B.Sc. Mathematics Honours C-2 Question Paper 2017 (CBCS)

UNIT - I

(Classical Algebra)

1. Answer any one question :
(a) If $x+iy$ moves on the straight line $3x+4y+5=0$, then find the minimum value of $|x+iy|$.
(b) Solve the equation $x^5+x^4+x^3+x^2+x+1=0$.

2. Answer any two questions :
(a) If $(1+i \tan \alpha)^{1+i \tan \beta}$ can have real values, then show that one of them is $(\sec \alpha)^{\sec^2 \beta}$.
(b) Show that the condition that the sum of two roots of the equation $x^4+mx^2+nx+p=0$ be equal to the product of the other two roots is $(2p-n)^2=(p-n)(p+m-n)^2$.
(c) If $a_1, a_2, ..., a_n$ be n real positive quantities then prove that $A.M. \ge G.M. \ge H.M$.

3. Answer any one question :
(a) (i) If $x+\frac{1}{x}=2 \cos \alpha$, $y+\frac{1}{y}=2 \cos \beta$, $z+\frac{1}{z}=2 \cos \gamma$ and $x+y+z=0$ then prove that $\sum \sin 4\alpha = 2 \sum \sin(\beta+\gamma)$ and $\sum \cos 4\alpha = 2 \sum \cos(\beta+\gamma)$.
(ii) If the equation whose roots are squares of the roots of the cubic $x^3-ax^2+bx-1=0$ is identical with this cubic, prove that either $a=b=0$ or $a=b=3$ or $a, b$ are the roots of the equation $t^2+t+2=0$.
(b) (i) If $a, b, c, x, y, z$ be all real numbers and $a^2+b^2+c^2=1$, $x^2+y^2+z^2=1$ then prove that $-1 \le ax+by+cz \le 1$.
If $a_1, a_2, ..., a_n$ be n positive rational numbers and $s=a_1+a_2+...+a_n$ prove that $(\frac{s}{a_1}-1)^{a_1}(\frac{s}{a_2}-1)^{a_2} \cdot \cdot \cdot (\frac{s}{a_n}-1)^{a_n} \le (n-1)^s$.
(ii) If the equation $x^3+px^2+qx+r=0$ has a root $a+ia$ where $p, q, r$ and $a$ are real, prove that $(p^2-2q)(q^2-2pr)=r^2$. Hence solve the equation $x^3-x^2-4x+24=0$.

UNIT - II

(Sets and Integers)

4. Answer any five questions :
(a) Prove that intersection of two equivalence relations is also an equivalence relation.
(b) Prove that square of any integer is of the form $3k$ or $3k+1$.
(c) Examine if the relation $\rho$ on the set $Z$ is an equivalence relation or not: $\rho = \{(a,b) \in \mathbb{Z} \times \mathbb{Z} : |a-b| \le 3\}$.
(d) Prove that, there exists no integer in between 0 and 1.
(e) Let $P = \{n \in \mathbb{Z} : 0 \le n \le 5\}, Q = \{n \in \mathbb{Z} : -5 \le n \le 0\}$ be two sets. Prove that cardinality of two sets are equal.
(f) If $s_n = 1 + \frac{1}{2} + \frac{1}{3} + \cdot \cdot \cdot + \frac{1}{n}$, then prove that $s_n > \frac{2n}{n+1}$ if $n > 1$.
(g) If $X$ and $Y$ are two non-empty sets and $f: X \rightarrow Y$ be an onto mapping, then for any subsets $A$ and $B$ of $Y$, prove that $f^{-1}(A \cup B) = f^{-1}(A) \cup f^{-1}(B)$.
(h) (i) State the Fundamental theorem of Arithmetic.
(ii) If $a$ divides $b$, then prove that every divisor of $a$ divides $b$.

5. Answer any one question :
(a) (i) Prove that $1^n - 3^n - 6^n + 8^n$ is divisible by $10 \forall n \in \mathbb{N}$.
(ii) Find integers $u$ and $v$ satisfying $20u + 63v = 1$.
(b) (i) State the division algorithm on the set of integers.
(ii) Find integers $s$ and $t$ such that $gcd(341, 1643) = 341s + 1643t$.
(iii) Using the theory of congruence for finding the remainder when the sum $1^5 + 2^5 + 3^5 + \cdot \cdot \cdot + 100^5$ is divided by 5.

UNIT - III

(System of Linear Equations)

6. Answer any two questions :
(a) Solve the system of equations :
$x+2y-z-3w=1$
$2x+4y+3z+w=3$
$3x+6y+4z-2w=5$
if possible.
(b) For what values of $k$ the system of equations
$x+2y+3z=kx$
$2x+y+3z=ky$
$x+3y+z=kz$
has a non-trivial solution.
(c) Determine $k$ so that the set $\{(1, 2, 1), (k, 3, 1), (2, k, 0)\}$ is linearly dependent.

7. Answer any one question :
(a) Determine the conditions for which the system
$x+y+z=1$
$x+2y-z=b$
$5x+7y+az=b^2$
admits of (i) only one solution, (ii) no solution, (iii) many solutions.
(b) (i) Obtain the fully row reduced normal form of the matrix :
$$\begin{pmatrix} 0 & 0 & 1 & 2 & 1 \\ 1 & 3 & 1 & 0 & 3 \\ 2 & 6 & 4 & 2 & 8 \\ 3 & 9 & 4 & 2 & 10 \end{pmatrix}$$
(ii) For what values of $k$, the planes $x-4y+5z=k, x-y+2z=3, 2x+y+z=0$ intersect in a line.

UNIT - IV

(Linear Transformation and Eigenvalues)

8. Answer any two questions :
(a) Find the rank of the matrix :
$$\begin{pmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \end{pmatrix}$$
if two straight lines $a_1x+b_1y+c_1=0$ and $a_2x+b_2y+c_2=0$ are coincident.
(b) Show that the rank of a skew symmetric matrix cannot be 1.
(c) State Cayley-Hamilton theorem and using theorem find $A^{-1}$ where
$A = \begin{pmatrix} 2 & 1 \\ 3 & 5 \end{pmatrix}$.

9. Answer any one question :
(a) (i) If $A = \begin{pmatrix} \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{pmatrix}$, $X = (x_1, x_2)^T$ and $Y = (y_1, y_2)^T$. Verify by means of the transformation $X=AY$ that $x_1^2+x_2^2$ is transformed to $y_1^2+y_2^2$. Find the dimension of the subspace $\mathbb{R}^3$ defined by $S = \{(x,y,z) \in \mathbb{R}^3 : x+2y=z, 2x+3z=y\}$.
(ii) Verify Caley-Hamilton's theorem for the matrix
$$A = \begin{pmatrix} 1 & 0 & 0 \\ 1 & 2 & 1 \\ 2 & 3 & 2 \end{pmatrix}$$
Hence compute $A^{-1}$.
(b) (i) Find all real $\lambda$ for which the rank of the matrix $A$ is 2, where
$$A = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 2 & -1 & \lambda \\ 5 & 7 & 1 & \lambda^2 \end{pmatrix}$$
(ii) If $X_1, X_2, ..., X_r$ be eigen vectors of an $n \times n$ matrix $A$ corresponding to $r$ distinct eigen values $\lambda_1, \lambda_2, ..., \lambda_r$ respectively, then prove that $X_1, X_2, ..., X_r$ are linearly independent.
(iii) $\lambda$ is an eigen value of a real skew symmetric matrix. Prove that $|\frac{1-\lambda}{1+\lambda}|=1$.