FEDERAL PUBLIC SERVICE COMMISSION
COMPETITIVE EXAMINATION2021
FOR RECRUITMENT TO POSTS IN BS17
UNDER THE FEDERAL GOVERNMENT
PURE MATHEMATICS
TIME ALLOWED: THREE HOURS
PARTI(MCQS): MAXIMUM 30 MINUTES
PARTI (MCQS) MAXIMUM MARKS = 20
PARTII MAXIMUM MARKS = 80
NOTE:
 (i) Attempt ONLY FIVE questions from PARTII. 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.
PARTII
SECTIONA
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 HKK11.
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 2x^2, x^3x, 23x^2 and 3x^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)
SECTIONB
Q. No. 4.
If
Show that.
It / (x,y)= tan1(.0 y2tanI(‘:::). Show that 21(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)

Write r = in rectangular coordinates. 2 — cos 9
 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 yaxis, the lines x=2 and the curve e’.
(b) Investigate the curve y = 3×2 + ]for points of inflexion.
SECTIONC
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).
**********