FEDERAL PUBLIC SERVICE COMMISSION
COMPETITIVE EXAMINATION-2020
FOR RECRUITMENT TO POSTS IN BS-17
UNDER THE FEDERAL GOVERNMENT
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 G and G′be two groups and :fGG′→ be a homomorphism then prove the
following:
(i) ()fee′=where e and e′are the identities of G and G′respectively
(ii) ()[]11(a)faf−−=, aG∀∈ (10)
(b)
Prove that every homomorphic image of a group is isomorphic to some quotient group. (10) (20)
Q. No. 2.
(a)
A ring R is without zero divisor if and only if the cancellation law hold. (10)
(b)
Prove that arbitrary intersection of subrings is a subring. (10) (20)
Q. No. 3.
(a)
Let T : R3 R3 be the linear transformation defined by
123121323(,,)(,,)Txxxxxxxxx=−++. Find a basis and dimension of Range of T. (10)
(b)
Prove that every finitely generated vector space has a basis.
(10) (20)
SECTION-B
Q. No. 4.
(a)
Find the critical points of ()3125fxxx=−− and identify the open intervals on which fis increasing and on which f is decreasing. (10)
(b)
Find the horizontal and vertical asymptotes of the graph of ()284fxx=−− (10) (20)
Q. No. 5.
(a)
Calculate 2224(1)(1)xdxxx−++−∫. (10)
(b)
Find wx∂∂at the point (x, y, z) = (2, -1, 1) if 222wxyz=++, 331zxyyzy−++=
and x and y are the independent variables.
(10) (20)
Q. No. 6.
(a)
Determine the focus, vertex and directrix of the parabola x2 + 6x -8y +17=0 (10)
(b)
Find polar coordinates of the point p whose rectangular coordinates are ()32,32−
(10) (20)
SECTION-C
Q. No. 7.
(a)
Show that ()cossincos(n)isin(n)niθθθθ+=+for all integers n. (10)
(b)
Find the n, nth roots of unity. (10) (20)
Q. No. 8.
(a)
Find the Taylor series generated by1(x)fx= at a = 2. Where, if anywhere, does the series converge to 1?x
(b)
Show that the p-series 11,pnn∞=Σ(p
a real constant) converges if p> 1, and diverges if P < 1
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