Questions — AQA (3620 questions)

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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]
AQA AS Paper 1 2019 June Q10
9 marks Moderate -0.3
On 18 March 2019 there were 12 hours of daylight in Inverness. On 16 June 2019, 90 days later, there will be 18 hours of daylight in Inverness. Jude decides to model the number of hours of daylight in Inverness, \(N\), by the formula $$N = A + B\sin t°$$ where \(t\) is the number of days after 18 March 2019.
    1. State the value that Jude should use for \(A\). [1 mark]
    2. State the value that Jude should use for \(B\). [1 mark]
    3. Using Jude's model, calculate the number of hours of daylight in Inverness on 15 May 2019, 58 days after 18 March 2019. [1 mark]
    4. Using Jude's model, find how many days during 2019 will have at least 17.4 hours of daylight in Inverness. [4 marks]
    5. Explain why Jude's model will become inaccurate for 2020 and future years. [1 mark]
  2. Anisa decides to model the number of hours of daylight in Inverness with the formula $$N = A + B\sin \left(\frac{360}{365}t\right)°$$ Explain why Anisa's model is better than Jude's model. [1 mark]
AQA AS Paper 1 2019 June Q11
1 marks Easy -1.8
A ball moves in a straight line and passes through two fixed points, \(A\) and \(B\), which are \(0.5 \text{m}\) apart. The ball is moving with a constant acceleration of \(0.39 \text{m s}^{-2}\) in the direction \(AB\). The speed of the ball at \(A\) is \(1.9 \text{m s}^{-1}\) Find the speed of the ball at \(B\). Circle your answer. [1 mark] \(2 \text{m s}^{-1}\) \(3.2 \text{m s}^{-1}\) \(3.8 \text{m s}^{-1}\) \(4 \text{m s}^{-1}\)
AQA AS Paper 1 2019 June Q12
1 marks Easy -1.8
A particle \(P\), of mass \(m\) kilograms, is attached to one end of a light inextensible string. The other end of this string is held at a fixed position, \(O\). \(P\) hangs freely, in equilibrium, vertically below \(O\). Identify the statement below that correctly describes the tension, \(T\) newtons, in the string as \(m\) varies. Tick (\(\checkmark\)) one box. [1 mark] \(T\) varies along the string, with its greatest value at \(O\) \(\square\) \(T\) varies along the string, with its greatest value at \(P\) \(\square\) \(T = 0\) because the system is in equilibrium \(\square\) \(T\) is directly proportional to \(m\) \(\square\)
AQA AS Paper 1 2019 June Q13
9 marks Moderate -0.3
A car, starting from rest, is driven along a horizontal track. The velocity of the car, \(v \text{m s}^{-1}\), at time \(t\) seconds, is modelled by the equation $$v = 0.48t^2 - 0.024t^3 \text{ for } 0 \leq t \leq 15$$
  1. Find the distance the car travels during the first 10 seconds of its journey. [3 marks]
  2. Find the maximum speed of the car. Give your answer to three significant figures. [4 marks]
  3. Deduce the range of values of \(t\) for which the car is modelled as decelerating. [2 marks]