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Total Number of Questions : 24

Time : 2.00 Hours Max. Marks : 100 | |
1. Prove : Characteristic of a field F is 0 then F is infinite. (3 Marks) ۲
2. Explain : 15 every field F has an infinite extension of F. (3 Marks)
3. Find the value of the improper integral (3 Marks)

(1
न eR.

4. What 15 the cardinality of the set of all complex functions f(z) = u(x, y) + iv(x, ५), 27೫3 iy

such that u is a harmonic conjugate of v and v is a harmonic conjugate of u ? Justify. (3 Marks)
5. Find the remainder when 673 + 7 is divided by 43. (3 Marks)
6. Prove : The center of the set of all ‏معدم‎ matrices over a field F is isomorphic to F. (4 Marks)

7. Find the number of similarity classes of idempotent matrices of order n over a field F.

Explain your answer. (4 Marks)
8, Prove : 5, is not solvable. (4 Marks)
⋅ ⋅ 1
9. Check whether the following sequence of functions 8५ (x)= nate) converges
uniformly or diverges on R. nlite ) (4 Marks)

10. Find the points on R? where the directional derivative of the function f : R? + R defined by
f(x, y) = ery + #2 exists. Also find the directional derivative(s) if it exists. (4 Marks)

11. Let f(z) be an entire function and M 15 3 constant such that for a positive real number R and for
an integern=1 ടി ‏كزع‎ for | 2 | > 8. Prove that f(z) 15 8 polynomial of degree less than
or equal ton. (4 Marks)

12. Let X be a normed linear space, 5 = {x € X/|| x 2 1} and f be a map from 5 into ಔ such that
flax + By) 5० f(x) + B fly) for all x, ۷ ‏ع‎ Sand ax + By € ऽ where 0 and $ are scalars. Is there is

an extension for f to all of X ? Justify. (4 Marks)
13. Let X, 1, 2 be topological space. If f: -+ ۷ and ‏زع‎ 1-2 Zare continuous then show that |
gof : X 215 continuous. (4 Marks) ச்‌
14. Ametric space X is connected iff every continuous function f:X (0, 1} is not onto. (4 Marks)

15. Prove or disprove if A and © are connected subsets of a metric space X andifAcBcCthen
8 15 connected. (4 Marks)

8.0,