CSS past paper

CSS Pure Mathematics Past Paper 2020

CSS Pure Mathematics 2020 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 $G$ and $G'$ be two groups and $f : G \rightarrow G'$ be a homomorphism then prove the following:
  (10)
  - (i)
    $f(e) = e'$ where $e$ and $e'$ are the identities of $G$ and $G'$ respectively
  - (ii)
    $f(a^{-1}) = [f(a)]^{-1}, \ \forall a \in G$
- (b)
  Prove that every homomorphic image of a group is isomorphic to some quotient group.
  (10)
Q. 2
(20)
- (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)
Q. 3
(20)
- (a)
  Let $\mathrm{T} : \mathrm{R}^{3} \longrightarrow \mathrm{R}^{3}$ be the linear transformation defined by $T(x_{1}, x_{2}, x_{3}) = (x_{1} - x_{2}, x_{1} + x_{3}, x_{2} + x_{3})$ . Find a basis and dimension of Range of T.
  (10)
- (b)
  Prove that every finitely generated vector space has a basis.
  (10)

SECTION-B

Q. 4
(20)
- (a)
  Find the critical points of $f(x) = x^{3} - 12x - 5$ and identify the open intervals on which $f$ is increasing and on which $f$ is decreasing.
  (10)
- (b)
  Find the horizontal and vertical asymptotes of the graph of $f(x) = -\frac{8}{x^{2} - 4}$
  (10)
Q. 5
(20)
- (a)
  Calculate $\int \frac{-2x + 4}{(x^{2} + 1)(x - 1)^{2}}\, dx$ .
  (10)
- (b)
  Find $\frac{\partial w}{\partial x}$ at the point $(x, y, z) = (2, -1, 1)$ if $w = x^{2} + y^{2} + z^{2}$ , $z^{3} - xy + yz + y^{3} = 1$ and $x$ and $y$ are the independent variables.
  (10)
Q. 6
(20)
- (a)
  Determine the focus, vertex and directrix of the parabola $x^{2} + 6x - 8y + 17 = 0$
  (10)
- (b)
  Find polar coordinates of the point $p$ whose rectangular coordinates are $(3\sqrt{2}, -3\sqrt{2})$
  (10)

SECTION-C

Q. 7
(20)
- (a)
  Show that $(\cos\theta + i\sin\theta)^{n} = \cos(\mathrm{n}\theta) + \mathrm{i}\sin(\mathrm{n}\theta)$ for all integers $n$ .
  (10)
- (b)
  Find the $n$ , $n$ th roots of unity.
  (10)
Q. 8
(20)
- (a)
  Find the Taylor series generated by $f(\mathrm{x}) = \frac{1}{x}$ at $a = 2$ . Where, if anywhere, does the series converge to $\frac{1}{x}$ ?
  (10)
- (b)
  Show that the p-series $\sum_{n=1}^{\infty} \frac{1}{n^{p}}$ , ( $p$ a real constant) converges if $p > 1$ , and diverges if $P < 1$
  (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.