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# Pure Mathematics 2021 — CSS Past Paper

## PURE MATHEMATICS

TIME ALLOWED: THREE HOURS
PART-I(MCQS): MAXIMUM 30 MINUTES
PART-I (MCQS) MAXIMUM MARKS = 20
PART-II MAXIMUM MARKS = 80

NOTE:

• (i) Attempt ONLY FIVE questions from PART-II. ALL questions carry EQUAL marks.
• (ii) All the parts (if any) of each Question must be attempted at one place instead of at different places.
• (iii) Candidate must write Q. No. in the Answer Book in accordance with Q. No. in the Q.Paper.
• (iv) No Page/Space be left blank between the answers. All the blank pages of Answer Book must be crossed.
• (v) Extra attempt of any question or any part of the attempted question will not be considered.
• (vi) Use of calculator is allowed.

PART-II

SECTION-A

Q. No. 1.

(a) Let ‘y’ be a homomorphism of group G into group G with kernel K. pros e that K is a normal subgroup of G. (10)

(b) Prove that if H and K are two subgroups of a group G, then HK is a subgroup of G if and only if HK-K1-1.

Q. No. 2.

(a) Find elements of the cyclic group generated by the permutation.

a    =       (1 2 3 4 5 6)

3 4 5 2 6 1 )

(1(,)T (10) (20)

(b) Verify that the polynomials 2-x^2, x^3-x, 2-3x^2 and 3-x^3 form a basis for the net the set of all polynomials of degree three. Also express the vectors 1+x2 and x+x^3 as a linear combination of these basis vectors.

Q. No. 3.

(a) Let V be the real vector space of all function from R to R. Show that (cos^2 x, sine^2 x, cos 2x) is linearly dependent while {cos, sins, coshx, sinhx} arc linearly independent. (10)

(b) Solve the system of linear equations: (10) (20)

SECTION-B

Q. No. 4.

If

Show that.

It / (x,y)= tan-1(.0- y2tan-I(‘-:::). Show that -2-1-(x, y) ayax x +y whenx < 2 (r)

(b) Evaluate

Q. No. 5.

(a) Let

where n is an integer. Prove that

Hence show that

(b)

1. Write r = in rectangular coordinates. 2 — cos 9

2. Write x4 + 2x2y2 + y4 — 6x2y + 2y2 = 0 in polar coordinates.

Q. No. 6.

Evaluate

if dydx and ffaxd’.

where D is the region bounded by the y-axis, the lines x=2 and the curve e’.

(b) Investigate the curve y = 3×2 + ]for points of inflexion.

SECTION-C

Q. No. 7.

(a) Sum the series

1 + 1/2 —cos 0 + 1.3/2.4— cos 20 + 1.3.5/2.4.6— cos30 + …

(b) Prove that

Q. No. 8.

(a) Construct the analytic function f whose real part isU = .x3 — 3xy2 +3x + 1 (10) (10) (20)

(b) Evaluate

Where C is a square with corners z2 + 2z + 2 (0,0),(-2,0),(-2,-2) and (0,-2).

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