Questions — AQA Paper 1 (128 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 Paper 1 2019 June Q1
1 marks Easy -1.8
Given that \(a > 0\), determine which of these expressions is not equivalent to the others. Circle your answer. [1 mark] $$-2\log_{10}\left(\frac{1}{a}\right) \quad 2\log_{10}(a) \quad \log_{10}(a^2) \quad -4\log_{10}(\sqrt{a})$$
AQA Paper 1 2019 June Q2
1 marks Easy -1.8
Given \(y = e^{kx}\), where \(k\) is a constant, find \(\frac{dy}{dx}\) Circle your answer. [1 mark] $$\frac{dy}{dx} = e^{kx} \quad \frac{dy}{dx} = ke^{kx} \quad \frac{dy}{dx} = kxe^{x-1} \quad \frac{dy}{dx} = \frac{e^{kx}}{k}$$
AQA Paper 1 2019 June Q3
1 marks Easy -1.8
The diagram below shows a sector of a circle. \includegraphics{figure_3} The radius of the circle is 4cm and \(\theta = 0.8\) radians. Find the area of the sector. Circle your answer. [1 mark] $$1.28\text{cm}^2 \quad 3.2\text{cm}^2 \quad 6.4\text{cm}^2 \quad 12.8\text{cm}^2$$
AQA Paper 1 2019 June Q4
4 marks Moderate -0.3
The point \(A\) has coordinates \((-1, a)\) and the point \(B\) has coordinates \((3, b)\) The line \(AB\) has equation \(5x + 4y = 17\) Find the equation of the perpendicular bisector of the points \(A\) and \(B\). [4 marks]
AQA Paper 1 2019 June Q5
7 marks Moderate -0.3
An arithmetic sequence has first term \(a\) and common difference \(d\). The sum of the first 16 terms of the sequence is 260
  1. Show that \(4a + 30d = 65\) [2 marks]
  2. Given that the sum of the first 60 terms is 315, find the sum of the first 41 terms. [3 marks]
  3. \(S_n\) is the sum of the first \(n\) terms of the sequence. Explain why the value you found in part (b) is the maximum value of \(S_n\) [2 marks]
AQA Paper 1 2019 June Q6
8 marks Moderate -0.3
The function f is defined by $$f(x) = \frac{1}{2}(x^2 + 1), \quad x \geq 0$$
  1. Find the range of f. [1 mark]
    1. Find \(f^{-1}(x)\) [3 marks]
    2. State the range of \(f^{-1}(x)\) [1 mark]
  3. State the transformation which maps the graph of \(y = f(x)\) onto the graph of \(y = f^{-1}(x)\) [1 mark]
  4. Find the coordinates of the point of intersection of the graphs of \(y = f(x)\) and \(y = f^{-1}(x)\) [2 marks]
AQA Paper 1 2019 June Q7
11 marks Standard +0.3
  1. By sketching the graphs of \(y = \frac{1}{x}\) and \(y = \sec 2x\) on the axes below, show that the equation $$\frac{1}{x} = \sec 2x$$ has exactly one solution for \(x > 0\) [3 marks] \includegraphics{figure_7a}
  2. By considering a suitable change of sign, show that the solution to the equation lies between 0.4 and 0.6 [2 marks]
  3. Show that the equation can be rearranged to give $$x = \frac{1}{2}\cos^{-1}x$$ [2 marks]
    1. Use the iterative formula $$x_{n+1} = \frac{1}{2}\cos^{-1}x_n$$ with \(x_1 = 0.4\), to find \(x_2\), \(x_3\) and \(x_4\), giving your answers to four decimal places. [2 marks]
    2. On the graph below, draw a cobweb or staircase diagram to show how convergence takes place, indicating the positions of \(x_2\), \(x_3\) and \(x_4\). [2 marks] \includegraphics{figure_7d}
AQA Paper 1 2019 June Q8
4 marks Standard +0.3
$$P(n) = \sum_{k=0}^{n} k^3 - \sum_{k=0}^{n-1} k^3 \text{ where } n \text{ is a positive integer.}$$
  1. Find P(3) and P(10) [2 marks]
  2. Solve the equation \(P(n) = 1.25 \times 10^8\) [2 marks]
AQA Paper 1 2019 June Q9
5 marks Standard +0.8
Prove that the sum of a rational number and an irrational number is always irrational. [5 marks]
AQA Paper 1 2019 June Q10
4 marks Standard +0.3
The volume of a spherical bubble is increasing at a constant rate. Show that the rate of increase of the radius, \(r\), of the bubble is inversely proportional to \(r^2\) Volume of a sphere = \(\frac{4}{3}\pi r^3\) [4 marks]
AQA Paper 1 2019 June Q11
4 marks Moderate -0.8
Jodie is attempting to use differentiation from first principles to prove that the gradient of \(y = \sin x\) is zero when \(x = \frac{\pi}{2}\) Jodie's teacher tells her that she has made mistakes starting in Step 4 of her working. Her working is shown below. \includegraphics{figure_11} Complete Steps 4 and 5 of Jodie's working below, to correct her proof. [4 marks] Step 4 \quad For gradient of curve at A, Step 5 \quad Hence the gradient of the curve at A is given by
AQA Paper 1 2019 June Q12
7 marks Standard +0.3
  1. Show that the equation $$2\cot^2 x + 2\cosec^2 x = 1 + 4\cosec x$$ can be written in the form $$a\cosec^2 x + b\cosec x + c = 0$$ [2 marks]
  2. Hence, given \(x\) is obtuse and $$2\cot^2 x + 2\cosec^2 x = 1 + 4\cosec x$$ find the exact value of \(\tan x\) Fully justify your answer. [5 marks]
AQA Paper 1 2019 June Q13
7 marks Challenging +1.2
A curve, C, has equation $$y = \frac{e^{3x-5}}{x^2}$$ Show that C has exactly one stationary point. Fully justify your answer. [7 marks]
AQA Paper 1 2019 June Q14
10 marks Standard +0.3
The graph of \(y = \frac{2x^3}{x^2 + 1}\) is shown for \(0 \leq x \leq 4\)
[diagram]
Caroline is attempting to approximate the shaded area, A, under the curve using the trapezium rule by splitting the area into \(n\) trapezia.
  1. When \(n = 4\)
    1. State the number of ordinates that Caroline uses. [1 mark]
    2. Calculate the area that Caroline should obtain using this method. Give your answer correct to two decimal places. [3 marks]
  2. Show that the exact area of \(A\) is $$16 - \ln 17$$ Fully justify your answer. [5 marks]
  3. Explain what would happen to Caroline's answer to part (a)(ii) as \(n \to \infty\) [1 mark]
AQA Paper 1 2019 June Q15
13 marks Standard +0.3
At time \(t\) hours after a high tide, the height, \(h\) metres, of the tide and the velocity, \(v\) knots, of the tidal flow can be modelled using the parametric equations $$v = 4 - \left(\frac{2t}{3} - 2\right)^2$$ $$h = 3 - 2\sqrt[3]{t - 3}$$ High tides and low tides occur alternately when the velocity of the tidal flow is zero. A high tide occurs at 2am.
    1. Use the model to find the height of this high tide. [1 mark]
    2. Find the time of the first low tide after 2am. [3 marks]
    3. Find the height of this low tide. [1 mark]
  2. Use the model to find the height of the tide when it is flowing with maximum velocity. [3 marks]
  3. Comment on the validity of the model. [2 marks]
AQA Paper 1 2019 June Q16
16 marks Standard +0.8
  1. \(y = e^{-x}(\sin x + \cos x)\) Find \(\frac{dy}{dx}\) Simplify your answer. [3 marks]
  2. Hence, show that $$\int e^{-x}\sin x \, dx = ae^{-x}(\sin x + \cos x) + c$$ where \(a\) is a rational number. [2 marks]
  3. A sketch of the graph of \(y = e^{-x}\sin x\) for \(x \geq 0\) is shown below. The areas of the finite regions bounded by the curve and the \(x\)-axis are denoted by \(A_1, A_2, \ldots, A_n, \ldots\) \includegraphics{figure_16c}
    1. Find the exact value of the area \(A_1\) [3 marks]
    2. Show that $$\frac{A_2}{A_1} = e^{-\pi}$$ [4 marks]
    3. Given that $$\frac{A_{n+1}}{A_n} = e^{-\pi}$$ show that the exact value of the total area enclosed between the curve and the \(x\)-axis is $$\frac{1 + e^\pi}{2(e^\pi - 1)}$$ [4 marks]
AQA Paper 1 2024 June Q1
1 marks Easy -1.8
Find the coefficient of \(x\) in the expansion of $$(4x^3 - 5x^2 + 3x - 2)(x^5 + 4x + 1)$$ Circle your answer. $$-5 \quad -2 \quad 7 \quad 11$$ [1 mark]
AQA Paper 1 2024 June Q2
1 marks Easy -1.8
The function f is defined by \(f(x) = e^x + 1\) for \(x \in \mathbb{R}\) Find an expression for \(f^{-1}(x)\) Tick \((\checkmark)\) one box. [1 mark] \(f^{-1}(x) = \ln(x - 1)\) \(\square\) \(f^{-1}(x) = \ln(x) - 1\) \(\square\) \(f^{-1}(x) = \frac{1}{e^x + 1}\) \(\square\) \(f^{-1}(x) = \frac{x - 1}{e}\) \(\square\)
AQA Paper 1 2024 June Q3
1 marks Easy -1.2
The expression $$\frac{12x^2 + 3x + 7}{3x - 5}$$ can be written as $$Ax + B + \frac{C}{3x - 5}$$ State the value of \(A\) Circle your answer. [1 mark] $$3 \quad 4 \quad 7 \quad 9$$
AQA Paper 1 2024 June Q4
1 marks Easy -1.8
One of the diagrams below shows the graph of \(y = \arccos x\) Identify the graph of \(y = \arccos x\) Tick \((\checkmark)\) one box. [1 mark] \includegraphics{figure_4}
AQA Paper 1 2024 June Q5
3 marks Easy -1.2
Solve the equation $$\sin^2 x = 1$$ for \(0° < x < 360°\) [3 marks]
AQA Paper 1 2024 June Q6
2 marks Easy -1.2
Use the chain rule to find \(\frac{dy}{dx}\) when \(y = (x^3 + 5x)^7\) [2 marks]
AQA Paper 1 2024 June Q7
4 marks Standard +0.3
Show that $$\frac{3 + \sqrt{8n}}{1 + \sqrt{2n}}$$ can be written as $$\frac{4n - 3 + \sqrt{2n}}{2n - 1}$$ where \(n\) is a positive integer. [4 marks]
AQA Paper 1 2024 June Q8
5 marks Moderate -0.8
\begin{enumerate}[label=(\alph*)] \item Find the first three terms, in ascending powers of \(x\), in the expansion of $$(2 + kx)^5$$ where \(k\) is a positive constant. [3 marks] \item Hence, given that the coefficient of \(x\) is four times the coefficient of \(x^2\), find the value of \(k\) [2 marks]
AQA Paper 1 2024 June Q9
5 marks Standard +0.8
  1. Show that, for small values of \(\theta\) measured in radians $$\cos 4\theta + 2 \sin 3\theta - \tan 2\theta \approx A + B\theta + C\theta^2$$ where \(A\), \(B\) and \(C\) are constants to be found. [3 marks]
  2. Use your answer to part (a) to find an approximation for $$\cos 0.28 + 2 \sin 0.21 - \tan 0.14$$ Give your answer to three decimal places. [2 marks]