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Below are the scanned copy of Kerala Public Service Commission (KPSC) Question Paper with answer keys of Exam Name 'ASSISTANT PROFESSOR -MATHEMATICS' And exam conducted in the year 21. And Question paper code was '001/21'. Medium of question paper was in Malayalam or English . Booklet Alphacode was 'A'. Answer keys are given at the bottom, but we suggest you to try answering the questions yourself and compare the key along wih to check your performance. Because we would like you to do and practice by yourself.

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+, ` ௫ 001/21

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,

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