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2 D 10667
Section B

Answer at least five questions.

Each question carries 6 marks.

All questions can be attended.
Overall Ceiling 30.

State and prove Cantor’s theorem.
Prove that there does not exist a rational number r such that r? = 2.
Solve the inequality | 2-1 | ‏دک‎ +1.
Let S be anon-empty set in R , that is bounded above. Prove that Sup (a +S)=a+SupS.
State and prove Archimedean property.
Prove that asequence in R can have atmost one limit.
Find the image of the half plane Re z > 2 under the mapping W = iZ.
Prove that |2] - | = [211-129].
(5 x 6 = 30 marks)
Section C

Answer any two questions.
Each question carries 10 marks.

(a) State and prove Arithmetic-geometric inequality.
(9) Let a,b,c e R. Then if ab < 0 then show that either a > 0 andb < 0ora< Oandb >0.

(c) If 1
(a) Prove that every contractive sequence is a Cauchy sequence.

(b) Prove that if asequence X of real numbers converges to a real number x, then any subsequence
of X also converge to x

(a) The polynomial equation x*— 7x + 2 = 0 has a solution between 0 and 1. Use an approximate
contractive sequence to calculate the solution correct to 4 decimal places.

1
(b) Show that ८८५

(a) Find an upperbound for | ‏سس‎ if | z|=2.
2 -52+ 1

(0) Find the image of the vertical strip 2
(c) Find the domain of /(2) = [मय
(2 x 10 = 20 marks)

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