Questions AS Paper 1 (378 questions)

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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]
AQA AS Paper 1 2021 June Q8
7 marks Standard +0.3
    1. Show that the equation $$3\sin\theta\tan\theta = 5\cos\theta - 2$$ is equivalent to the equation $$(4\cos\theta - 3)(2\cos\theta + 1) = 0$$ [3 marks]
    2. Solve the equation $$3\sin\theta\tan\theta = 5\cos\theta - 2$$ for \(-180° \leq \theta \leq 180°\) [2 marks]
  2. Hence, deduce all the solutions of the equation $$3\sin\left(\frac{1}{2}\theta\right)\tan\left(\frac{1}{2}\theta\right) = 5\cos\left(\frac{1}{2}\theta\right) - 2$$ for \(-180° \leq \theta \leq 180°\), giving your answers to the nearest degree. [2 marks]
AQA AS Paper 1 2021 June Q9
7 marks Standard +0.8
A curve has equation $$y = \frac{a}{\sqrt{x}} + bx^2 + \frac{c}{x^3} \quad \text{for } x > 0$$ where \(a\), \(b\) and \(c\) are positive constants. The curve has a single turning point. Use the second derivative of \(y\) to determine the nature of this turning point. You do not need to find the coordinates of the turning point. Fully justify your answer. [7 marks]
AQA AS Paper 1 2021 June Q10
1 marks Easy -2.0
Two forces \(\begin{bmatrix}3\\-2\end{bmatrix}\) N and \(\begin{bmatrix}-7\\-5\end{bmatrix}\) N act on a particle. Find the resultant force. Circle your answer. [1 mark] \(\begin{bmatrix}-21\\10\end{bmatrix}\) N \(\begin{bmatrix}-4\\-7\end{bmatrix}\) N \(\begin{bmatrix}4\\3\end{bmatrix}\) N \(\begin{bmatrix}10\\7\end{bmatrix}\) N
AQA AS Paper 1 2021 June Q11
1 marks Easy -2.5
Jackie says: "A person's weight on Earth is directly proportional to their mass." Tom says: "A person's weight on Earth is different to their weight on the moon." Only one of the statements below is correct. Identify the correct statement. Tick (✓) one box. [1 mark] Jackie and Tom are both wrong. \(\square\) Jackie is right but Tom is wrong. \(\square\) Jackie is wrong but Tom is right. \(\square\) Jackie and Tom are both right. \(\square\)
AQA AS Paper 1 2021 June Q12
4 marks Easy -1.2
A particle P lies at rest on a smooth horizontal table. A constant resultant force, F newtons, is then applied to P. As a result P moves in a straight line with constant acceleration \(\begin{bmatrix}8\\6\end{bmatrix}\) m s⁻²
  1. Show that the magnitude of the acceleration of P is 10 m s⁻² [1 mark]
  2. Find the speed of P after 3 seconds. [1 mark]
  3. Given that \(\mathbf{F} = \begin{bmatrix}2\\1.5\end{bmatrix}\) N, find the mass of P. [2 marks]
AQA AS Paper 1 2021 June Q13
5 marks Easy -1.2
A car, initially at rest, is driven along a straight horizontal road. The graph below is a simple model of how the car's velocity, \(v\) metres per second, changes with respect to time, \(t\) seconds. \includegraphics{figure_13}
  1. Find the displacement of the car when \(t = 45\) [3 marks]
  2. Shona says: "This model is too simple. It is unrealistic to assume that the car will instantaneously change its acceleration." On the axes below sketch a graph, for the first 10 seconds of the journey, which would represent a more realistic model. [2 marks] \includegraphics{figure_13b}
AQA AS Paper 1 2021 June Q14
6 marks Moderate -0.3
A particle, P, is moving along a straight line such that its acceleration \(a\) m s⁻², at any time, \(t\) seconds, may be modelled by $$a = 3 + 0.2t$$ When \(t = 2\), the velocity of P is \(k\) m s⁻¹
  1. Show that the initial velocity of P is given by the expression \((k - 6.4)\) m s⁻¹ [4 marks]
  2. The initial velocity of P is one fifth of the velocity when \(t = 2\) Find the value of \(k\). [2 marks]
AQA AS Paper 1 2021 June Q15
10 marks Moderate -0.3
In this question, use \(g = 10\) m s⁻² A box, B, of mass 4 kg lies at rest on a fixed rough horizontal shelf. One end of a light string is connected to B. The string passes over a smooth peg, attached to the end of the shelf. The other end of the string is connected to particle, P, of mass 1 kg, which hangs freely below the shelf as shown in the diagram below. \includegraphics{figure_15} B is initially held at rest with the string taut. B is then released. B and P both move with constant acceleration \(a\) m s⁻² As B moves across the shelf it experiences a total resistance force of 5 N
  1. State one type of force that would be included in the total resistance force. [1 mark]
  2. Show that \(a = 1\) [4 marks]
  3. When B has moved forward exactly 20 cm the string breaks. Find how much further B travels before coming to rest. [4 marks]
  4. State one assumption you have made when finding your solutions in parts (b) or (c). [1 mark]