Times are the two masses at the same height 2 total

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Unformatted text preview: n # is both commutative and associative. (4) 416 (b) Show that the set of real numbers forms a group under the operation #. (4) (Total 8 marks) 623. (a) Determine with reasons which of the following functions is a bijection from p(x) = x2 + 1, q(x) = x3, r(x) = to . x2 + 1 x2 + 2 (4) (b) Let t be a function from set A to set B, and s be a function from set B to set C. Show that if both s and t are bijective then is s ° t is also bijective. (3) (Total 7 marks) 624. (a) Describe how the integral test is used to show that a series is convergent. Clearly state all the necessary conditions. (3) ∞ (b) Test the series n ∑e n =1 n2 for convergence. (5) (Total 8 marks) 625. (a) Find the first four non-zero terms of the Maclaurin series for (i) sin x; (ii) ex . 2 (4) 2 (b) x Hence find the Maclaurin series for e sin x, up to the term containing x5. (2) 417 (c) e x 2 sin x – Use the result of part (b) to find lim x →0 x3 x . (2) (Total 8 marks) 626. Let f(x) = x3 – 2x2 – 1. (a) Find f′(x). (b) Find the gradient of the curve of f(x) at the point (2, –1). Working: Answers: (a) ………………………………………… (b) ………………………………………… (Total 6 marks) 418 627. The diagram below shows two circles which have the same centre O and radii 16 cm and 10 cm respectively. The two arcs AB and CD have the same sector angle θ = 1.5 radians. A B D C O Find the area of the shaded region. Working: Answer: ………………………………………….. (Total 6 marks) 419 628. The cumulative frequency curve below shows the marks obtained in an examination by a group of 200 students. 200 190 180 170 160 150 140 Number of 130 students 120 110 100 90 80 70 60 50 40 30 20 10 0 10 20 30 40 50 60 Mark obtained 70 80 90 100 420 (a) Use the cumulative frequency curve to complete the frequency table below. Mark (x) Number of students (b) 0 ≤ x < 20 20 ≤ x < 40 40 ≤ x < 60 22 60 ≤ x < 80 80 ≤ x < 100 20 Forty percent of the students fail. Find the pass mark. Working: Answer: (b) ………………………………………….. (Total 6 marks) 421 629. Let f(x) = 2x + 1. (a) On the grid below draw the graph of f(x) for 0 ≤ x ≤ 2. (b) Let g(x) = f(x +3) –2. On the grid below draw the graph of g(x) for –3 ≤ x ≤ –1. y 6 5 4 3 2 1 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6x –1 –2 –3 –4 –5 –6 422 Working: (Total 6 marks) 630. Let A and B be events such that P(A) = 1 3 7 , P(B) = and P(A ∪ B) = . 2 4 8 (a) Calculate P(A ∩ B). (b) Calculate P(AB). (c) Are the events A and B independent? Give a reason for your answer. Working: Answers: (a) ………………………………………….. (b) ………………………………………….. (c) …………………………………….......... (Total 6 marks) 423 631. Let f(x) = sin(2x + 1), 0 ≤ x ≤ π. (a) Sketch the curve of y = f(x) on the grid below. y 2 1.5 1 0.5 0 0.5 1 1.5 2 2.5 3 3.5 x –0.5 –1 –1.5 –2 (b) Find the x-coordinates of the maximum and minimum points of f(x), g...
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