Embed Size (px)
Citation preview
Page 1 of 29
12.3 Turning points in Physics – Special relativity – Qs
Q1. (a) The theory of special relativity is based on two postulates. One of these postulates
is that the speed of light in free space is invariant. State the other postulate.
___________________________________________________________________
___________________________________________________________________ (1)
(b) An electron in the Stanford linear accelerator is accelerated to an energy of 24.0 GeV.
(i) An electron travelling with this energy has a velocity v.
Show that the value of is about 2.1 × 10−5.
(3)
(ii) The Stanford linear accelerator has a length of 3.0 km. Assume that the electron travels for the full length of the accelerator with an energy of 24 GeV. Calculate the length, in m, of the accelerator in the reference frame of the electron.
length of accelerator = ____________________ m (1)
(c) Draw a graph to show how the relativistic mass of an electron varies with speed as it is accelerated from rest.
Rest mass of an electron = m0
Page 2 of 29
(2)
(Total 7 marks)
Q2. Cosmic rays mostly consist of high-energy protons. These protons can collide with atomic nuclei in the Earth’s upper atmosphere producing pions (π−). Pions are unstable and decay into high-energy muons (μ−).
(a) (i) Which of the following is the particle group for pions (π−)?
Tick (✔) the correct answer.
Baryons
Leptons
Mesons
Photons
(1)
(ii) Complete the equation for the decay of a pion (π−).
π− ⟶ μ− + ____________________ (1)
Page 3 of 29
(b) 2.5 × 108 muons are created simultaneously above the EarthÙs surface. These muons are unstable and have a half-life of 2.2 μs. They are created at a height of 10.7 km and travel towards the Earth’s surface with a constant vertical velocity of 2.85 × 108 m s−1.
(i) Show that, for the reference frame of an observer on Earth, the time taken for the muons to reach the Earth’s surface is approximately 17 muon half-lives.
(2)
(ii) Estimate the number of these muons that an observer on Earth would expect to remain after 17 half-lives.
number ____________________ (2)
(iii) The number of muons that reach the Earth’s surface is considerably different from the estimated number in part (b)(ii).
Identify the theory that explains the difference between the estimated and observed number of muons.
______________________________________________________________ (1)
(iv) Outline why the number of muons that actually reach the Earth’s surface is different from the estimated number in part (b)(ii).
______________________________________________________________
______________________________________________________________
______________________________________________________________ (1)
(v) Calculate, for the reference frame of a muon, the time taken for the muons to travel this distance.
time ____________________ s (3)
(vi) Calculate the number of muons that remain at the end of the time interval calculated in part (b)(v).
number ____________________ (3)
(Total 14 marks)
Q3. Cosmic rays detected on a spacecraft are protons with a total energy of 3.7 × 109 eV.
Calculate the velocity of the protons as a fraction of the speed of light.
Page 4 of 29
proton velocity = ____________________ c (Total 3 marks)
Q4. A muon is an unstable particle produced by cosmic rays in the Earth’s atmosphere. Muons that are produced at a height of 10.7 km above the Earth’s surface, travel at a speed of 0.996c toward Earth, where c is the speed of light. In the frame of reference of the muons, the muons have a half-life of 1.60 × 10–6 s.
(a) (i) Calculate how many muons will reach the Earth’s surface for every 1000 that are produced at a height of 10.7 km.
number of muons ____________________ (3)
(ii) Which of the following statements is correct? Tick (✓) the correct answer.
✓if correct
For an observer in a laboratory on Earth, the distance travelled by a muon that reaches the Earth is greater than the distance travelled by a muon in its frame of reference
For an observer in a laboratory on Earth, time passes more slowly than it does for a muon in its frame of reference
For an observer in a laboratory on Earth, the probability of a muon decaying each second is lower than it is for a muon in its frame of reference
(1)
(b) (i) Show that the total energy of an electron that has been accelerated to a speed of 0.98c is about 4 × 10–13 J.
Page 5 of 29
(2)
(ii) The total energy of an electron travelling at a speed of 0.97c is 3.37 × 10–13 J. Calculate the potential difference required to accelerate an electron from a speed of 0.97c to a speed of 0.98c.
potential difference = ____________________ V (1)
(Total 7 marks)
Q5. (a) One of the two postulates of Einstein’s theory of special relativity is that the speed
of light in free space is invariant.
(i) Explain what is meant by this statement.
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
(ii) What is the other postulate?
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________ (3)
(b) K+ mesons are sub-atomic particles of half-life 86 ns when at rest. In an accelerator experiment, a beam of K+ mesons travelling at a speed of 0.95c is created, where c is the speed of light.
Page 6 of 29
(i) Calculate the half-life of the K+ mesons in the beam measured in the laboratory frame of reference.
______________________________________________________________
______________________________________________________________
______________________________________________________________
(ii) What is the greatest distance that a detector could be sited from the point of production of the K+ mesons to detect at least 25% of the K+ mesons produced?
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________ (6)
(Total 9 marks)
Q6. The diagram represents the Michelson-Morley interferometer.
Page 7 of 29
(a) (i) Name the object labelled A.
______________________________________________________________
(ii) Name the object labelled B and explain its purpose.
______________________________________________________________
______________________________________________________________
______________________________________________________________ (3)
(b) Describe and explain what is observed through the viewing telescope
(i) when distances l1 and l2 are equal.
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
(ii) as distance l1 is made slightly longer than distance l2.
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
Page 8 of 29
(4)
(c) Michelson and Morley used the interferometer to try to detect the motion of the Earth through the hypothetical ether.
(i) Outline how the apparatus was used and state what the result was.
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
(ii) Explain the significance of the result.
______________________________________________________________
______________________________________________________________
______________________________________________________________ (3)
(Total 10 marks)
Q7. (a) One of the two postulates of Einstein’s theory of special relativity is that the speed
of light in free space, c, is invariant.
Explain what is meant by this statement.
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________ (1)
(b) A beam of identical particles moving at a speed of 0.98c is directed along a straight line between two detectors 25 m apart.
Page 9 of 29
The particles are unstable and the intensity of the beam at the second detector is a quarter of the intensity at the first detector.
Calculate the half-life of the particles in their rest frame.
answer = ______________________ s (4)
(Total 5 marks)
Q8. (a) One of the two postulates of Einstein’s theory of special relativity is that physical
laws have the same form in all inertial frames of reference.
Explain, with the aid of a suitable example, what is meant by an inertial frame of reference.
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________ (2)
(b) A certain type of sub-atomic particle has a half-life of 18 ns when at rest. A beam of these particles travelling at a speed of 0.995c is produced in an accelerator.
(i) Calculate the half-life of these particles in the laboratory frame of reference.
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
(ii) Calculate the time taken by these particles to travel a distance of 108 m in the
Page 10 of 29
laboratory at a speed of 0.995c and hence show that the intensity of the beam is reduced to 25% of its original value over this distance.
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________ (5)
(Total 7 marks)
Q9. (a) In a science fiction film, a space rocket travels away from the Earth at a speed of
0.994 c, where c is the speed of light in free space. A radio message of duration 800 s is transmitted by the space rocket.
(i) Calculate the duration of the message when it is received at the Earth.
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
(ii) Calculate the distance moved by the rocket in the Earth’s frame of reference in the time taken to send the message.
______________________________________________________________
______________________________________________________________
______________________________________________________________ (4)
(b) A student claims that a twin who travels at a speed close to the speed of light from Earth to a distant star and back would, on return to Earth, be a different age to the twin who stayed on Earth. Discuss whether or not this claim is correct.
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
Page 11 of 29
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________ (3)
(Total 7 marks)
Q10. (a) Calculate the speed at which a matter particle has a mass equal to 10 times its rest
mass.
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________ (3)
(b) Explain why a matter particle can not travel as fast as a photon in free space even though its kinetic energy can be increased without limit.
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________ (3)
(Total 6 marks)
Q11. π mesons, travelling in a straight line at a speed of 0.95 c, pass two detectors 34 m apart, as shown in the figure below.
Page 12 of 29
(i) Calculate the time taken, in the frame of reference of the detectors, for a π meson to travel between the two detectors.
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
(ii) π mesons are unstable and decay with a half-life of 18 ns when at rest. Show that approximately 75% of the π mesons passing the first detector decay before they reach the second detector.
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________ (Total 5 marks)
Q12. (a) Calculate the speed of a particle at which its mass is twice its rest mass.
speed ____________________ m s–1
(2)
(b) Use the axes below to show how the mass m of a particle changes from its rest mass mo as its speed v increases from zero.
Mark and label on the graph the point P where the mass of the particle is twice its rest mass.
Page 13 of 29
(3)
(c) By considering the relationship between the energy of a particle and its mass, explain why the theory of special relativity does not allow a matter particle to travel as fast as light.
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________ (2)
(Total 7 marks)
Q13. In an experiment, a beam of protons moving along a straight line at a constant speed of 1.8 × 108ms–1 took 95 ns to travel between two detectors at a fixed distance d0 apart, as shown in the figure below.
Page 14 of 29
(a) (i) Calculate the distance d0 between the two detectors in the frame of reference of the detectors.
answer = ______________________ m (1)
(ii) Calculate the distance between the two detectors in the frame of reference of the protons.
answer = ______________________ m (2)
(b) A proton is moving at a speed of 1.8 × 108ms–1
Calculate the ratio
Page 15 of 29
answer = ______________________ (5)
(Total 8 marks)
Q14. The figure below represents the Michelson-Morley interferometer. Interference fringes are seen by an observer looking through the viewing telescope.
(a) Explain why the interference fringes shift their position if the distance from either of the two mirrors to the semi-silvered block is changed.
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________ (2)
(b) Michelson and Morley predicted that the interference fringes would shift when the apparatus was rotated through 90°. When they tested their prediction, no such fringe shift was observed.
Page 16 of 29
(i) Why was it predicted that a shift of the fringes would be observed?
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________ (3)
(ii) What conclusion was drawn from the observation that the fringes did not shift?
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________ (1)
(Total 6 marks)
Q15. (a) A student models a spacecraft journey that takes one year. The spacecraft travels
directly away from an observer at a speed of 1.2 × 107 m s–1. The student predicts that a clock stationary relative to the observer will record a time several days longer than an identical clock on the spacecraft.
Comment on the student’s prediction. Support your answer with a time dilation calculation.
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
Page 17 of 29
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________ (4)
(b) In practice, the gravitational field of the Sun affects the motion of the spacecraft and it does not travel directly away from the Earth throughout the journey.
Explain why this means that the theory of special relativity cannot be applied to the journey.
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________ (2)
(Total 6 marks)
Q16. (a) One of the two postulates of Einstein’s theory of special relativity is that physical
laws have the same form in all inertial frames of reference. Explain in terms of velocity what is meant by an inertial frame of reference.
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________ (1)
(b) Light takes 4.3 years to reach the Earth from the star Alpha Centauri.
(i) A space probe is to be sent from the Earth to the star to arrive 5.0 years later, according to an observer on Earth.
Assuming the space probe’s velocity is constant, calculate its speed in ms–1 on this journey.
speed ____________________ ms–1
(1)
(ii) Calculate the time taken for this journey in years registered by a clock in the space probe.
Page 18 of 29
time taken ____________________ years (3)
(Total 5 marks)
Q17. (a) Bertozzi’s experiment was designed to test the relationship between the kinetic
energy of an electron and its speed as predicted by the theory of special relativity.
Describe Bertozzi’s experiment.
Your answer should include: • a diagram of the experimental arrangement • details of how the kinetic energy and the speed were measured.
(6)
(b) A scientist conducts an experiment similar to Bertozzi’s experiment and reports that when the electron speed is 2.93 × 108 m s−1 the measured kinetic energy is 2.4 MeV.
Determine whether these data are consistent with the result expected using the theory of special relativity.
(4) (Total 10 marks)
Q18. (i) Calculate the kinetic energy, in J, of a proton accelerated in a straight line from rest
through a potential difference of 1.1 × 109 V.
___________________________________________________________________
___________________________________________________________________
(ii) Show that the mass of a proton at this energy is 2.2 m0, where m0 is the proton rest mass.
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
(iii) Hence calculate the speed of a proton of mass 2.2 m0.
___________________________________________________________________
Page 19 of 29
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________ (Total 7 marks)
Q19. (a) In a particle beam experiment, a short pulse of 1 ns duration of particles moving at
constant speed passed directly between 2 detectors at a fixed distance apart of 240 m. The pulse took 0.84 μs to travel from one detector to the other.
(i) Calculate the speed of the particles.
______________________________________________________________
______________________________________________________________
(ii) Calculate the distance between the two detectors in the frame of reference of the particles.
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________ (4)
(b) In a 'thought experiment' about relativity, a student stated that a twin who travelled from the Earth to a distant planet and back at a speed close to the speed of light would be the same age on return as the twin who stayed on Earth. Explain why this statement is not correct.
___________________________________________________________________
Page 20 of 29
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________ (4)
(Total 8 marks)
Q20. A particle passes through a detector and 152 ns later hits a target 45.0 m away from the detector.
(i) Calculate the speed of the particle between the detector and the target.
___________________________________________________________________
___________________________________________________________________
(ii) Calculate the transit time of the particle from the detector to the target, in the frame of reference of the particle.
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________ (Total 4 marks)
Page 21 of 29
Q21. The Michelson-Morley experiment represented in the diagram was designed to find out if the speed of light depended on its direction relative to the Earth’s motion through space. Interference fringes were seen by the observer.
(a) (i) Explain why interference fringes were seen.
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
(ii) The interference fringe pattern did not shift when the apparatus was rotated by 90°. Explain the significance of this null observation.
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________ (5)
(b) Einstein postulated that the speed of light in free space is invariant. Explain what is meant by this postulate.
Page 22 of 29
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________ (2)
(Total 7 marks)
Q22. (a) One of the two postulates of Einstein’s theory of special relativity is that the speed
of light in free space is invariant.
(i) Explain what is meant by this postulate.
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
(ii) State and explain the other postulate.
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________ (4)
(b) A stationary muon has a rest mass of 1.88 × 10–28 kg and a half-life of 2.2 × 10–6 s.
Calculate
(i) the mass of a muon travelling at 0.996 c, where c is the speed of light in a vacuum,
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
Page 23 of 29
(ii) the distance, in a laboratory frame of reference, travelled in one half-life by a muon moving at 0.996 c.
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________ (6)
(Total 10 marks)
Q23. (a) The speed of an object cannot be greater than or equal to the speed of light yet its
kinetic energy can be increased without limit. Explain the apparent contradiction that the speed of an object is limited whereas its kinetic energy is not limited.
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________ (3)
(b) Protons are accelerated from rest through a potential difference of 2.1 × 1010 V.
(i) Show that the kinetic energy of a proton after it has been accelerated from rest through this potential difference is 3.4 × 10–9 J.
______________________________________________________________
______________________________________________________________
(ii) Show that the mass of a proton with the kinetic energy value calculated in part (a) is approximately 23m0, where m0 is its rest mass.
______________________________________________________________
______________________________________________________________
______________________________________________________________
Page 24 of 29
______________________________________________________________
______________________________________________________________ (4)
(c) Calculate the speed of a proton which has a mass equal to 23m0.
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________ (3)
(Total 10 marks)
Q24. (a) Michelson and Morley attempted to detect absolute motion by investigating whether
or not the speed of light in a direction parallel to the Earth’s motion differs from the speed of light perpendicular to the Earth’s motion.
Discuss what resulted from this experiment and what was concluded.
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________ (3)
(b) In a science fiction story, a space rocket left the Earth in 2066 and travelled out of the Solar System at a speed of 0.80c, where c is the speed of light in vacuo, to a star 16 light years from the Earth.
(i) How many years, in the frame of reference of the Earth, did the spacecraft take to reach the star?
______________________________________________________________
______________________________________________________________
(ii) What was the distance, in the frame of reference of the spacecraft, between the Earth and the star?
______________________________________________________________
Page 25 of 29
______________________________________________________________
______________________________________________________________
______________________________________________________________
(iii) A member of the crew was 21 years old on leaving the Earth. How old was this person on arrival at the star?
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________ (6)
(Total 9 marks)
Q25. In a particle beam experiment, a pulsed beam of protons at a speed of 1.00 × 108 m s–1 passed through a stationary detector in a time of 15.0 ns.
(a) Calculate the length of the pulsed beam in
(i) the frame of reference of the detector,
______________________________________________________________
______________________________________________________________
(ii) the frame of reference of the protons.
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________ (3)
(b) (i) Calculate the kinetic energy of each proton in the beam, in J.
______________________________________________________________
Page 26 of 29
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
(ii) The beam consisted of 107 protons. It passed through the detector and was stopped by a stationary target. Calculate the average power which the proton beam delivered to the target during the pulse.
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________ (5)
(Total 8 marks)
Q26. One of the two postulates of Einstein’s theory of special relativity is that the speed of light in free space is invariant.
(a) Explain what is meant by this postulate.
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________ (1)
(b) State the other postulate.
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________ (1)
(c) Two detectors are measured to be 34 m apart by an observer in a stationary frame of reference. A beam of π mesons travel in a straight line at a speed of 0.95 c past
Page 27 of 29
the two detectors, as shown in the figure below.
Calculate the time taken, in the frame of reference of the observer, for a π meson to travel between the two detectors.
time = ____________________ (1)
(d) π mesons are unstable and decay with a half-life of 18 ns. It is found in experiments that approximately 75% of the π mesons that pass the first detector decay before reaching the second detector.
Show how this provides evidence to support the theory of special relativity. In your answer compare the percentage expected by the laboratory observer with and without application of the theory of special relativity.
(5)
(Total 8 marks)
Q27.
Page 28 of 29
The diagram shows the paths of light rays through a simplified version of the apparatus used by Michelson and Morley.
In the apparatus, light waves reflected by the mirrors M1 and M2, meet at P so that they superpose and produce interference fringes. These are observed using the microscope.
Michelson and Morley predicted that the fringes would shift when the apparatus was rotated through 90°. They thought that this shift would enable them to measure the speed of the Earth through a substance, called the aether, that was thought to fill space.
(a) Explain why Michelson and Morley expected that the fringe positions would shift when the apparatus was rotated through 90°.
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________ (2)
(b) In their apparatus they made the distances PM1 and PM2 the same and equal to d. They used light of wavelength (λ) about 550 nm and knew that the speed of light c was 3.0 × 108 m s-1. Using known astronomical data, they calculated the speed v at which they thought the Earth moved through the aether. They were then able to predict that when the apparatus was rotated through 90o the fringes should shift by a distance 0.4f, where f was the fringe spacing.
(i) To determine v, Michelson and Morley assumed that the Sun was stationary with respect to the aether as the Earth moved through it. Suggest, using this assumption, how the speed v of the Earth through the aether could be determined. You do not need to do the calculation.
______________________________________________________________
Page 29 of 29
______________________________________________________________
______________________________________________________________
______________________________________________________________ (1)
(ii) Michelson and Morley calculated v to be 3.0 × 104 m s−1. They worked out ∆f, the magnitude of the expected shift of the fringes, using the
formula ∆f =
Calculate the distance d they used in their experiment.
d = ____________________ m (1)
(c) Although a shift of 0.4 f was easily detectable, no shift was observed. Explain what this null result demonstrated and its significance for Einstein in his special theory of relativity.
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________ (2)
(Total 6 marks)
