CSS past paper

CSS Pure Mathematics Past Paper 2018

CSS Pure Mathematics 2018 carries 100 marks over 3 hours. It has 8 questions.

Marks: 100
Time: 3 hours
Questions: 8

Years

How to attempt: Attempt FIVE questions in all by selecting TWO Questions each from SECTION-A&B and ONE Question from SECTION-C. ALL questions carry EQUAL marks.

Questions

SECTION-A

Q. 1
(20)
- (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 $H \subseteq K$ . Then show that
  (10)
  ${}^{(G/H)}\!\big/_{(K/H)} \cong G/K$
Q. 2
(20)
- (a)
  Show that every finite integral domain is a field.
  (10)
- (b)
  Consider the following linear system,
  (10)
  $\begin{aligned} x + 2y + z &= 3 \\ ay + 5z &= 10 \\ 2x + 7y + az &= b \end{aligned}$
  - (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.
Q. 3
(20)
- (a)
  Find condition on $a,b,c$ so that vector $(a,b,c)$ in $\mathrm{R}^{3}$ belongs to $W = span\ \{u_{1}, u_{2}, u_{3}\}$ where $u_{1} = (1,2,0)$ , $u_{2} = (-1,1,2)$ , $u_{3} = (3,0,-4)$ .
  (10)
- (b)
  Let $W_{1}$ and $W_{2}$ be finite dimensional subspaces of a vector space $V$ . Show that
  (10)
  $dimW_{1} + dimW_{2} = dim\,(W_{1} \cap W_{2}) + dim\,(W_{1} + W_{2})$

SECTION-B

Q. 4
(20)
- (a)
  Let $f(x) = \begin{cases} x^{2} & if\ \ x \leq 1 \\ x & if\ \ x > 1 \end{cases}$
  (10)
  Does the Mean Value Theorem hold for $f$ on $\left[\frac{1}{2}, 2\right]$ .
- (b)
  Calculate the. $\lim_{x \to 0} \frac{ln sin3x}{ln sinx}$
  (10)
Q. 5
(20)
- (a)
  Evaluate $\int_{-1}^{5} |x - 2|\, dx$ .
  (10)
- (b)
  Prove that $f_{xy}(0,0) \neq f_{yx}(0,0)$ if
  (10)
  $f(x, y) = \begin{cases} x^{2} y \sin \frac{1}{x} & \textit{when } x, y \textit{ are not both } 0 \\ 0 & \textit{when } x, y \textit{ are both } 0 \end{cases}$
Q. 6
(20)
- (a)
  Find the area of the region bounded by the cycloid $x = a(\theta - \sin\theta)$ , $y = a(1 - \cos\theta)$ and its base.
  (10)
- (b)
  Find the equation of a plane through (5,-1,4) and perpendicular to each of the planes
  (10)
  $x + y - 2z - 3 = 0 \text{ and } 2x - 3y + z = 0$

SECTION-C

Q. 7
(20)
- (a)
  Express $\cos^{5}\theta \sin^{3}\theta$ in a series of sines of multiples of $\theta$ .
  (10)
- (b)
  Use Cauchy’s Residue Theorem to evaluate the integral $\int_{C} \frac{5z - 2}{Z(Z - 1)}\, dz$ where $C$ is the circle $|z| = 2$ , described counter clock wise.
  (10)
Q. 8
(20)
- (a)
  Find the Laurent series that represent the function $f(z) = \frac{z + 1}{z - 1}$ in the domain $1 < |z| < \infty$ .
  (10)
- (b)
  Expand $f(x) = sinx$ in a Fourier cosine series in the interval $0 \leq x \leq \pi$ .
  (10)

Attempt FIVE questions in all by selecting TWO Questions each from SECTION-A&B and ONE Question from SECTION-C. ALL questions carry EQUAL marks.
All the parts (if any) of each Question must be attempted at one place instead of at different places.
Candidate must write Q. No. in the Answer Book in accordance with Q. No. in the Q.Paper.
No Page/Space be left blank between the answers. All the blank pages of Answer Book must be crossed.
Extra attempt of any question or any part of the attempted question will not be considered.
Use of Calculator is allowed.