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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 nonzero 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 xcoordinates of the maximum and minimum points of f(x), g...
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 Fall '13
 Apple
 The Land

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