Questions Paper 1 (379 questions)

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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]
AQA Paper 1 Specimen Q15
8 marks Standard +0.3
The height \(x\) metres, of a column of water in a fountain display satisfies the differential equation \(\frac{dx}{dt} = \frac{8\sin 2t}{3\sqrt{x}}\), where \(t\) is the time in seconds after the display begins.
  1. Solve the differential equation, given that initially the column of water has zero height. Express your answer in the form \(x = f(t)\) [7 marks]
  2. Find the maximum height of the column of water, giving your answer to the nearest cm. [1 mark]
AQA Paper 1 Specimen Q16
5 marks Standard +0.8
A student argues that when a rational number is multiplied by an irrational number the result will always be an irrational number.
  1. Identify the rational number for which the student's argument is not true. [1 mark]
  2. Prove that the student is right for all rational numbers other than the one you have identified in part (a). [4 marks]
AQA Paper 1 Specimen Q17
6 marks Challenging +1.2
\(f(x) = \sin x\) Using differentiation from first principles find the exact value of \(f'\left(\frac{\pi}{6}\right)\) Fully justify your answer. [6 marks]