RGU Question Paper Mathematics Semester IV 2022: Algebra-II and Analysis-I

 

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B.Sc (IV Semester) Examination

MATHEMATICS

Paper : MATH–241 (Old Course)

( Algebra—II and Analysis—I )

Full Marks : 80

Pass Marks : 35%

Time : Three Hours
Note : 1. Answer all questions.
2. The figures in the margin indicate full marks for the questions.

1. Answer any five parts : 2×5=10
(a) Define binary operation. Explain it with a suitable example.

(b) Define odd and even permutations. Whether or not, the following is an even permutation?
$f=\left(\begin{array}{cccc}1& 2& 3& 4\\ 1& 2& 4& 3\end{array}\right)$

(c) What do you mean by order of an element and order of a group?

(d) Define subring and ideal. What are the differences between an ideal and a subring? 

(e) Prove that the following function
$f\left(x\right)=\left|x\right|$
is continuous but not differentiable at $x=0$.

(f) Define infimum and supremum. Find infimum and supremum of the
following set :
$A=\left\{x\right| \frac{1}{n}<x\le n\}$
where $n$ is an arbitrary natural number.

(g) Define sequence and series. What are the fundamental differences between a sequence and a series?

2. Answer any four parts : 5×4=20

(a) State Lagrange’s theorem for finite group. Prove that order of an element of a finite group divides the order of the group.

(b) Prove that the set of all $2\times \mathrm{2 }$matrices over integers under matrix addition is an Abelian group.

(c) State and prove Cayley’s theorem of group.

(d) If $A$ and $B$ are two ideals of a ring $R$, then prove that $A+B$ is also an ideal and .

(e) Define convergence of a sequence. Show that every convergent sequence is bounded.

(f) Prove that the series $1+\frac{1}{2!}+\frac{1}{3!}+...$is convergent.

(g) State Rolle’s theorem. Explain it geometrically.

(h) Show that the function $f\left(x\right)=\left|x\right|+\left|x-1\right| \forall x\in \mathbb{R}$ is continuous but not derivable at $x=\mathrm{0 }$and $x=1$.

3. Answer any five parts : 10×5=50

(a) (i) Prove that the function
$f\left(x\right)=\left\{\begin{array}{ll}\sin \frac{1}{x}& for x\ne 0\\ 0& for x=0\end{array}\right.$
is not uniformly continuous on $[0,\infty )$ . 5

(ii) Prove that if a function is continuous in a closed interval, then it is bounded therein. 5

(b) (i) State and prove Lagrange’s mean value theorem. 6

(ii) Show that the function $f\left(x\right)=x^{\mathrm{2 }}$is uniformly continuous on $[-1,1]$ . Is it continuous also? 4

(c) Show that the set of all permutations $S_n$ on a set of $n$-symbols form a group under permutation multiplication. What is the order of this group? List all elements of $S_3$ and separate the even permutations of $S_{\mathrm{3 }}$from odd permutations. 6+1+3=10

(d) State and prove Lagrange’s theorem of group. Is the converse of Lagrange’s theorem true? Show that the set of even integers is an ideal of the ring integer $\mathbb{Z}$. 5+1+4=10

(e) (i) Define integral domain and field. Show that the ring of integers is an integral domain but not a field. 5

(ii) Define ring homomorphism. Prove that kernel of a ring homomorphism is an ideal. 5

(f) Let $R\left[x\right]$ be the ring of polynomial of a ring $R$ and suppose
$f\left(x\right)=a_0+a_{1}x+a_2{x}^2+...+a_n{x}^n$
$g\left(x\right)=b_0+b_{1}x+b_2{x}^2+...+b_m{x}^m$
are two non-zero polynomials of degrees $n$ and $m$, respectively. Prove that—
(i) $deg\left(f\right(x)+g(x\left)\right)\le max\{m,n\}$, provided $f\left(x\right)+g\left(x\right)\ne 0$;
(ii) if $f\left(x\right)g\left(x\right)\ne 0$, then $deg\left(f\right(x\left)g\right(x\left)\right)\le m+n$;
(iii) if $R$ is an integral domain, then $deg\left(f\right(x\left)g\right(x\left)\right)=m+n$. 10

(g) Examine the convergence of the following series : 5+5=10
(i)$\frac{1}{{\left(log2\right)}^p}+\frac{1}{{\left(log3\right)}^p}+...+\frac{1}{{\left(logn\right)}^p}+...$for $p>0$
(ii)$\sum _{}^{}\frac{n^2-1}{n^2+1}{x}^n$

(h) Prove that a sequence $\left\{x_n\right\}$ is convergent if and only if for every $\epsilon >0$, there exists a positive integer $\mathrm{m }$such that $\left|S_{n+p}-S_n\right|<\epsilon \forall n\ge \mathrm{m }$and $p\ge 1$. Using it, show that the sequence $\left\{x_n\right\}$, where $x_n=1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n}$ is not convergent. 7+3=10

(i) Test for convergence of the following sequences : 5+5=10
(i) $x_n=\frac{n}{n+1}$
(ii) $x_n={\left(1+\frac{1}{n}\right)}^n$