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Unformatted text preview: ot to B.
(a) Use Venn diagrams to verify that (A – B) ∪ (B – A) = (A ∪ B) – (A ∩ B).
(2) (b) Use De Morgan’s laws to prove that (A – B) ∪ (B – A) = (A ∪ B) – (A ∩ B).
(4)
(Total 6 marks) a b 225. Show that the set H = 0 1 a = ±1, and b ∈ Z forms a group under matrix multiplication. (You may assume that matrix multiplication is associative).
(Total 6 marks) 154 226. (a)
(b) State Lagrange’s theorem.
Let (G, °) be a group of order 24 with identity element e. Let a ∈ G, and
suppose that a12 ≠ e and a8 ≠ e. Prove that (G, °) is a cyclic group
with generator a.
(Total 7 marks) { } 227. Let S = x  x = a + b 2 ; a, b ∈ Q, a 2 − 2b 2 ≠ 0
(a) Prove that S is a group under multiplication, ×, of numbers. (b) For x = a + b 2 , define f(x) = a – b 2 . Prove that f is an isomorphism
from (S, ×) onto (S, ×).
(Total 11 marks) 228. (a) Use the Euclidean algorithm to prove that for n ∈
relatively prime. , (8n + 3) and (5n + 2) are
(4) (b) Any integer a with (n + 1) digits can be written as
a = 10nrn + 10n–1rn–1 +... + 10r1 + r0, where 0 ≤ ri ≤ 9 for 0 ≤ i ≤ n, and
(i) Show that a ≡ (r0 + r1 +... + rn) mod 3.
(3) (ii) Hence or otherwise, find all values of the single digit x such that the
number a = 137486x225 is a multiple of 3.
(6)
(Total 13 marks) 155 229. Let G = (V, E) be a connected planar graph with v vertices and e edges, in which each cycle has
a length of at least c.
(a) Use Euler’s theorem and the fact that the degree of each face is the length of the cycle
enclosing it to prove that
e≤ c
(v – 2).
c−2
(5) (b) Find the minimum cycle length in a к3,3 graph and use it to prove that the graph is not
planar.
(4)
(Total 9 marks) 230. (a) (b) Express f(x) = x2 – 6x + 14 in the form f(x) = (x – h)2 + k, where h and k are to be
determined.
Hence, or otherwise, write down the coordinates of the vertex of the parabola with
equation y – x2 – 6x + 14. Working: Answers:
(a) …………………………………………..
(b) ..................................................................
(Total 4 marks) 156 231. Let f′ (x) = 1 – x2. Given that f(3) = 0, find f(x).
Working: Answers:
…………………………………………..
(Total 4 marks) 232. Town A is 48 km from town B and 32 km from town C as shown in the diagram. C
32km
A
48km B ˆ
Given that town B is 56 km from town C, find the size of angle CAB to the nearest degree. Working: Answers: …………………………………………..
(Total 4 marks) 157 158 233. In a survey of 200 people, 90 of whom were female, it was found that 60 people were
unemployed, including 20 males.
(a) Using this information, complete the table below.
Males Females Totals Unemployed
Employed
Totals 200 159 (b) If a person is selected at random from this group of 200, find the probability that this
person is
(i) an unemployed female; (ii) a male, given that the person is employed. Working: Answers:
(b) (1) ………...........……...
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 Fall '13
 Apple
 The Land

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