Questions AS Paper 1 (378 questions)

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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]
AQA AS Paper 1 2019 June Q14
7 marks Moderate -0.8
Two particles, \(A\) and \(B\), lie at rest on a smooth horizontal plane. \(A\) has position vector \(\mathbf{r}_A = (13\mathbf{i} - 22\mathbf{j})\) metres \(B\) has position vector \(\mathbf{r}_B = (3\mathbf{i} + 2\mathbf{j})\) metres
  1. Calculate the distance between \(A\) and \(B\). [2 marks]
  2. Three forces, \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\) are applied to particle \(A\), where \(\mathbf{F}_1 = (-2\mathbf{i} + 4\mathbf{j})\) newtons \(\mathbf{F}_2 = (6\mathbf{i} - 10\mathbf{j})\) newtons Given that \(A\) remains at rest, explain why \(\mathbf{F}_3 = (-4\mathbf{i} + 6\mathbf{j})\) newtons [1 mark]
  3. A force of \((5\mathbf{i} - 12\mathbf{j})\) newtons, is applied to \(B\), so that \(B\) moves, from rest, in a straight line towards \(A\). \(B\) has a mass of \(0.8 \text{kg}\)
    1. Show that the acceleration of \(B\) towards \(A\) is \(16.25 \text{m s}^{-2}\) [2 marks]
    2. Hence, find the time taken for \(B\) to reach \(A\). Give your answer to two significant figures. [2 marks]
AQA AS Paper 1 2019 June Q15
9 marks Standard +0.3
A tractor and its driver have a combined mass of \(m\) kilograms. The tractor is towing a trailer of mass \(4m\) kilograms in a straight line along a horizontal road. The tractor and trailer are connected by a horizontal tow bar, modelled as a light rigid rod. A driving force of \(11080 \text{N}\) and a total resistance force of \(160 \text{N}\) act on the tractor. A total resistance force of \(600 \text{N}\) acts on the trailer. The tractor and the trailer have an acceleration of \(0.8 \text{m s}^{-2}\)
  1. Find \(m\). [3 marks]
  2. Find the tension in the tow bar. [2 marks]
  3. At the instant the speed of the tractor reaches \(18 \text{km h}^{-1}\) the tow bar breaks. The total resistance force acting on the trailer remains constant. Starting from the instant the tow bar breaks, calculate the time taken until the speed of the trailer reduces to \(9 \text{km h}^{-1}\) [4 marks]
AQA AS Paper 1 2020 June Q1
1 marks Easy -1.8
At the point \((1, 0)\) on the curve \(y = \ln x\), which statement below is correct? Tick (\(\checkmark\)) one box. [1 mark] The gradient is negative and decreasing The gradient is negative and increasing The gradient is positive and decreasing The gradient is positive and increasing
AQA AS Paper 1 2020 June Q2
1 marks Easy -1.8
Given that \(f(x) = 10\) when \(x = 4\), which statement below must be correct? Tick (\(\checkmark\)) one box. [1 mark] \(f(2x) = 5\) when \(x = 4\) \(f(2x) = 10\) when \(x = 2\) \(f(2x) = 10\) when \(x = 8\) \(f(2x) = 20\) when \(x = 4\)
AQA AS Paper 1 2020 June Q3
4 marks Moderate -0.3
Jia has to solve the equation $$2 - 2\sin^2 \theta = \cos \theta$$ where \(-180° \leq \theta \leq 180°\) Jia's working is as follows: $$2 - 2(1 - \cos^2 \theta) = \cos \theta$$ $$2 - 2 + 2\cos^2 \theta = \cos \theta$$ $$2\cos^2 \theta = \cos \theta$$ $$2\cos \theta = 1$$ $$\cos \theta = 0.5$$ $$\theta = 60°$$ Jia's teacher tells her that her solution is incomplete.
  1. Explain the two errors that Jia has made. [2 marks]
  2. Write down all the values of \(\theta\) that satisfy the equation $$2 - 2\sin^2 \theta = \cos \theta$$ where \(-180° \leq \theta \leq 180°\) [2 marks]
AQA AS Paper 1 2020 June Q4
3 marks Moderate -0.5
In the binomial expansion of \((\sqrt{3} + \sqrt{2})^4\) there are two irrational terms. Find the difference between these two terms. [3 marks]
AQA AS Paper 1 2020 June Q5
4 marks Moderate -0.5
Differentiate from first principles $$y = 4x^2 + x$$ [4 marks]
AQA AS Paper 1 2020 June Q6
9 marks Moderate -0.3
  1. It is given that $$f(x) = x^3 - x^2 + x - 6$$ Use the factor theorem to show that \((x - 2)\) is a factor of \(f(x)\). [2 marks]
  2. Find the quadratic factor of \(f(x)\). [1 mark]
  3. Hence, show that there is only one real solution to \(f(x) = 0\) [3 marks]
  4. Find the exact value of \(x\) that solves $$e^{3x} - e^{2x} + e^x - 6 = 0$$ [3 marks]