CSS past paper

CSS Pure Mathematics Past Paper 2025

CSS Pure Mathematics 2025 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
- (a)
  Let $Q^{+}$ be the set of positive rational numbers and define $*$ by $a * b = \frac{ab}{2}$ then prove that $(Q^{+}, *)$ is a group.
  (10)
- (b)
  Find all the cyclic subgroups of $Z_{18}$ .
  (10)
Q. 2
- (a)
  A homomorphism $\varphi: Z_{6} \to Z_{6}$ is a one to one mapping then calculate Ker( $\varphi$ ).
  (10)
- (b)
  Check whether the vectors $a = (1, 2, 3), b = (2, 5, 7)\,and\ c = (1, 3, 5)$ are linearly dependent or independent.
  (10)
Q. 3
- (a)
  Let $V$ be a vector space of all 2 $\times$ 2 matrices. W be a subspace of V which consists of all symmetric matrices then find two different basis of W.
  (10)
- (b)
  Consider the transformation $T: R^{3} \to R^{2}$ given by $T(x, y, z) = (|x|, y + z)$ . Check whether T is linear or not.
  (10)

SECTION-B

Q. 4
- (a)
  Find all the real numbers $x \in R$ such that $x^{2} + x > 2$ .
  (10)
- (b)
  Use the definition of limit to establish that $\lim_{x \to 2} \frac{x^{3} - 4}{x^{2} + 1} = \frac{4}{5}$ .
  (10)
Q. 5
- (a)
  Use Mean Value Theorem to show that $e^{x} \geq 1 + x\ \forall\ x \varepsilon R$ .
  (10)
- (b)
  Find the absolute extrema of the function $f(x, y) = xy - 2x$ on the region R given by vertices (0, 4), (4, 0) and (0, 0).
  (10)
Q. 6
- (a)
  Change the order of integration in double integral $\int_{0}^{2}\int_{0}^{\sqrt{x}} f(x, y)\,dydx$ .
  (10)
- (b)
  Draw the graph of the conic $r = 2Cos\theta$ .
  (10)

SECTION-C

Q. 7
- (a)
  Check that the Cauchy-Riemann Equations are satisfied in polar coordinates for $f(Z) = \frac{1}{Z}$ .
  (10)
- (b)
  Evaluate the contour integral $\oint_{C} f(Z)dZ$ where $f(Z) = y - x - 3ix^{2}$ and C is simple closed contour OABO with O = 0+0i, A = 0+i and B = 1+i.
  (10)
Q. 8
- (a)
  Use Cauchy Residue Theorem to evaluate the integral $\int_{C} \frac{5Z - 2}{Z(Z - 1)}dZ$ where C is the circle $|Z| = 2$ .
  (10)
- (b)
  Find the Maclaurin series for the function $f(Z) = Z^{2}e^{3Z}$ .
  (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.