6 marks 323 ln 2 giving the answer in the form a

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Unformatted text preview: S. (7) (Total 10 marks) 506. (a) Explain how to use Euclid’s algorithm to obtain the greatest common divisor (gcd) of two positive integers a, b with b < a. (3) (b) Let d be the gcd of 364 and 154. Use Euclid’s algorithm to find d, and hence find integers x and y so that d = 364x +154y. (5) (Total 8 marks) 336 507. The floor plan of a certain building is shown below. There are four rooms A, B, C, D and doorways are indicated between the rooms and to the exterior O. B A O D (a) C Draw a graph by associating a vertex with each room using the letters A, B, C, D, and O. If there is a door between the two rooms, draw an edge joining the corresponding two vertices. (2) (b) Does the graph in part (a) possess an Eulerian trail? Give a reason for your answer. What does your answer mean about the floor plan? (4) (c) Does the graph in part (a) possess a Hamiltonian cycle? Give a reason for your answer. What does your answer mean about the floor plan? (4) (Total 10 marks) 508. Let T = (V, E) denote a tree T, with V being the set of the vertices and E the set of the edges. Show that V = E + 1, i.e. that the number of vertices is one more than the number of edges. (Total 2 marks) 337 509. Consider the sequence of partial sums {Sn} given by n Sn = 1 ∑k, n = 1, 2,… k =1 (a) Show that for all positive integers n, S2n ≥ Sn + 1 . 2 (2) (b) Hence prove that the sequence {Sn} is not convergent. (5) (Total 7 marks) 510. Gwendolyn added the multiples of 3, from 3 to 3750 and found that 3 + 6 + 9 + … + 3750 = s. Calculate s. Working: Answer: ………………………………………….. (Total 6 marks) 338 511. Find the term containing x10 in the expansion of (5 + 2x2)7. Working: Answer: ………………………………………….. (Total 6 marks) 339 512. The number of hours of sleep of 21 students are shown in the frequency table below Hours of sleep Number of students 4 2 5 5 6 4 7 3 8 4 10 2 12 1 Find (a) the median; (b) the lower quartile; (c) the interquartile range. Working: Answers: (a) ………………………………………….. (b) ………………………………………….. (c) .................................................................. (Total 6 marks) 340 513. Part of the graph of y = p + q cos x is shown below. The graph passes through the points (0, 3) and (π, –1). y 3 2 1 0 π 2π x –1 Find the value of (a) p; (b) q. Working: Answers: (a) ………………………………………….. (b) .................................................................. (Total 6 marks) 341 x 514. Let f(x) = e 3 + 5cos2x. Find f′ (x). Working: Answer: ………………………………………….. (Total 6 marks) 515. Find all solutions of the equation cos3x = cos(0.5x), for 0 ≤ x ≤ π. Working: Answer: ………………………………………….. (Total 6 marks) 342 516. The vector equations of two lines are given below. 5 3 r1 = + λ , r2 = 1 – 2 – 2 4 +t 2 1 The lines intersect at the point P. Find the position vector of P. Working: Answer: ……………...
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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.

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