14.

15.

16.

18.

19.

20.

21.

22.

23.

24.
25.

246470

2 D 30569

Find the reciprocal of z = 2 - अ.
Express — /3 —i in polar form.

Section B

Answer any number of questions.
Each question carries 5 Marks.
Maximum 35 marks.

State and prove Cantor’s Theorem.
Prove that there does not exist a rational number r such that r? = 2.

(a) Define supremum of a set of real numbers.

(9) Prove that there can be only one supremum of a given subset S of R, ifit exists.

Prove that lim 5 =0.
n

If 0 <6 <1, then prove that lim (b”) = 0.
Prove that the intersection of an arbitrary collection of closed sets in R is closed.

Show that the complex function ¢ (2) = 2 + 3i is one-to-one on the entire complex plane and find a

formula for its inverse function.
If f (z)= 3 then show by two path test that lim f(z) does not exist.
2 220

Section C

Answer any two questions.
Each question carries 10 marks.

State and prove Monotone Subsequence Theorem.

Prove that a monotone sequence of real numbers is convergent if and only if it is bounded. If
X = (x,,) is a bounded increasing sequence, then prove that :

lim (७, ) =sup {x, :n = N}.

246470