RGU Question Paper Mathematics Semester I 2021: Higher Algebra

 

DECEMBER—2021

BA/B.Sc (I Semester) Examination

(CBCS)

MATHEMATICS

Paper : MAT–CC–112

( Higher Algebra )

Full Marks : 80

Pass Marks : 35%

Time : Three Hours

Note : 1. Answer the questions according to the instructions given.

2. The figures in the margin indicate full marks for the questions.

Section—A

1. Answer any four of the following questions : 5×4=20

(a) Define complex number. Explain the polar representation of complex numbers. Find the cube roots of unity.

(b) Write the Euler’s expansion of cosine and sine. Show that

(i) coth2 x = 1+ cosech2 x

(ii) cosh x = cos (ix) and $\sin hx=\frac{\sin \left(ix\right)}{i}$ 

(c) Show that the real part of $i^{\log (1+i)}$ is

(d) If p, q, r are positive, find the nature of roots of the equation

(e) Show that

(f) Show that any square matrix can be expressed as the sum of a symmetric and a skew-symmetric matrix.

Section—B

2. Answer any three of the following questions : 10×3=30

(a) (i) If the roots of the equation $x^n-1$ are $1,{\alpha }_1,{\alpha }_2,...,{\alpha }_{n-\mathrm{1, }}$then prove that

(ii) Use de Moivre’s theorem to solve the equation

(b) (i) For what values of $\lambda$ the system of linear equations

will have a unique solution?

(ii) Show that every square matrix can be expressed in the form R + iS; where R and S are Hermitian matrices.

(c) (i) If a, b, c are all different and if

then prove that abc = -1.

(ii) Find the rank of the matrix

(d) (i) Solve the biquadratic equation :

(ii) Express $x^5+5x^3+3x$ as a polynomial in $x-1$.

(e) (i) Write the Cardan’s method to solve $x^3-px+q=0, p,q>\mathrm{0 }$with the suitable conditions.

(ii) If $\alpha ,\beta ,\gamma$ are the roots of the equation $x^3-p{x}^2+qx-r=0$, then form an equation whose roots are

Section—C

3. Answer the following questions : 15×2=30

(a) (i) State and prove the Gregory series.

(ii) Express $\cos 7\theta$ in terms of cosines of multiples of $\theta$.

(iii) If $\tan (x+iy)=u+iv$, prove that

(b) (i) If

then show that 1+ abc = 0.

(ii) Using elementary row transformation, find the inverse of the matrix

(iii) Using Cramer’s rule, find the solution of the system of equations

provided $a\ne b, b\ne c,$ and $c\ne a$.