Questions — AQA Paper 1 (128 questions)

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AQA Paper 1 2024 June Q10
6 marks Moderate -0.8
  1. An arithmetic sequence has 300 terms. The first term of the sequence is \(-7\) and the last term is 32 Find the sum of the 300 terms. [2 marks]
  2. A school holds a raffle at its summer fair. There are nine prizes. The total value of the prizes is £1260 The values of the prizes form an arithmetic sequence. The top prize has the highest value, and the bottom prize has the least value. The value of the top prize is six times the value of the bottom prize. Find the value of the top prize. [4 marks]
AQA Paper 1 2024 June Q11
5 marks Standard +0.3
It is given that $$f(x) = x(x - a)(x - 6)$$ where \(0 < a < 6\)
  1. Sketch the graph of \(y = f(x)\) on the axes below. [3 marks] \includegraphics{figure_11a}
  2. Sketch the graph of \(y = f(-2x)\) on the axes below. [2 marks] \includegraphics{figure_11b}
AQA Paper 1 2024 June Q12
5 marks Moderate -0.8
The terms, \(u_n\), of a periodic sequence are defined by $$u_1 = 3 \quad \text{and} \quad u_{n+1} = \frac{-6}{u_n}$$
  1. Find \(u_2\), \(u_3\) and \(u_4\) [2 marks]
  2. State the period of the sequence. [1 mark]
  3. Find the value of \(\sum_{n=1}^{101} u_n\) [2 marks]
AQA Paper 1 2024 June Q13
6 marks Standard +0.3
  1. It is given that $$P(x) = 4x^3 + 8x^2 + 11x + 4$$ Use the factor theorem to show that \((2x + 1)\) is a factor of \(P(x)\) [2 marks]
  2. Express \(P(x)\) in the form $$P(x) = (2x + 1)(ax^2 + bx + c)$$ where \(a\), \(b\) and \(c\) are constants to be found. [2 marks]
  3. Given that \(n\) is a positive integer, use your answer to part (b) to explain why \(4n^3 + 8n^2 + 11n + 4\) is never prime. [2 marks]
AQA Paper 1 2024 June Q14
10 marks Standard +0.3
  1. The equation $$x^3 = e^{6-2x}$$ has a single solution, \(x = \alpha\) By considering a suitable change of sign, show that \(\alpha\) lies between 0 and 4 [2 marks]
  2. Show that the equation \(x^3 = e^{6-2x}\) can be rearranged to give $$x = 3 - \frac{3}{2}\ln x$$ [3 marks]
    1. Use the iterative formula $$x_{n+1} = 3 - \frac{3}{2}\ln x_n$$ with \(x_1 = 4\), to find \(x_2\), \(x_3\) and \(x_4\) Give your answers to three decimal places. [2 marks]
    2. Figure 1 below shows a sketch of parts of the graphs of $$y = 3 - \frac{3}{2}\ln x \quad \text{and} \quad y = x$$ On Figure 1, draw a staircase or cobweb diagram to show how convergence takes place. Label, on the \(x\)-axis, the positions of \(x_2\), \(x_3\) and \(x_4\) [2 marks]
      [diagram]
    3. Explain why the iterative formula $$x_{n+1} = 3 - \frac{3}{2}\ln x_n$$ fails to converge to \(\alpha\) when the starting value is \(x_1 = 0\) [1 mark]
AQA Paper 1 2024 June Q15
6 marks Standard +0.3
  1. Show that the expression $$\sin 2\theta \cosec \theta + \cos 2\theta \sec \theta$$ can be written as $$4 \cos \theta - \sec \theta$$ where \(\sin \theta \neq 0\) and \(\cos \theta \neq 0\) [4 marks]
  2. A student is attempting to solve the equation $$\sin 2\theta \cosec \theta + \cos 2\theta \sec \theta = 3 \quad \text{for } 0° \leq \theta \leq 360°$$ They use the result from part (a), and write the following incorrect solution: \(\sin 2\theta \cosec \theta + \cos 2\theta \sec \theta = 3\) Step 1: \(4 \cos \theta - \sec \theta = 3\) Step 2: \(4 \cos \theta - \frac{1}{\cos \theta} - 3 = 0\) Step 3: \(4 \cos^2 \theta - 3 \cos \theta - 1 = 0\) Step 4: \(\cos \theta = 1\) or \(\cos \theta = -0.25\) Step 5: \(\theta = 0°, 104.5°, 255.5°, 360°\)
    1. Explain why the student should reject one of their values for \(\cos \theta\) in Step 4. [1 mark]
    2. State the correct solutions to the equation $$\sin 2\theta \cosec \theta + \cos 2\theta \sec \theta = 3 \quad \text{for } 0° \leq \theta \leq 360°$$ [1 mark]
AQA Paper 1 2024 June Q16
5 marks Moderate -0.8
Figure 2 below shows a 1.5 metre length of pipe. \includegraphics{figure_16} The symmetrical cross-section of the pipe is shown below, in Figure 3, where \(x\) and \(y\) are measured in centimetres. \includegraphics{figure_16_cross_section} Use the trapezium rule, with the values shown in the table below, to find the best estimate for the volume of the pipe. \begin{array}{|c|c|c|c|c|c|c|} \hline x & 0 & 0.4 & 0.8 & 1.2 & 1.6 & 2
\hline y & -3 & -2.943 & -2.752 & -2.353 & -1.572 & 0
\hline \end{array} [5 marks]
AQA Paper 1 2024 June Q17
6 marks Moderate -0.8
The function f is defined by $$f(x) = |x| + 1 \quad \text{for } x \in \mathbb{R}$$ The function g is defined by $$g(x) = \ln x$$ where g has its greatest possible domain.
  1. Using set notation, state the range of f [2 marks]
  2. State the domain of g [1 mark]
  3. The composite function h is given by $$h(x) = g f(x) \quad \text{for } x \in \mathbb{R}$$
    1. Write down an expression for \(h(x)\) in terms of \(x\) [1 mark]
    2. Determine if h has an inverse. Fully justify your answer. [2 marks]
AQA Paper 1 2024 June Q18
11 marks Standard +0.3
  1. Use a suitable substitution to show that $$\int_0^1 (4x + 1)(2x + 1)^{\frac{1}{2}} dx$$ can be written as $$\frac{1}{2}\int_a^9 (2u^{\frac{1}{2}} - u^{\frac{1}{2}}) du$$ where \(a\) is a constant to be found. [5 marks]
  2. Hence, or otherwise, show that $$\int_0^1 (4x + 1)(2x + 1)^{\frac{1}{2}} dx = \frac{1322}{15}$$ [4 marks]
  3. A graph has the equation $$y = (4x + 1)\sqrt{2x + 1}$$ A student uses four rectangles to approximate the area under the graph between the lines \(x = 0\) and \(x = 4\) The rectangles are all the same width. All the rectangles are drawn under the curve as shown in the diagram below. \includegraphics{figure_18c} The total area of the four rectangles is \(A\) The student decides to improve their approximation by increasing the number of rectangles used. Explain why the value of the student's improved approximation will be greater than \(A\), but less than \(\frac{1322}{15}\) [2 marks]
AQA Paper 1 2024 June Q19
7 marks Challenging +1.2
A curve has equation $$y^3e^{2x} + 2y - 16x = k$$ where \(k\) is a constant. The curve has a stationary point on the \(y\)-axis. Determine the value of \(k\) [7 marks]
AQA Paper 1 2024 June Q20
10 marks Standard +0.3
A gardener stores rainwater in a cylindrical container. The container has a height of 130 centimetres. The gardener empties the water from the container through a hose. The hose is attached 5 centimetres from the bottom of the container. At time \(t\) minutes after the hose is switched on, the depth of water, \(h\) centimetres, in the container decreases at a rate which is proportional to \(h - 5\) Initially the container of water is full, and the depth of water is decreasing at a rate of 1.5 centimetres per minute.
  1. Show that $$\frac{dh}{dt} = -0.012(h - 5)$$ [3 marks]
  2. Solve the differential equation $$\frac{dh}{dt} = -0.012(h - 5)$$ to find an expression for \(h\) in terms of \(t\) [5 marks]
  3. Find the time taken for the container to be half empty. Give your answer to the nearest minute. [2 marks]
AQA Paper 1 Specimen Q1
1 marks Easy -2.0
Find the gradient of the line with equation \(2x + 5y = 7\) Circle your answer. [1 mark] \(\frac{2}{5}\) \quad \(\frac{5}{2}\) \quad \(-\frac{2}{5}\) \quad \(-\frac{5}{2}\)
AQA Paper 1 Specimen Q2
1 marks Easy -1.8
A curve has equation \(y = \frac{2}{\sqrt{x}}\) Find \(\frac{dy}{dx}\) Circle your answer. [1 mark] \(\frac{\sqrt{x}}{3}\) \quad \(\frac{1}{x\sqrt{x}}\) \quad \(-\frac{1}{x\sqrt{x}}\) \quad \(-\frac{1}{2x\sqrt{x}}\)
AQA Paper 1 Specimen Q3
3 marks Standard +0.3
When \(\theta\) is small, find an approximation for \(\cos 3\theta + \theta \sin 2\theta\), giving your answer in the form \(a + b\theta^2\) [3 marks]
AQA Paper 1 Specimen Q4
6 marks Moderate -0.3
\(p(x) = 2x^3 + 7x^2 + 2x - 3\)
  1. Use the factor theorem to prove that \(x + 3\) is a factor of \(p(x)\) [2 marks]
  2. Simplify the expression \(\frac{2x^3 + 7x^2 + 2x - 3}{4x^2 - 1}\), \(x \neq \pm \frac{1}{2}\) [4 marks]
AQA Paper 1 Specimen Q5
8 marks Moderate -0.3
The diagram shows a sector \(AOB\) of a circle with centre \(O\) and radius \(r\) cm. \includegraphics{figure_5} The angle \(AOB\) is \(\theta\) radians The sector has area 9 cm\(^2\) and perimeter 15 cm.
  1. Show that \(r\) satisfies the equation \(2r^2 - 15r + 18 = 0\) [4 marks]
  2. Find the value of \(\theta\). Explain why it is the only possible value. [4 marks]
AQA Paper 1 Specimen Q6
4 marks Standard +0.3
Sam goes on a diet. He assumes that his mass, \(m\) kg after \(t\) days, decreases at a rate that is inversely proportional to the cube root of his mass.
  1. Construct a differential equation involving \(m\), \(t\) and a positive constant \(k\) to model this situation. [3 marks]
  2. Explain why Sam's assumption may not be appropriate. [1 mark]
AQA Paper 1 Specimen Q7
4 marks Moderate -0.3
Find the values of \(k\) for which the equation \((2k - 3)x^2 - kx + (k - 1) = 0\) has equal roots. [4 marks]
AQA Paper 1 Specimen Q8
7 marks Challenging +1.2
  1. Given that \(u = 2^x\), write down an expression for \(\frac{du}{dx}\) [1 mark]
  2. Find the exact value of \(\int_0^1 2^x \sqrt{3 + 2^x}\) dx Fully justify your answer. [6 marks]
AQA Paper 1 Specimen Q9
8 marks Standard +0.3
A curve has equation \(y = \frac{2x + 3}{4x^2 + 7}\)
    1. Find \(\frac{dy}{dx}\) [2 marks]
    2. Hence show that \(y\) is increasing when \(4x^2 + 12x - 7 < 0\) [4 marks]
  2. Find the values of \(x\) for which \(y\) is increasing. [2 marks]
AQA Paper 1 Specimen Q10
10 marks Standard +0.3
The function f is defined by $$f(x) = 4 + 3^{-x}, \quad x \in \mathbb{R}$$
  1. Using set notation, state the range of f [2 marks]
  2. The inverse of f is \(f^{-1}\)
    1. Using set notation, state the domain of \(f^{-1}\) [1 mark]
    2. Find an expression for \(f^{-1}(x)\) [3 marks]
  3. The function g is defined by $$g(x) = 5 - \sqrt{x}, \quad (x \in \mathbb{R} : x > 0)$$
    1. Find an expression for \(gf(x)\) [1 mark]
    2. Solve the equation \(gf(x) = 2\), giving your answer in an exact form. [3 marks]
AQA Paper 1 Specimen Q11
8 marks Moderate -0.3
A circle with centre \(C\) has equation \(x^2 + y^2 + 8x - 12y = 12\)
  1. Find the coordinates of \(C\) and the radius of the circle. [3 marks]
  2. The points \(P\) and \(Q\) lie on the circle. The origin is the midpoint of the chord \(PQ\). Show that \(PQ\) has length \(n\sqrt{3}\), where \(n\) is an integer. [5 marks]
AQA Paper 1 Specimen Q12
8 marks Challenging +1.8
A sculpture formed from a prism is fixed on a horizontal platform, as shown in the diagram. The shape of the cross-section of the sculpture can be modelled by the equation \(x^2 + 2xy + 2y^2 = 10\), where \(x\) and \(y\) are measured in metres. The \(x\) and \(y\) axes are horizontal and vertical respectively. \includegraphics{figure_12} Find the maximum vertical height above the platform of the sculpture. [8 marks]
AQA Paper 1 Specimen Q13
3 marks Moderate -0.8
Prove the identity \(\cot^2 \theta - \cos^2 \theta = \cot^2 \theta \cos^2 \theta\) [3 marks]
AQA Paper 1 Specimen Q14
10 marks Standard +0.3
An open-topped fish tank is to be made for an aquarium. It will have a square horizontal base, rectangular vertical sides and a volume of 60 m\(^3\) The materials cost:
  • £15 per m\(^2\) for the base
  • £8 per m\(^2\) for the sides.
  1. Modelling the sides and base of the fish tank as laminae, use calculus to find the height of the tank for which the overall cost of the materials has its minimum value. Fully justify your answer. [8 marks]
    1. In reality, the thickness of the base and sides of the tank is 2.5 cm Briefly explain how you would refine your modelling to take account of the thickness of the sides and base of the tank. [1 mark]
    2. How would your refinement affect your answer to part (a)? [1 mark]