# Pure Mathematics 2020 — 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 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

**********