## 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

**********