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# gpsc.gujarat.gov.in Administrative Service Main Exam Mathematics Sample Paper : Public Commission

Organisation : Gujarat Public Service Commission
Post : Main Exam, Gujarat Administrative Service Class-1 & Class-2
Document Type : Sample Paper
Subject : Mathematics
Website : https://gpsc.gujarat.gov.in/Archieved/sample-papers.html

## GPSC Main Exam Mathematics Sample Paper

Main Examination :
English Medium :

Related : GPSC Gujarat Administrative Service Main Exam Economics Sample Paper : www.pdfquestion.in/25633.html

Time : 3 Hours
Total Marks : 200

## Instruction

(1) The question paper has been divided into three parts, A, B and C. The number of questions to be attempted and their marks are indicated in each part.

(2) Answers of all the questions of each part should be written continuously in the answer sheet and should not be mixed with other parts’ Answer. In the event of answer found, which are belongs to other part, such answers will not be assessed by examiner.

(3) The candidate should write the answer within the limit of words prescribed in the parts A, B and C.

(4) If there is any difference in English language question and its Gujarati Translation, then English language question will be considered as valid.

(5) Answer should be written in one of the two languages. Write in the language (English or Gujarati) preference given by you. Answer should not be written in both the languages in the same paper.

## Sample Question

Instructions : (40 Marks)
1) Question No. 1 to 20.
2) Attempt all 20 questions.
3) Each question carries 2 marks.
4) Answer should be given approximately in 20 to 30 words.

### Part – A

1. Decompose the permutation 1 2 3 4 5 6 7 6 5 2 4 3 1 7 into transpositions.
2. Prove that if for every element a in a group G, a2 = e, then G is an Abelian group.
3. Prove that the only idempotent elements of an integral domain with unity are 0 and 1.
4. If U is an ideal of a ring R with unity 1 and 1 ? U, then prove that U = R.

5. Find the dimension of the subspace W = {(s, s – t, t) : s, t ?ú} of ú3 with standard operations.
7. Solve : (D2 + 4)y = sin 3x + ex, where D = d/dx.
8. Solve : (D2 – 2D + 4) y = ex cos x, where D =d/dx.

9. Find the nth derivative of log [(ax + b) (cx + d)].
11. If y = etan–1x, prove that (1 + x2)yn+2 + [2(n + 1)x – 1]yn+1 + n(n + 1)yn = 0.
15. Define a Cauchy sequence in a metric space. When is a metric space said to be complete ? Give an example of a complete metric space.

16. Show that the series cos x + cos 2x 22 + cos 3x 32 + ······ converges uniformly on ú.
17. Show that the function f(z) = xy + iy is everywhere continuous but is not analytic in.
18. Find the Taylor series expansion of 4 + x. Where is the expansion valid ?

19. Let T : ú3 ? ú2 be the linear transformation given by T(x, y, z) = (x + y, y – z). Find the standard matrix of T.
20. Show that the function f(x, y) = 2x2y / x4 + y2 has no limit as (x, y) approaches (0, 0).

### Part – B

Instructions :
(1) Question No. 21 to 32.
(2) Attempt all 12 questions.
(3) Each question carries 5 marks.
(4) Answer should be given approximately in 50 to 60 words.

21. State the rank-nullity theorem and verify it for the linear transformation T : ú3 ? ú3, T(x, y, z) = (x – y, x – y, 0).
22. If G is a group and H is a subgroup of index 2 in G, prove that H is a normal subgroup of G.

23. Prove that set I( ) 2 of numbers of the form a + b 2, where a and b are integers is an integral domain with respect to ordinary addition and multiplication. Is it a field ?
24. Prove that the function f(x) = 1/1 + e1x has a jump discontinuity at the origin.
26. Solve : (yz + z2) dx – xzdy + xydz = 0.

27. The region bounded by y = x2 and y = 2 – x2 is revolved about the line x = – 2. Find the volume of the solid generated.
28. Solve the following differential equation : dy /dx – 2y tan x = y2tan2 x.
29. Use Lagrange’s mean value theorem to prove that 1 + x < ex < 1 + xex ? x > 0.

30. Prove that the sequence <fn> defined by fn(x) = x 1 + nx , 0 = x < 8 is uniformly convergent to 0 on [0, 8).
31. Prove that the three vectors (1, 1, –1), (2, –3, 5) and (–2, 1, 4) of ú3 are linearly independent.

32. A point moves in a straight line so that its distance from a fixed point in that line is the square root of the quadratic function of the time. Prove that its acceleration varies inversely as the cube of the distance from the fixed point.

### Part – C

Instructions :
(1) Question No. 33 to 39.
(2) Attempt any 5 out of 7 questions.
(3) Each question carries 20 marks.
(4) Answer should be given approximately in 200 words.

33. A uniform rod, of length a hangs against a smooth vertical wall being supported by means of a string, of length l, tied to one end of the rod, the other end of the string attached to a point in the wall. Show that the rod can rest inclined to the wall at an angle ? given by What are the limits of the ratio of a : l that the equilibrium may be possible ?

34. Six equal rods AB, BC, CD, DE, EF and FA are each of weight W and are freely joined at their extremities so as to form a hexagon; the rod AB is fixed in a horizontal position and the middle points of AB and DE are joined by a string; Prove that its tension is 3W.

35. Prove that in a parabolic orbit the time taken to move from the vertex to a point distant r from the focus is 1 3 µ (r + l) 2r – l where 2l is the length of latus rectum.
36. How many generators are there of the cyclic group of order 10 ?

37. Define (i) an absolutely convergent series (ii) a conditionally convergent series. Prove that 8S2 (– 1)n logn n is conditionally convergent.
38. Find the critical points and classify them as local maxima, minima and saddle points for the function f(x, y) = x3 – 3x + y3 – 3y.

39. Use Green’s theorem and evaluate the line integral ?I C y2 dx + x2dy, where C is the closed curve which is the boundary of the triangle with vertices (0, 0), (1, 1) and (1, 0) with the counter clockwise orientation.