Questions — AQA AS Paper 1 (133 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 PURE 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 PURE S1 S2 S3 S4 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 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 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
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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 \(F - R = 0.6\)
AQA AS Paper 1 2018 June Q13
6 marks Moderate -0.8
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 \(-1.2\,\mathrm{m}\,\mathrm{s}^{-2}\) The vehicle then immediately stops accelerating, and travels a further \(33\,\mathrm{m}\) at constant speed.
  1. Draw a velocity-time graph for this journey on the grid below. [3 marks] \includegraphics{figure_13}
  2. Find the distance of the car from \(P\) after \(20\) seconds. [3 marks]
AQA AS Paper 1 2018 June Q14
6 marks Moderate -0.8
In this question use \(g = 9.81\,\mathrm{m}\,\mathrm{s}^{-2}\) Two particles, of mass \(1.8\,\mathrm{kg}\) and \(1.2\,\mathrm{kg}\), are connected by a light, inextensible string over a smooth peg. \includegraphics{figure_14}
  1. Initially the particles are held at rest \(1.5\,\mathrm{m}\) above horizontal ground and the string between them is taut. The particles are released from rest. Find the time taken for the \(1.8\,\mathrm{kg}\) particle to reach the ground. [5 marks]
  2. State one assumption you have made in answering part (a). [1 mark]
AQA AS Paper 1 2018 June Q15
6 marks Moderate -0.3
A cyclist, Laura, is travelling in a straight line on a horizontal road at a constant speed of \(25\,\mathrm{km}\,\mathrm{h}^{-1}\) A second cyclist, Jason, is riding closely and directly behind Laura. He is also moving with a constant speed of \(25\,\mathrm{km}\,\mathrm{h}^{-1}\)
  1. The driving force applied by Jason is likely to be less than the driving force applied by Laura. Explain why. [1 mark]
  2. Jason has a problem and stops, but Laura continues at the same constant speed. Laura sees an accident \(40\,\mathrm{m}\) ahead, so she stops pedalling and applies the brakes. She experiences a total resistance force of \(40\,\mathrm{N}\) Laura and her cycle have a combined mass of \(64\,\mathrm{kg}\)
    1. Determine whether Laura stops before reaching the accident. Fully justify your answer. [4 marks]
    2. State one assumption you have made that could affect your answer to part (b)(i). [1 mark]
AQA AS Paper 1 2018 June Q16
7 marks Moderate -0.8
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 $$v = 0.06(2 + t - t^2)$$
  1. Find an expression for the displacement, \(r\) metres, of the toy from \(A\) at time \(t\) seconds. [4 marks]
  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. [3 marks]
AQA AS Paper 1 2019 June Q1
1 marks Easy -1.8
State the number of solutions to the equation \(\tan 4\theta = 1\) for \(0° < \theta < 180°\) Circle your answer. [1 mark] 1 2 4 8
AQA AS Paper 1 2019 June Q2
1 marks Easy -1.2
Dan believes that for every positive integer \(n\), at least one of \(2^n - 1\) and \(2^n + 1\) is prime. Which value of \(n\) shown below is a counter example to Dan's belief? Circle your answer. [1 mark] \(n = 3\) \(n = 4\) \(n = 5\) \(n = 6\)
AQA AS Paper 1 2019 June Q3
5 marks Moderate -0.3
It is given that \((x + 1)\) and \((x - 3)\) are two factors of \(f(x)\), where $$f(x) = px^3 - 3x^2 - 8x + q$$
  1. Find the values of \(p\) and \(q\). [3 marks]
  2. Fully factorise \(f(x)\). [2 marks]
AQA AS Paper 1 2019 June Q4
4 marks Moderate -0.8
Show that \(\frac{\sqrt{6}}{\sqrt{3} - \sqrt{2}}\) can be expressed in the form \(m\sqrt{n} + n\sqrt{m}\), where \(m\) and \(n\) are integers. Fully justify your answer. [4 marks]
AQA AS Paper 1 2019 June Q5
5 marks Moderate -0.8
  1. Sketch the curve \(y = g(x)\) where $$g(x) = (x + 2)(x - 1)^2$$ [3 marks]
  2. Hence, solve \(g(x) \leq 0\) [2 marks]
AQA AS Paper 1 2019 June Q6
6 marks Moderate -0.3
    1. Show that \(\cos \theta = \frac{1}{2}\) is one solution of the equation $$6\sin^2 \theta + 5\cos \theta = 7$$ [2 marks]
    2. Find all the values of \(\theta\) that solve the equation $$6\sin^2 \theta + 5\cos \theta = 7$$ for \(0° \leq \theta \leq 360°\) Give your answers to the nearest degree. [2 marks]
  2. Hence, find all the solutions of the equation $$6\sin^2 2\theta + 5\cos 2\theta = 7$$ for \(0° \leq \theta \leq 360°\) Give your answers to the nearest degree. [2 marks]
AQA AS Paper 1 2019 June Q7
6 marks Challenging +1.2
Given that \(y \in \mathbb{R}\), prove that $$(2 + 3y)^4 + (2 - 3y)^4 \geq 32$$ Fully justify your answer. [6 marks]
AQA AS Paper 1 2019 June Q8
6 marks Standard +0.3
Prove that the curve with equation $$y = 2x^5 + 5x^4 + 10x^3 - 8$$ has only one stationary point, stating its coordinates. [6 marks]
AQA AS Paper 1 2019 June Q9
10 marks Moderate -0.3
A curve cuts the \(x\)-axis at \((2, 0)\) and has gradient function $$\frac{dy}{dx} = \frac{24}{x^3}$$
  1. Find the equation of the curve. [4 marks]
  2. Show that the perpendicular bisector of the line joining \(A(-2, 8)\) to \(B(-6, -4)\) is the normal to the curve at \((2, 0)\) [6 marks]