# Pure Mathematics 2018 — 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 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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