Questions Paper 2 (402 questions)

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AQA Paper 2 Specimen Q16
11 marks
16 In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
The diagram shows a box, of mass 8.0 kg , being pulled by a string so that the box moves at a constant speed along a rough horizontal wooden board. The string is at an angle of \(40 ^ { \circ }\) to the horizontal.
The tension in the string is 50 newtons.
\includegraphics[max width=\textwidth, alt={}, center]{a57b0526-cf9c-44d6-a349-cac392f85a70-26_334_862_884_575} The coefficient of friction between the box and the board is \(\mu\)
Model the box as a particle.
16
  1. Show that \(\mu = 0.83\)
    [0pt] [4 marks] Question 16 continues on the next page 16
  2. One end of the board is lifted up so that the board is now inclined at an angle of \(5 ^ { \circ }\) to the horizontal. The box is pulled up the inclined board.
    The string remains at an angle of \(40 ^ { \circ }\) to the board.
    The tension in the string is increased so that the box accelerates up the board at \(3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
    \includegraphics[max width=\textwidth, alt={}, center]{a57b0526-cf9c-44d6-a349-cac392f85a70-28_385_858_778_577} 16
    1. Draw a diagram to show the forces acting on the box as it moves. 16
  3. (ii) Find the tension in the string as the box accelerates up the slope at \(3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
    [0pt] [7 marks]
AQA Paper 2 Specimen Q20
8 marks
20
40
200 3 A curve is defined by the parametric equations $$x = t ^ { 3 } + 2 , \quad y = t ^ { 2 } - 1$$ 3
  1. Find the gradient of the curve at the point where \(t = - 2\)
    [0pt] [4 marks]
    3
  2. Find a Cartesian equation of the curve.
    4 The equation \(x ^ { 3 } - 3 x + 1 = 0\) has three real roots. 4
  3. Show that one of the roots lies between - 2 and - 1
    4
  4. Taking \(x _ { 1 } = - 2\) as the first approximation to one of the roots, use the Newton-Raphson method to find \(x _ { 2 }\), the second approximation.
    [0pt] [3 marks]
    4
  5. Explain why the Newton-Raphson method fails in the case when the first approximation is \(x _ { 1 } = - 1\)
    [0pt] [1 mark]