## FEDERAL PUBLIC SERVICE COMMISSION

### COMPETITIVE EXAMINATION-2018

### 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 H and K be normal subgroups of a group G. Show that HK is a normal subgroup of G. (10)

(b)

Let H and K be normal subgroups of a group G such that H ⊆ K. Then show that KGHKHG/)/()/(≅

(10) (20)

**Q. No. 2.**

(a)

Show that every finite integral domain is a field. (10)

(b)

Consider the following linear system,

32=++zyx

105=+zay

bazyx=++72

- (i) Find the values of a for which the system has unique solution.
- (ii) Find the values of the pair (a, b) for which the system has more than one solution.

(10) (20)

**Q. No. 3.**

(a)

Find condition on a,b,c so that vector (a,b,c) in R3 belongs to

W= span {u1,u2,u3} where u1 = (1,2,0), u2 = (-1,1,2), u3 = (3,0,-4). (10)

(b)Let W1 and W2 be finite dimensional subspaces of a vector space V. Show that

dimW1 + dimW2 = dim ( W1 W2) + dim ( W1 + W2)

(10) (20)

**SECTION-B**

**Q. No. 4.**

(a) (10)

Let 11)({2>≤=xifxifxxxf

Does the Mean Value Theorem hold for f on

2,21.

(b)

Calculate the. lnsinxlnsin3xxlim0→

(10) (20)

**Q. No. 5.**

(a) (10)

Evaluate .251dxx∫−−

(b)

Prove that )0,0()0,0(ffyxxy≠ if =0,00,x1sin),(2bothareyxwhenbothnotareyxwhenyyxfx

(10) (20)

**Q. No. 6.**

(a)

Find the area of the region bounded by the cycloid )cos1(),sin(θθθ−=−=ayax and its base. (10)

(b)

Find the equation of a plane through (5,-1,4) and perpendicular to each of the planes 032=−−+zyx and 032=+−zyx

(10) (20)

**SECTION-C**

**Q. No. 7.**

(a)

Express cos5 θ sin3 θ in a series of sines of multiples of θ. (10)

(b)

Use Cauchy’s Residue Theorem to evaluate the integral ∫−−cdzZZz)1(25 where C is the circle2=z, described counter clock wise.

(10) (20)

**Q. No. 8.**

(a)

Find the Laurent series that represent the function 11)(−+=zzzf in the domain

∞<<z1. (10)

(b)

Expand sinxf(x)= in a Fourier cosine series in the interval π≤≤x0. (10)

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