RGU Question Paper Mathematics Semester II 2022: Elementary Algebra

 

2022

( May/June )

BA/B.Sc (Il Semester) Examination (CBCS)

MATHEMATICS

Paper : MAT-GE-002

( Elementary Algebra)

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 the following questions : 2×5=10

(a) If $\alpha ,\beta ,\gamma$ are the roots of the equation $x^3+qx+r=0$, then show that

(b) Simplify $(sin\alpha +icos\alpha {)}^n$.

(c) Find the value of

(d) Define a group.

(e) Distinguish between relation and function.

2. Answer any four questions : 5×4=20

(a) Solve the equation

given that the roots are in AP.

(b) Simplify :

(c) If x, y, z are different and

then show that 1+ xyz = 0.

(d) Find the rank of the matrix:

(e) If (G, *) be a group and S is a subset of G, then show that S will be a sub- group of G if and only if

3. Answer any five questions: 10×5=50

(a) State and prove De Moivre's theorem.

(b) (i) Solve $x^3-7x^2+36=0$, given that one root is double the other.

(ii) Apply Cardan's method to solve the equation $x^3-12x+65=0$.

(c) (i) Show that

ii) Prove that any square matrix A is invertible if and only if A is non-singular.

(d) Using matrix method, show that the equations

are consistent and hence obtain the solution for x, y and z.

(e) (i) Show that $f:\mathbb{N}\to \mathbb{N}$, given by

is both one-one and onto.

(ii) Show that the relation R in the set Z of integers given by

R = {(a, b): 2 divides a - b)

is an equivalence relation.

(f) (i) Given a *x*a =b in a group G, then find x.

(ii) Prove that a multiplicative group G is Abelian, if and only if

(g) (i) Prove that

(i) Expand $\tan 8\theta$ in powers of $\tan \theta$.