# The probability that dumisani will be late for school

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Unformatted text preview: raph. (ii) Giving a reason, determine whether or not there exists an isomorphism between graphs G and H. 455 (iii) Using the result in part (b), or otherwise, determine whether or not graph H is isomorphic to graph J. (7) (Total 14 marks) 681. (a) State the well-ordering principle. (2) (b) Use the well-ordering principle to prove that, given any two positive integers a and b, (a < b), there exists a positive integer n such that na > b. (3) (Total 5 marks) 682. Find the Maclaurin series of the function f(x) = ln (1 + sin x) up to and including the term in x4. (Total 8 marks) ˆ ˆ 683. The following diagram shows a triangle ABC, where BC = 5 cm, B = 60°, C = 40°. A B (a) 40° 60° 5 cm C Calculate AB. 456 (b) Find the area of the triangle. Working: Answers: (a) ………………………………………….. (b) ………………………………………….. (Total 6 marks) 684. Let f(x) = 6 3 x 2 . Find f′(x). Working: Answer: …………………………………………........ (Total 6 marks) 457 685. The cumulative frequency curve below shows the heights of 120 basketball players in centimetres. 120 110 100 90 80 70 60 Number of players 50 40 30 20 10 0 160 165 170 175 180 185 190 195 200 Height in centimetres 458 Use the curve to estimate (a) the median height; (b) the interquartile range. Working: Answers: (a) ………………………………………….. (b) ………………………………………….. (Total 6 marks) 686. Find the term containing x3 in the expansion of (2 – 3x)8. Working: Answer: …………………………………………........ (Total 6 marks) 459 687. The following diagram shows a circle divided into three sectors A, B and C. The angles at the centre of the circle are 90°, 120° and 150°. Sectors A and B are shaded as shown. A C 150° 90° 120° B The arrow is spun. It cannot land on the lines between the sectors. Let A, B, C and S be the events defined by A B C S : : : : Arrow lands in sector A Arrow lands in sector B Arrow lands in sector C Arrow lands in a shaded region. 460 Find (a) P(B); (b) P(S); (c) P(AS). Working: Answers: (a) ………………………………………….. (b) ………………………………………….. (c) ………………………………………….. (Total 6 marks) 688. Let a = log x, b = log y, and c = log z. x2 y Write log 3 in terms of a, b and c. z Working: Answer: …………………………………………........ (Total 6 marks) 461 689. Let a, b, c and d be integers such that a < b, b < c and c = d. The mode of these four numbers is 11. The range of these four numbers is 8. The mean of these four numbers is 8. Calculate the value of each of the integers a, b, c, d. Working: Answers: a = ............................., b = ............................. c = ............................., d = ............................. (Total 6 marks) 462 690. Let f(x) = 2x + 1 and g(x) = 3x2 – 4. Find (a) f–1(x); (b) (g ° f )(–2);...
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