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AQA FP2 2016 June Q4
6 marks Challenging +1.2
4
  1. Given that \(y = \tan ^ { - 1 } \sqrt { ( 3 x ) }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), giving your answer in terms of \(x\).
  2. Hence, or otherwise, show that \(\int _ { \frac { 1 } { 3 } } ^ { 1 } \frac { 1 } { ( 1 + 3 x ) \sqrt { x } } \mathrm {~d} x = \frac { \sqrt { 3 } \pi } { n }\), where \(n\) is an integer.
    [0pt] [4 marks]
AQA FP2 2016 June Q5
12 marks Standard +0.3
5
  1. Find the modulus of the complex number \(- 4 \sqrt { 3 } + 4 \mathrm { i }\), giving your answer as an integer.
  2. The locus of points, \(L\), satisfies the equation \(| z + 4 \sqrt { 3 } - 4 \mathrm { i } | = 4\).
    1. Sketch the locus \(L\) on the Argand diagram below.
    2. The complex number \(w\) lies on \(L\) so that \(- \pi < \arg w \leqslant \pi\). Find the least possible value of arg \(w\), giving your answer in terms of \(\pi\).
  3. Solve the equation \(z ^ { 3 } = - 4 \sqrt { 3 } + 4 \mathrm { i }\), giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
    [0pt] [5 marks]
AQA FP2 2016 June Q6
14 marks Standard +0.3
6
  1. Given that \(y = \sinh x\), use the definition of \(\sinh x\) in terms of \(\mathrm { e } ^ { x }\) and \(\mathrm { e } ^ { - x }\) to show that \(x = \ln \left( y + \sqrt { y ^ { 2 } + 1 } \right)\).
  2. A curve has equation \(y = 6 \cosh ^ { 2 } x + 5 \sinh x\).
    1. Show that the curve has a single stationary point and find its \(x\)-coordinate, giving your answer in the form \(\ln p\), where \(p\) is a rational number.
    2. The curve lies entirely above the \(x\)-axis. The region bounded by the curve, the coordinate axes and the line \(x = \cosh ^ { - 1 } 2\) has area \(A\). Show that $$A = a \cosh ^ { - 1 } 2 + b \sqrt { 3 } + c$$ where \(a\), \(b\) and \(c\) are integers.
      [0pt] [5 marks] \(7 \quad\) Given that \(p \geqslant - 1\), prove by induction that, for all integers \(n \geqslant 1\), $$( 1 + p ) ^ { n } \geqslant 1 + n p$$ [6 marks]
AQA FP2 2016 June Q8
13 marks Challenging +1.2
8
  1. By applying de Moivre's theorem to \(( \cos \theta + i \sin \theta ) ^ { 4 }\), where \(\cos \theta \neq 0\), show that $$( 1 + i \tan \theta ) ^ { 4 } + ( 1 - i \tan \theta ) ^ { 4 } = \frac { 2 \cos 4 \theta } { \cos ^ { 4 } \theta }$$
  2. Hence show that \(z = \mathrm { i } \tan \frac { \pi } { 8 }\) satisfies the equation \(( 1 + z ) ^ { 4 } + ( 1 - z ) ^ { 4 } = 0\), and express the three other roots of this equation in the form \(\mathrm { i } \tan \phi\), where \(0 < \phi < \pi\).
  3. Use the results from part (b) to find the values of:
    1. \(\tan ^ { 2 } \frac { \pi } { 8 } \tan ^ { 2 } \frac { 3 \pi } { 8 }\);
    2. \(\tan ^ { 2 } \frac { \pi } { 8 } + \tan ^ { 2 } \frac { 3 \pi } { 8 }\).
      \includegraphics[max width=\textwidth, alt={}]{a629b09d-3633-4dbd-83db-7eb89577438c-23_2488_1709_219_153}
      \section*{DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED}
AQA Further Paper 3 Statistics Specimen Q2
1 marks Easy -1.2
2 The continuous random variable \(Y\) has cumulative distribution function defined by $$\mathrm { F } ( y ) = \left\{ \begin{array} { c c } 0 & y < 0 \\ \frac { y ^ { 2 } } { 36 } & 0 \leq y \leq 6 \\ 1 & y > 6 \end{array} \right.$$ Find the value of \(\mathrm { P } ( Y > 4 )\) Circle your answer. \(\frac { 4 } { 9 }\) \(\frac { 5 } { 9 }\) \(\frac { 16 } { 27 }\) \(\frac { 11 } { 27 }\)
AQA Further Paper 3 Statistics Specimen Q3
4 marks Moderate -0.5
3 The continuous random variable \(R\) follows a rectangular distribution with probability density function given by $$f ( r ) = \begin{cases} k & - a \leq r \leq b \\ 0 & \text { otherwise } \end{cases}$$ Prove, using integration, that \(\mathrm { E } ( R ) = \frac { 1 } { 2 } ( b - a )\) [0pt] [4 marks]
AQA Further Paper 3 Statistics Specimen Q4
6 marks Standard +0.3
4 David, a zoologist, is investigating a particular species of monitor lizard. He measures the lengths, in centimetres, of a random sample of this particular species of lizard. His measured lengths are $$\begin{array} { l l l l l l l l l l } 53.2 & 57.8 & 55.3 & 58.9 & 59.0 & 60.2 & 61.8 & 62.3 & 65.4 & 66.5 \end{array}$$ The lengths may be assumed to be normally distributed.
David correctly constructed a 90\% confidence interval for the mean length of lizard using the measured lengths given and the formula \(\bar { x } \pm \left( b \times \frac { s } { \sqrt { n } } \right)\) This interval had limits of 57.63 and 62.45, correct to two decimal places.
4
  1. State the value for \(b\) used in David's formula. 4
  2. David interprets his interval and states,
    "My confidence interval indicates that exactly 90\% of the population of lizard lengths for this particular species lies between 57.63 cm and \(62.45 \mathrm {~cm} ^ { \prime \prime }\). Do you think David's statement is true? Explain your reasoning. 4
  3. David's assistant, Amina, correctly constructs a \(\beta \%\) confidence interval from David's random sample of measured lengths. Amina informs David that the width of her confidence interval is 8.54 .
    Find the value of \(\beta\).
    [0pt] [3 marks]
    Turn over for the next question
AQA Further Paper 3 Statistics Specimen Q5
8 marks Standard +0.3
5 Students at a science department of a university are offered the opportunity to study an optional language module, either German or Mandarin, during their second year of study. From a sample of 50 students who opted to study a language module, 31 were female. Of those who opted to study Mandarin, 8 were female and 12 were male. Test, using the \(5 \%\) level of significance, whether choice of language is independent of gender. The sample of students may be regarded as random.
[0pt] [8 marks] Turn over for the next question
AQA Further Paper 3 Statistics Specimen Q6
9 marks Challenging +1.8
6 The random variable \(T\) can take the value \(T = - 2\) or any value in the range \(0 \leq T < 12\) The distribution of \(T\) is given by \(\mathrm { P } ( T = - 2 ) = c , \mathrm { P } ( 0 \leq T \leq t ) = 225 k - k ( 15 - t ) ^ { 2 }\) 6
    1. Show that \(1 - c = 216 k\) [0pt] [3 marks] 6
  2. (ii) Given that \(c = 0.1\), find the value of \(\mathrm { E } ( T )\) [0pt] [3 marks]
    6
  3. Show that \(\mathrm { E } ( \sqrt { | T | } ) = \frac { 5 \sqrt { 2 } + 52 \sqrt { 3 } } { 50 }\) [0pt] [3 marks]
AQA Further Paper 3 Statistics Specimen Q7
10 marks Standard +0.3
7 Petroxide Industries produces a chemical used in the production of mobile phone covers for a mobile phone company. The chemical becomes less effective when the mean level of impurity is greater than 3 per cent.
Sunita is the Quality Control manager at Petroxide Industries. After a complaint from the mobile phone company, Sunita obtains a random sample of this chemical from 9 batches. She measures the level of impurity, \(X\) per cent, in each sample.
The summarised results are as follows. $$\sum x = 28.8 \quad \sum ( x - \bar { x } ) ^ { 2 } = 0.6$$ 7
    1. Investigate using the \(5 \%\) level of significance whether the mean level of impurity in the chemical is greater than 3 per cent.
      [0pt] [7 marks]
      7
  2. (ii) State the assumption that it was necessary for you to make in order for the test in part (a)(i) to be valid.
    7
  3. State the changes that would be required to your test in part (a) if you were told that the standard deviation of the level of impurity is known to be 0.25 per cent.
    [0pt] [2 marks]
    Turn over for the next question
AQA Further Paper 3 Statistics Specimen Q8
11 marks Standard +0.3
8 The time in hours to failure of a component may be modelled by an exponential distribution with parameter \(\lambda = 0.025\) In a manufacturing process, the machine involved uses one of these components continuously until it fails. The component is then immediately replaced.
8
  1. Write down the mean time to failure for a component. 8
  2. Find the probability that a component will fail during a 12-hour shift. 8
  3. A component has not failed for 30 hours. Find the probability that this component lasts for at least another 30 hours.
    [0pt] [2 marks] 8
  4. Find the probability that a component does not fail during 4 consecutive 12-hour shifts.
    [0pt] [3 marks]
    8
    1. State the distribution that can be used to model the number of components that fail during one hour of the manufacturing process.
      [0pt] [2 marks]
      8
  6. (ii) Hence, or otherwise, find the probability that no components fail during 5 consecutive 12-hour shifts.
    [0pt] [2 marks]
AQA Further Paper 3 Mechanics 2024 June Q2
1 marks Easy -1.3
2 As a particle moves along a straight horizontal line, it is subjected to a force \(F\) newtons that acts in the direction of motion of the particle. At time \(t\) seconds, \(F = \frac { t } { 5 }\) Calculate the magnitude of the impulse on the particle between \(t = 0\) and \(t = 3\) Circle your answer.
[0pt] [1 mark] \(0.3 \mathrm {~N} \mathrm {~s} \quad 0.6 \mathrm {~N} \mathrm {~s} \quad 0.9 \mathrm {~N} \mathrm {~s} \quad 1.8 \mathrm {~N} \mathrm {~s}\) A conical pendulum consists of a light string and a particle of mass \(m \mathrm {~kg}\) The conical pendulum completes horizontal circles with radius \(r\) metres and angular speed \(\omega\) radians per second. The string makes an angle \(\theta\) with the downward vertical. The tension in the string is \(T\) newtons. The conical pendulum and the forces acting on the particle are shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{86817115-46a1-4702-8a33-8f9b05d69bb9-03_616_593_689_703} Which one of the following statements is correct?
Tick ( ✓ ) one box. \(T \cos \theta = m r \omega ^ { 2 }\) \includegraphics[max width=\textwidth, alt={}, center]{86817115-46a1-4702-8a33-8f9b05d69bb9-03_108_108_1567_900} \(T \sin \theta = m r \omega ^ { 2 }\) \includegraphics[max width=\textwidth, alt={}, center]{86817115-46a1-4702-8a33-8f9b05d69bb9-03_109_108_1726_900} \(T \cos \theta = \frac { m \omega ^ { 2 } } { r }\) \includegraphics[max width=\textwidth, alt={}, center]{86817115-46a1-4702-8a33-8f9b05d69bb9-03_109_108_1886_900} \(T \sin \theta = \frac { m \omega ^ { 2 } } { r }\) □
AQA Further Paper 3 Mechanics 2024 June Q4
5 marks Easy -1.2
4 A particle of mass 3 kg is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point on a smooth horizontal surface.
The particle is set into motion so that it moves with a constant speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a circular path with radius 0.8 metres on the horizontal surface. 4
  1. Find the acceleration of the particle.
    4
  2. Find the tension in the string.
    4
  3. Show that the angular speed of the particle is 48 revolutions per minute correct to two significant figures.
AQA Further Paper 3 Mechanics 2024 June Q5
4 marks Moderate -0.5
5 When a sphere of radius \(r\) metres is falling at \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) it experiences an air resistance force \(F\) newtons. The force is to be modelled as $$F = k r ^ { \alpha } { } _ { V } { } ^ { \beta }$$ where \(k\) is a constant with units \(\mathrm { kg } \mathrm { m } ^ { - 2 }\) 5
  1. State the dimensions of \(F\) 5
  2. Use dimensional analysis to find the value of \(\alpha\) and the value of \(\beta\)
AQA Further Paper 3 Mechanics 2024 June Q6
10 marks Standard +0.3
6 In this question use \(\boldsymbol { g } = 9.8 \mathbf { m ~ s } ^ { - 2 }\) A light elastic string has natural length 3 metres and modulus of elasticity 18 newtons.
One end of the elastic string is attached to a particle of mass 0.25 kg
The other end of the elastic string is attached to a fixed point \(O\) The particle is released from rest at a point \(A\), which is 4.5 metres vertically below \(O\) 6
  1. Calculate the elastic potential energy of the string when the particle is at \(A\) 6
  2. The point \(B\) is 3 metres vertically below \(O\) Calculate the gravitational potential energy gained by the particle as it moves from \(A\) to \(B\) 6
  3. Find the speed of the particle at \(B\) 6
  4. The point \(C\) is 3.6 metres vertically below \(O\) Explain, showing any calculations that you make, why the speed of the particle is increasing the first time that the particle is at \(C\)
AQA Further Paper 3 Mechanics 2024 June Q7
10 marks Standard +0.3
7 A sphere, of mass 0.2 kg , moving on a smooth horizontal surface, collides with a fixed wall. Before the collision the sphere moves with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(60 ^ { \circ }\) to the wall. After the collision the sphere moves with speed \(\nu \mathrm { m } \mathrm { s } ^ { - 1 }\) at an angle of \(\theta ^ { \circ }\) to the wall. The velocities are shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{86817115-46a1-4702-8a33-8f9b05d69bb9-08_303_762_735_625} The coefficient of restitution between the wall and the sphere is 0.7 7
  1. Assume that the wall is smooth. 7
    1. Find the value of \(v\) Give your answer to two significant figures.
      7
  3. (ii) Find the value of \(\theta\) Give your answer to the nearest whole number.
    7
  4. (iii) Find the magnitude of the impulse exerted on the sphere by the wall.
    Give your answer to two significant figures.
    7
  5. In reality the wall is not smooth.
    Explain how this would cause a change in the magnitude of the impulse calculated in part (a)(iii).
AQA Further Paper 3 Mechanics 2024 June Q8
10 marks Challenging +1.2
8 The finite region enclosed by the line \(y = k x\), the \(x\)-axis and the line \(x = 5\) is rotated through \(360 ^ { \circ }\) around the \(x\) axis to form a solid cone. 8
    1. Use integration to show that the position of the centre of mass of the cone is independent of \(k\) 8
  2. (ii) State the distance between the base of the cone and its centre of mass.
    8
  3. State one assumption that you have made about the cone. 8
  4. The plane face of the cone is placed on a rough inclined plane.
    The coefficient of friction between the cone and the plane is 0.8
    The angle between the plane and the horizontal is gradually increased from \(0 ^ { \circ }\) Find the range of values of \(k\) for which the cone slides before it topples.
    [0pt] [4 marks]
AQA Further Paper 3 Mechanics 2024 June Q9
8 marks Challenging +1.2
9 A small sphere, of mass \(m\), is attached to one end of a light inextensible string of length \(a\) The other end of the string is attached to a fixed point \(O\) The sphere is at rest in equilibrium directly below \(O\) when it is struck, giving it a horizontal impulse of magnitude \(m U\) After the impulse, the sphere follows a circular path in a vertical plane containing the point \(O\) until the string becomes slack at the point \(C\) At \(C\) the string makes an angle of \(30 ^ { \circ }\) with the upward vertical through \(O\), as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{86817115-46a1-4702-8a33-8f9b05d69bb9-12_583_331_875_901} 9
  1. Show that $$U ^ { 2 } = \frac { a g } { 2 } ( 4 + 3 \sqrt { 3 } )$$ where \(g\) is the acceleration due to gravity.
    9
  2. With reference to any modelling assumptions that you have made, explain why giving your answer as an inequality would be more appropriate, and state this inequality. \includegraphics[max width=\textwidth, alt={}, center]{86817115-46a1-4702-8a33-8f9b05d69bb9-14_2491_1755_173_123} \begin{center} \begin{tabular}{|l|l|} \hline Question number & Additional page, if required. Write the question numbers in the left-hand margin.
    \hline & \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \hline & \begin{tabular}{l}
AQA Further Paper 3 Mechanics Specimen Q1
1 marks Easy -1.8
1 A ball of mass 0.2 kg is travelling horizontally at \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it hits a vertical wall.
It rebounds horizontally at \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Find the magnitude of the impulse exerted on the ball by the wall.
Circle your answer.
[0pt] [1 mark]
0.4 N s
1.4 N s
AQA Further Paper 3 Mechanics Specimen Q2
1 marks Easy -1.2
2 Ns
2.4 N s 2 In this question
\(a\)represents acceleration,
\(T\)represents time,
\(l\)represents length,
\(m\)represents mass,
\(v\)represents velocity,
\(F\)represents force.
One of these formulae is dimensionally consistent.
Circle your answer.
[0pt] [1 mark] $$T = 2 \pi \sqrt { \frac { a } { l } } \quad v ^ { 2 } = \frac { 2 a l } { T } \quad F l = m v ^ { 2 } \quad F T = m \sqrt { a }$$ Turn over for the next question
AQA Further Paper 3 Mechanics Specimen Q3
6 marks Standard +0.3
3 In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
A composite body consists of a uniform rod, \(A B\), and a particle.
The rod has length 4 metres and mass 22.5 kilograms.
The particle, \(P\), has mass 20 kilograms and is placed on the rod at a distance of 0.3 metres from \(B\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{4fdb2637-6368-422c-99da-85b80efe31c5-04_163_1323_767_402} 3
  1. Find the distance of the centre of mass of the body from \(A\). 3
  2. The body rests in equilibrium in a horizontal position on two supports, \(C\) and \(D\).
    The support \(C\) is 0.5 metres from \(A\) and the support \(D\) is 1 metre from \(B\). Find the magnitudes of the forces exerted on the body by the supports.
    [0pt] [4 marks]
AQA Further Paper 3 Mechanics Specimen Q4
6 marks Moderate -0.3
4 Two discs, \(A\) and \(B\), have equal radii and masses 0.8 kg and 0.4 kg respectively. The discs are placed on a horizontal surface. The discs are set in motion when they are 3 metres apart, so that they move directly towards each other, each travelling at a speed of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The discs collide directly with each other. After the collision \(A\) moves in the opposite direction with a speed of \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) The coefficient of restitution between the two discs is \(e\). 4
  1. Assuming that the surface is smooth, show that \(e = 0.8\) 4
  2. Describe one way in which the model you have used could be refined. Turn over for the next question
AQA Further Paper 3 Mechanics Specimen Q5
6 marks Moderate -0.5
5 In this question use \(\boldsymbol { g } = 9.8 \mathbf { m ~ s } ^ { \mathbf { - 2 } }\).
A conical pendulum consists of a string of length 60 cm and a particle of mass 400 g . The string is at an angle of \(30 ^ { \circ }\) to the vertical, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{4fdb2637-6368-422c-99da-85b80efe31c5-08_501_606_644_854} 5
  1. Show that the tension in the string is 4.5 N . 5
  2. Find the angular speed of the particle.
    [0pt] [3 marks]
    5
  3. State two assumptions that you have made about the string.
AQA Further Paper 3 Mechanics Specimen Q6
7 marks Challenging +1.8
6 A uniform solid is formed by rotating the region enclosed by the positive \(x\)-axis, the line \(x = 2\) and the curve \(y = \frac { 1 } { 2 } x ^ { 2 }\) through \(360 ^ { \circ }\) around the \(x\)-axis. 6
  1. Find the centre of mass of this solid.
    6
  2. The solid is placed with its plane face on a rough inclined plane and does not slide. The angle between the inclined plane and the horizontal is gradually increased. When the angle between the inclined plane and the horizontal is \(\alpha\), the solid is on the point of toppling. Find \(\alpha\), giving your answer to the nearest \(0.1 ^ { \circ }\)
AQA Further Paper 3 Mechanics Specimen Q7
5 marks Standard +0.3
7 In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
When a car, of mass 1200 kg , travels at a speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) it experiences a total resistive force which can be modelled as being of magnitude \(36 v\) newtons.
The maximum power of the car is 90 kilowatts.
The car starts to descend a hill, inclined at \(5.2 ^ { \circ }\) to the horizontal, along a straight road.
Find the maximum speed of the car down this hill.
[0pt] [5 marks]