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 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.
AQA AS Paper 1 2018 June Q1
1 Three of the following points lie on the same straight line.
Which point does not lie on this line?
Tick one box.
(-2, 14) □
(-1, 8)
\includegraphics[max width=\textwidth, alt={}, center]{c982106c-b742-444f-aeed-6f59ff3fae56-02_109_113_1082_813}
\(( 1 , - 1 )\) □
\(( 2 , - 6 )\) □
AQA AS Paper 1 2018 June Q2
1 marks
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.
[0pt] [1 mark]
\(- \frac { 3 } { 2 }\)
\(- \frac { 2 } { 3 }\)
\(\frac { 2 } { 3 }\)
\(\frac { 3 } { 2 }\)
AQA AS Paper 1 2018 June Q3
2 marks
3 State the interval for which \(\sin x\) is a decreasing function for \(0 ^ { \circ } \leq x \leq 360 ^ { \circ }\)
[0pt] [2 marks]
AQA AS Paper 1 2018 June Q4
4
  1. Find the first three terms in the expansion of \(( 1 - 3 x ) ^ { 4 }\) in ascending powers of \(x\). 4
  2. Using your expansion, approximate \(( 0.994 ) ^ { 4 }\) to six decimal places.
AQA AS Paper 1 2018 June Q5
5 marks
5 Point \(C\) has coordinates \(( c , 2 )\) and point \(D\) has coordinates \(( 6 , d )\). The line \(y + 4 x = 11\) is the perpendicular bisector of \(C D\).
Find \(c\) and \(d\).
[0pt] [5 marks]
\(6 \quad A B C\) is a right-angled triangle.
\includegraphics[max width=\textwidth, alt={}, center]{c982106c-b742-444f-aeed-6f59ff3fae56-06_693_426_315_808}
\(D\) is the point on hypotenuse \(A C\) such that \(A D = A B\).
The area of \(\triangle A B D\) is equal to half that of \(\triangle A B C\).
AQA AS Paper 1 2018 June Q6
6
  1. Show that \(\tan A = 2 \sin A\)
    6
    1. Show that the equation given in part (a) has two solutions for \(0 ^ { \circ } \leq A \leq 90 ^ { \circ }\)
      6
  3. (ii) State the solution which is appropriate in this context.
AQA AS Paper 1 2018 June Q8
8 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 $$P = c V ^ { d }$$ 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[max width=\textwidth, alt={}, center]{c982106c-b742-444f-aeed-6f59ff3fae56-10_1386_1076_792_482} 8
  1. Find the value of \(P\) and the value of \(V\) for the data point labelled \(A\) on the graph.
    8
  2. Calculate the value of each of the constants \(c\) and \(d\).
    8
  3. Estimate the pressure of the gas when the volume is 2 litres.
AQA AS Paper 1 2018 June Q9
5 marks
9 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
3-614-12-6
3-60.13.1-6.51-5.1
3-60.01
3-60.001
3-60.0001
9
  1. Show how the value - 5.1 has been calculated. 9
  2. Complete the third row of the table above.
    9
  		3. State the limit suggested by Craig's investigation for the gradient of these chords as \(h\) tends to 0
    [1 mark]
    9
  		4. Using differentiation from first principles, verify that your result in part (c) is correct. [4 marks] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)
AQA AS Paper 1 2018 June Q10
2 marks
10 A curve has equation \(y = 2 x ^ { 2 } - 8 x \sqrt { x } + 8 x + 1\) for \(x \geq 0\) 10
  1. Prove that the curve has a maximum point at ( 1,3 )
    Fully justify your answer.
    10
  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
11 In this question use \(g = 9.8 \mathrm {~ms} ^ { - 2 }\)
A ball, initially at rest, is dropped from a height of 40 m above the ground.
Calculate the speed of the ball when it reaches the ground.
Circle your answer.
\(- 28 \mathrm {~ms} ^ { - 1 }\)
\(28 \mathrm {~ms} ^ { - 1 }\)
\(- 780 \mathrm {~ms} ^ { - 1 }\)
\(780 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
AQA AS Paper 1 2018 June Q12
1 marks
12 An object of mass 5 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.
[0pt] [1 mark]
\(F - R = 0\)
\(F - R = 5\)
\(F - R = 3\)
\(F - R = 0.6\)
AQA AS Paper 1 2018 June Q13
3 marks
13 A vehicle, which begins at rest at point \(P\), is travelling in a straight line. For the first 4 seconds the vehicle moves with a constant acceleration of \(0.75 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
For the next 5 seconds the vehicle moves with a constant acceleration of \(- 1.2 \mathrm {~ms} ^ { - 2 }\) The vehicle then immediately stops accelerating, and travels a further 33 m at constant speed. 13
  1. Draw a velocity-time graph for this journey on the grid below.
    [0pt] [3 marks]
    \includegraphics[max width=\textwidth, alt={}, center]{c982106c-b742-444f-aeed-6f59ff3fae56-17_739_1670_790_185} 13
  2. Find the distance of the car from \(P\) after 20 seconds.
AQA AS Paper 1 2018 June Q14
14 In this question use \(g = 9.81 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) Two particles, of mass 1.8 kg and 1.2 kg , are connected by a light, inextensible string over a smooth peg.
\includegraphics[max width=\textwidth, alt={}, center]{c982106c-b742-444f-aeed-6f59ff3fae56-18_556_680_488_680} 14
  1. Initially the particles are held at rest 1.5 m above horizontal ground and the string between them is taut. The particles are released from rest.
    Find the time taken for the 1.8 kg particle to reach the ground.
    14
  2. State one assumption you have made in answering part (a).
AQA AS Paper 1 2018 June Q15
1 marks
15 (b) (ii) State one assumption you have made that could affect your answer to part (b)(i).
[0pt] [1 mark] Turn over for the next question
AQA AS Paper 1 2018 June Q16
16 A remote-controlled toy car is moving over a horizontal surface. It moves in a straight line through a point \(A\). The toy is initially at the point with displacement 3 metres from \(A\). Its velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), at time \(t\) seconds is defined by $$v = 0.06 \left( 2 + t - t ^ { 2 } \right)$$ 16
  1. Find an expression for the displacement, \(r\) metres, of the toy from \(A\) at time \(t\) seconds.
    16
  2. In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) At time \(t = 2\) seconds, the toy launches a ball which travels directly upwards with initial speed \(3.43 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Find the time taken for the ball to reach its highest point.
    \includegraphics[max width=\textwidth, alt={}, center]{c982106c-b742-444f-aeed-6f59ff3fae56-24_2496_1721_214_150}
AQA AS Paper 1 2019 June Q1
1 State the number of solutions to the equation \(\tan 4 \theta = 1\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\)
Circle your answer. 1288