Questions — AQA (3620 questions)

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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]
AQA AS Paper 1 2020 June Q7
6 marks Standard +0.3
Curve C has equation \(y = x^2\) C is translated by vector \(\begin{pmatrix} 3 \\ 0 \end{pmatrix}\) to give curve \(C_1\) Line L has equation \(y = x\) L is stretched by scale factor 2 parallel to the \(x\)-axis to give line \(L_1\) Find the exact distance between the two intersection points of \(C_1\) and \(L_1\) [6 marks]
AQA AS Paper 1 2020 June Q8
8 marks Standard +0.3
  1. Find the equation of the tangent to the curve \(y = e^{4x}\) at the point \((a, e^{4a})\). [3 marks]
  2. Find the value of \(a\) for which this tangent passes through the origin. [2 marks]
  3. Hence, find the set of values of \(m\) for which the equation $$e^{4x} = mx$$ has no real solutions. [3 marks]
AQA AS Paper 1 2020 June Q9
5 marks Challenging +1.2
The diagram below shows a circle and four triangles.
[diagram]
\(AB\) is a diameter of the circle. \(C\) is a point on the circumference of the circle. Triangles \(ABK\), \(BCL\) and \(CAM\) are equilateral. Prove that the area of triangle \(ABK\) is equal to the sum of the areas of triangle \(BCL\) and triangle \(CAM\). [5 marks]
AQA AS Paper 1 2020 June Q10
12 marks Moderate -0.8
Raj is investigating how the price, \(P\) pounds, of a brilliant-cut diamond ring is related to the weight, \(C\) carats, of the diamond. He believes that they are connected by a formula $$P = aC^n$$ where \(a\) and \(n\) are constants.
  1. Express \(\ln P\) in terms of \(\ln C\). [2 marks]
  2. Raj researches the price of three brilliant-cut diamond rings on a website with the following results.
    \(C\)0.601.151.50
    \(P\)49512001720
    1. Plot \(\ln P\) against \(\ln C\) for the three rings on the grid below. [2 marks] \includegraphics{figure_10b}
    2. Explain which feature of the plot suggests that Raj's belief may be correct. [1 mark]
    3. Using the graph on page 15, estimate the value of \(a\) and the value of \(n\). [4 marks]
  3. Explain the significance of \(a\) in this context. [1 mark]
  4. Raj wants to buy a ring with a brilliant-cut diamond of weight 2 carats. Estimate the price of such a ring. [2 marks]
AQA AS Paper 1 2020 June Q11
1 marks Easy -1.8
A go-kart and driver, of combined mass 55 kg, move forward in a straight line with a constant acceleration of \(0.2\text{ m s}^{-2}\) The total driving force is 14 N Find the total resistance force acting on the go-kart and driver. Circle your answer. [1 mark] 0N 3N 11N 14N
AQA AS Paper 1 2020 June Q12
1 marks Easy -1.8
One of the following is an expression for the distance between the points represented by position vectors \(5\mathbf{i} - 3\mathbf{j}\) and \(18\mathbf{i} + 7\mathbf{j}\) Identify the correct expression. Tick (\(\checkmark\)) one box. [1 mark] \(\sqrt{13^2 + 4^2}\) \(\sqrt{13^2 + 10^2}\) \(\sqrt{23^2 + 4^2}\) \(\sqrt{23^2 + 10^2}\)
AQA AS Paper 1 2020 June Q13
3 marks Easy -1.8
An object is moving in a straight line, with constant acceleration \(a\text{ m s}^{-2}\), over a time period of \(t\) seconds. It has an initial velocity \(u\) and final velocity \(v\) as shown in the graph below. \includegraphics{figure_13} Use the graph to show that $$v = u + at$$ [3 marks]
AQA AS Paper 1 2020 June Q14
5 marks Moderate -0.3
A particle of mass 0.1 kg is initially stationary. A single force \(\mathbf{F}\) acts on this particle in a direction parallel to the vector \(7\mathbf{i} + 24\mathbf{j}\) As a result, the particle accelerates in a straight line, reaching a speed of \(4\text{ m s}^{-1}\) after travelling a distance of 3.2 m Find \(\mathbf{F}\). [5 marks]
AQA AS Paper 1 2020 June Q15
7 marks Standard +0.3
A particle, \(P\), is moving in a straight line with acceleration \(a\text{ m s}^{-2}\) at time \(t\) seconds, where $$a = 4 - 3t^2$$
  1. Initially \(P\) is stationary. Find an expression for the velocity of \(P\) in terms of \(t\). [2 marks]
  2. When \(t = 2\), the displacement of \(P\) from a fixed point, O, is 39 metres. Find the time at which \(P\) passes through O, giving your answer to three significant figures. Fully justify your answer. [5 marks]
AQA AS Paper 1 2020 June Q16
10 marks Standard +0.3
A simple lifting mechanism comprises a light inextensible wire which is passed over a smooth fixed pulley. One end of the wire is attached to a rigid triangular container of mass 2 kg, which rests on horizontal ground. A load of \(m\) kg is placed in the container. The other end of the wire is attached to a particle of mass 5 kg, which hangs vertically downwards. The mechanism is initially held at rest as shown in the diagram below. \includegraphics{figure_16} The mechanism is released from rest, and the container begins to move upwards with acceleration \(a\text{ m s}^{-2}\) The wire remains taut throughout the motion.
  1. Show that $$a = \left(\frac{3 - m}{m + 7}\right)g$$ [4 marks]
  2. State the range of possible values of \(m\). [1 mark]
  3. In this question use \(g = 9.8\text{ m s}^{-2}\) The load reaches a height of 2 metres above the ground 1 second after it is released. Find the mass of the load. [4 marks]
  4. Ignoring air resistance, describe one assumption you have made in your model. [1 mark]
AQA AS Paper 1 2021 June Q1
1 marks Easy -1.8
Find the coefficient of the \(x\) term in the binomial expansion of \((3 + x)^4\) Circle your answer. [1 mark] 12 27 54 108
AQA AS Paper 1 2021 June Q2
1 marks Easy -1.8
Given that \(\frac{dy}{dx} = \frac{1}{x}\) find \(\frac{d^2y}{dx^2}\) Circle your answer. [1 mark] \(-\frac{2}{x^2}\) \(-\frac{1}{x^2}\) \(\frac{1}{x^2}\) \(\frac{2}{x^2}\)
AQA AS Paper 1 2021 June Q3
3 marks Easy -1.3
The graph of the equation \(y = \frac{1}{x}\) is translated by the vector \(\begin{bmatrix}3\\0\end{bmatrix}\)
  1. Write down the equation of the transformed graph. [1 mark]
  2. State the equations of the asymptotes of the transformed graph. [2 marks]
AQA AS Paper 1 2021 June Q4
9 marks Moderate -0.3
\(ABCD\) is a trapezium where \(A\) is the point \((1, -2)\), \(B\) is the point \((7, 1)\) and \(C\) is the point \((3, 4)\) \(DC\) is parallel to \(AB\). \(AD\) is perpendicular to \(AB\).
    1. Find the equation of the line \(CD\). [2 marks]
    2. Show that point \(D\) has coordinates \((-1, 2)\) [3 marks]
    1. Find the sum of the length of \(AB\) and the length of \(CD\) in simplified surd form. [2 marks]
    2. Hence, find the area of the trapezium \(ABCD\). [2 marks]
AQA AS Paper 1 2021 June Q5
6 marks Moderate -0.3
  1. Sketch the curve $$y = (x - a)^2(3 - x) \quad \text{where } 0 < a < 3$$ indicating the coordinates of the points where the curve and the axes meet. [4 marks] \includegraphics{figure_5}
  2. Hence, solve $$(x - a)^2(3 - x) > 0$$ giving your answer in set notation form. [2 marks]
AQA AS Paper 1 2021 June Q6
7 marks Standard +0.3
A curve has the equation \(y = e^{-2x}\) At point \(P\) on the curve the tangent is parallel to the line \(x + 8y = 5\) Find the coordinates of \(P\) stating your answer in the form \((\ln p, q)\), where \(p\) and \(q\) are rational. [7 marks]
AQA AS Paper 1 2021 June Q7
12 marks Moderate -0.8
Scientists observed a colony of seabirds over a period of 10 years starting in 2010. They concluded that the number of birds in the colony, its population \(P\), could be modelled by a formula of the form $$P = a(10^{bt})$$ where \(t\) is the time in years after 2010, and \(a\) and \(b\) are constants.
  1. Explain what the value of \(a\) represents. [1 mark]
  2. Show that \(\log_{10} P = bt + \log_{10} a\) [2 marks]
  3. The table below contains some data collected by the scientists.
    Year20132015
    \(t\)3
    \(P\)1020012800
    \(\log_{10} P\)4.0086
    1. Complete the table, giving the \(\log_{10} P\) value to 5 significant figures. [1 mark]
    2. Use the data to calculate the value of \(a\) and the value of \(b\). [4 marks]
    3. Use the model to estimate the population of the colony in 2024. [2 marks]
    1. State an assumption that must be made in using the model to estimate the population of the colony in 2024. [1 mark]
    2. Hence comment, with a reason, on the reliability of your estimate made in part (c)(iii). [1 mark]