to half that of \(\triangle ABC\).
  1. Show that \(\tan A = 2 \sin A\) [4 marks]
    1. Show that the equation given in part (a) has two solutions for \(0° \leq A \leq 90°\) [2 marks]
    2. State the solution which is appropriate in this context. [1 mark]
AQA AS Paper 1 2018 June Q7
5 marks Standard +0.8
Prove that \(n\) is a prime number greater than \(5 \Rightarrow n^4\) has final digit \(1\) [5 marks]
AQA AS Paper 1 2018 June Q8
8 marks Moderate -0.3
Maxine measures the pressure, \(P\) kilopascals, and the volume, \(V\) litres, in a fixed quantity of gas. Maxine believes that the pressure and volume are connected by the equation $$P = cV^d$$ where \(c\) and \(d\) are constants. Using four experimental results, Maxine plots \(\log_{10} P\) against \(\log_{10} V\), as shown in the graph below. \includegraphics{figure_8}
  1. Find the value of \(P\) and the value of \(V\) for the data point labelled \(A\) on the graph. [2 marks]
  2. Calculate the value of each of the constants \(c\) and \(d\). [4 marks]
  3. Estimate the pressure of the gas when the volume is \(2\) litres. [2 marks]
AQA AS Paper 1 2018 June Q9
8 marks Moderate -0.3
Craig is investigating the gradient of chords of the curve with equation \(\mathrm{f}(x) = x - x^2\) Each chord joins the point \((3, -6)\) to the point \((3 + h, \mathrm{f}(3 + h))\) The table shows some of Craig's results.
\(x\)\(\mathrm{f}(x)\)\(h\)\(x + h\)\(\mathrm{f}(x + h)\)Gradient
\(3\)\(-6\)\(1\)\(4\)\(-12\)\(-6\)
\(3\)\(-6\)\(0.1\)\(3.1\)\(-6.51\)\(-5.1\)
\(3\)\(-6\)\(0.01\)
\(3\)\(-6\)\(0.001\)
\(3\)\(-6\)\(0.0001\)
  1. Show how the value \(-5.1\) has been calculated. [1 mark]
  2. Complete the third row of the table above. [2 marks]
  3. State the limit suggested by Craig's investigation for the gradient of these chords as \(h\) tends to \(0\) [1 mark]
  4. Using differentiation from first principles, verify that your result in part (c) is correct. [4 marks]
AQA AS Paper 1 2018 June Q10
11 marks Standard +0.3
A curve has equation \(y = 2x^2 - 8x\sqrt{x} + 8x + 1\) for \(x \geq 0\)
  1. Prove that the curve has a maximum point at \((1, 3)\) Fully justify your answer. [9 marks]
  2. Find the coordinates of the other stationary point of the curve and state its nature. [2 marks]
AQA AS Paper 1 2018 June Q11
1 marks Easy -2.0
In this question use \(g = 9.8\,\mathrm{m}\,\mathrm{s}^{-2}\) A ball, initially at rest, is dropped from a height of \(40\,\mathrm{m}\) above the ground. Calculate the speed of the ball when it reaches the ground. Circle your answer. [1 mark] \(-28\,\mathrm{m}\,\mathrm{s}^{-1}\) \quad \(28\,\mathrm{m}\,\mathrm{s}^{-1}\) \quad \(-780\,\mathrm{m}\,\mathrm{s}^{-1}\) \quad \(780\,\mathrm{m}\,\mathrm{s}^{-1}\)
AQA AS Paper 1 2018 June Q12
1 marks Easy -1.8
An object of mass \(5\,\mathrm{kg}\) is moving in a straight line. As a result of experiencing a forward force of \(F\) newtons and a resistant force of \(R\) newtons it accelerates at \(0.6\,\mathrm{m}\,\mathrm{s}^{-2}\) Which one of the following equations is correct? Circle your answer. [1 mark] \(F - R = 0\) \quad \(F - R = 5\) \quad \(F - R = 3\) \quad \(F - R = 0.6\)