CSS past paper

CSS Pure Mathematics Past Paper 2024

CSS Pure Mathematics 2024 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 N be a normal subgroup of a group G. If H is a subgroup of G, then prove that
  (10)
  $NH = \{nh : n \in N\ \text{and}\ h \in H\}$ is a subgroup of G.
- (b)
  If $\boldsymbol{E}$ is an epimorphism from a group $G$ onto a group $H$ then prove that $G/K$ is isomorphic to $H$ , where $K = Ker\boldsymbol{E}$ .
  (10)
Q. 2
(20)
- (a)
  Let $R$ be a ring. If every $x \in R$ satisfies $x^{2} = x$ then prove that $R$ is a commutative.
  (10)
- (b)
  For which value(s) of $a$ will the following system have no solution? Exactly one solution? Infinitely many solutions?
  (10)
  $\begin{array}{c} x + 2y - 3z = 4 \\ 3x - y + 5z = 2 \\ 4x + y + (a^{2} - 14)\,z = a + 2. \end{array}$
Q. 3
(20)
- (a)
  Determine a basis for and the dimension of the solution space of the system
  (10)
  $\begin{array}{c} x - 2z + w = 0 \\ 3x + y - 5z = 0 \\ x + 2y - 5w = 0. \end{array}$
- (b)
  Let $v_{1} = (1, 1, 1)$ , $v_{2} = (1, 1, 0)$ and $v_{3} = (1, 0, 0)$ be a basis for $\mathbb{R}^{3}$ . Find a linear transformation $T: \mathbb{R}^{3} \to \mathbb{R}^{2}$ such that $T(v_{1}) = (1, 0)$ , $T(v_{2}) = (2, -1)$ and $T(v_{3}) = (4, 3)$ .
  (10)

SECTION-B

Q. 4
(20)
- (a)
  Evaluate the limit:
  (10)
  - (i)
    $\lim_{x \to \frac{\pi}{2}} (1 + \cos x)^{\tan x}$
  - (ii)
    $\lim_{x \to 0} (\frac{1}{\sin x} - \frac{1}{x})$
- (b)
  State and prove the Mean Value Theorem.
  (10)
Q. 5
(20)
- (a)
  If $w = f(x^{2} + y^{2})$ then show that $y\left(\frac{\partial w}{\partial x}\right) - x\left(\frac{\partial w}{\partial y}\right) = 0$ .
  (10)
- (b)
  Find all the local maxima, local minima and saddle points of the given function $2x^{3} + y^{2} - 9x^{2} - 4y + 12x - 2$ .
  (10)
Q. 6
(20)
- (a)
  Evaluate the integral $\int_{0}^{\infty} x^{\frac{3}{2}}((1 + 2x))^{-5}\,dx$ and show that the result is $\frac{9\pi}{384}$ , using Beta function.
  (10)
- (b)
  Find the vertices and foci of the hyperbola
  (10)
  $25x^{2} - 16y^{2} + 250x + 32y + 109 = 0.$

SECTION-C

Q. 7
(20)
- (a)
  Verify that $u(x, y) = \cos x\cosh y$ is harmonic function and find a corresponding analytic function $f(z) = u(x, y) + iv(x, y)$ .
  (10)
- (b)
  Use Residue theorem to evaluate the integral $\int_{C} \frac{3z^{2} + z - 1}{(z^{2} - 1)(z - 3)}dz$ , where $C$ is the circle $|z| = 4$ .
  (10)
Q. 8
(20)
- (a)
  Use the Cauchy’s integral formula to evaluate the integral
  (10)
  $\int_{C} \frac{z + 4}{z^{2} + 2z + 5}dz, \text{ where } C \text{ is the circle } |z + 1 - i| = 2.$
- (b)
  Find the three cube roots of $\sqrt{3} + i$ .
  (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.
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.