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Unformatted text preview: i vehicle and the other in a Chryssault vehicle. 18 The Toyundai has velocity vector km h–1, and the Chryssault has velocity 24 36 − 16 vector km h–1. (a) Find the speed of each vehicle.
(2) (b) (i) Find the position vectors of each vehicle at 06:30.
(2) (ii) Hence, or otherwise, find the distance between the vehicles at 06:30.
(3) (c) At this time (06:30) the Chryssault stops and its crew begin their day’s work, laying
cable in a northerly direction. The Toyundai continues travelling in the same direction at
the same speed until it is exactly north of the Chryssault. The Toyundai crew then begin
their day’s work, laying cable in a southerly direction. At what time does the Toyundai
crew begin laying cable?
(4) (d) Each crew lays an average of 800 m of cable in an hour. If they work non-stop until their
lunch break at 11:30, what is the distance between them at this time?
(4) (e) How long would the Toyundai take to return to base camp from its lunch-time position,
assuming it travelled in a straight line and with the same average speed as on the morning
journey? (Give your answer to the nearest minute.)
(Total 20 marks) 97 140. The lifespan of a particular species of insect is normally distributed with a mean of 57 hours
and a standard deviation of 4.4 hours.
(a) The probability that the lifespan of an insect of this species lies between 55 and 60
hours is represented by the shaded area in the following diagram. This diagram
represents the standard normal curve. a0 (i) b Write down the values of a and b.
(2) (ii) Find the probability that the lifespan of an insect of this species is
(a) more than 55 hours;
(1) (b) between 55 and 60 hours.
(2) (b) 90% of the insects die after t hours.
(i) Represent this information on a standard normal curve diagram, similar to the
one given in part (a), indicating clearly the area representing 90%.
(2) (ii) Find the value of t.
(Total 10 marks) 98 141. The diagram shows part of the graph of the curve with equation
y = e2x cos x.
y P(a, b) 0 (a) Show that x dy
= e2x(2 cos x – sin x).
(2) (b) Find d2 y
(4) There is an inflexion point at P(a, b).
(c) Use the results from parts (a) and (b) to prove that:
(i) tan a = 3 ;
(3) (ii) the gradient of the curve at P is e2x.
(Total 14 marks) 99 142. (a)
(b) Sketch the graph of f(x) = sin 3x + sin 6x, 0 < x < 2π.
Write down the exact period of the function f. Working: Answers:
(Total 3 marks) 143. Let z1 = a cos π + i sin π and z2 = b cos π + i sin π 4
3 3 z Express 1 in the form z = x + yi. z2 Working: Answers:
(Total 3 marks) 100 144. A sample of 70 batteries was tested to see how long they last. The results were:
Time (hours) Number of batteries
(frequency) 0 ≤ t ≤ 10 2 10 ≤ t ≤ 20 4 20 ≤ t ≤ 30 8 30 ≤ t ≤ 40 9 40 ≤ / ≤ 50 12 50 ≤ t ≤ 60 13 60 ≤ t ≤ 70 8 70 ≤ t ≤ 80 7 80 ≤ t ≤ 90 6 90 ≤ t...
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This note was uploaded on 06/24/2013 for the course HISTORY exam taught by Professor Apple during the Fall '13 term at Berlin Brothersvalley Shs.
- Fall '13
- The Land