B.Sc. Mathematics Honours C-2 Question Paper 2018 (CBCS)

1. Answer any one question :
(a) If the complex numbers $z_{1}$, $z_{2}$ and $z_{3}$ represent the three points P, Q, R and be such that $lz_{1}+mz_{2}+nz_{3}=0$ where $l+m+n=0$, then show that the points P, Q, R lie on a straight line.
(b) Apply Descarte's rule of signs to ascertain the minimum number of complex roots of the equation $x^{6}-3x^{2}-2x-3=0$.

2. Answer any two questions :
(a) Prove that : $x^{n}+1=\prod_{k=0}^{\frac{n-2}{2}}[x^{2}-2x \cos\frac{(2k+1)\pi}{n}+1]$ if n be an even positive integer. Deduce the $sin\frac{\pi}{16}sin\frac{3\pi}{16}sin\frac{5\pi}{16}sin\frac{7\pi}{16}=\frac{1}{8\sqrt{2}}$.
(b) Solve the equation $x^{3}-15x^{2}-33x+847=0$ by Cardan's method.
(c) State and Prove Cauchy Schwarz's inequality.

3. Answer any one question :
(a) (i) Show that the solution of the equation $(1+x)^{n}-(1-x)^{n}=0$ are $x=i \tan\frac{\pi r}{n}$ where $r=0,1,2,...,n-1$, if n be odd; $r=0,1,2,...,\frac{n}{2}-1,\frac{n}{2}+1,...,n-1$, if n be even.
(ii) If x, y, z be positive and $x+y+z=1$, then show that $8xyz \le (1-x)(1-y)(1-z) \le \frac{8}{27}$. Show that $3x(3x+1)^{2} > 4(n!)^{\frac{1}{n}}$ where n be a positive integer (>1).
(b) (i) Solve the equation $x^{4}-12x^{3}+47x^{2}-72x+36=0$ given that the product of two of the roots is equal to the product of the other two.
(ii) State Descartes' rule of signs. Obtain the equation whose roots exceed the roots of the equation $x^{4}+3x^{2}+8x+3=0$ by 1. Use Descartes' rule of signs to both the equations to find the exact number of real and complex roots of the given equation.


4. Answer any five questions :
(a) Find $f \circ g$, if $f:\mathbb{R} \rightarrow \mathbb{R}$ is defined by $f(x)=|x|+x, x\in\mathbb{R}$ and $g:\mathbb{R} \rightarrow \mathbb{R}$ is defined by $g(x)=|x|-x, x\in\mathbb{R}$.
(b) Let $f:A \rightarrow B$ and $P \subseteq A$. Prove that $P \subseteq f^{-1}f(P)$.
(c) Let $f:A \rightarrow B$. If $S \subset A$, then show that $S \subset f^{-1}[f(S)]$. If further f be one-one and onto, then prove that $f^{-1}[f(S)]=S$.
(d) Find integers u and v satisfying $52u-91v=78$.
(e) Using the principle of induction, prove that $2.7^{n}+3.5^{n}-5$ is divisible by 24 for $n \in N$.
(f) Find the remainder when $1!+2!+3!+...+50!$ is divided by 15.
(g) If a is prime to b and a is prime to c then prove that a is prime to bc.
(h) Find the units digit in $7^{99}$.

5. Answer any one question :
(a) (i) Use division algorithm to prove that the square of an odd integer is of the form $(8k+1)$, where k is an integer.
(ii) Use Euclidean algorithm to find integers u and v such that $gcd(72,120)=72u+120v$.
(b) Define equivalence relation. A relation $\rho$ is defined on $\mathbb{N} \times \mathbb{N}$ by $(a,b)\rho(c,d)$ if and only if $ad=bc$ for $(a, b), (c,d) \in \mathbb{N} \times \mathbb{N}$. Show that $\rho$ is an equivalence relation.


6. Answer any two questions :
(a) Find a row echelon matrix which is row equivalent to
$$\begin{pmatrix} 0 & 0 & 2 & 0 \\ 1 & 3 & 2 & 4 & 1 \\ 2 & 6 & 2 & 6 & 2 \\ 3 & 9 & 1 & 10 & 6 \end{pmatrix}$$
(b) Show that the planes $2x-y+z=5$, $x+2y+4z=7$, $5x+3y-z=0$ are concurrent.
(c) Let x, y, z be elements of a vector space V over F and let $a, b \in F$. Show that x, y, z are linearly dependent, if $(x+ay+bz)$, y, z be linearly dependent.

7. Answer any one question :
(a) Investigate, for what values of $\lambda$ and $\mu$, the following equations $x+y+z=6, x+2y+3z=10, x+2y+\lambda z=\mu$ have i) No solution ii) a unique solution and iii) an infinite number of solutions.
(b) i) Obtain the fully row reduced normal form of the matrix:
$$\begin{bmatrix} 2 & 3 & -1 & -1 \\ 1 & -1 & -2 & -4 \\ 3 & 1 & 3 & -2 \\ 6 & 3 & 0 & -7 \end{bmatrix}$$
ii) Find the value of k, such that the following system of linear equation is consistent : $2x+y-z=12, x-y-2z=-3, 3y+3z=k$.


8. Answer any two questions :
(a) A and B are any two $2 \times 2$ matrices and E is the corresponding unit matrix. Show that $AB-BA=E$ cannot hold under any circumstances.
(b) If $\lambda$ be an eigen value of a non-singular matrix A, then prove that $\lambda^{-1}$ is an eigen value of $A^{-1}$.
(c) If $A = \begin{pmatrix} 1 & -1 & 1 \\ 2 & -1 & 0 \\ 1 & 0 & 0 \end{pmatrix}$ then show that $A^{2}=A^{-1}$.

9. Answer any one question :
(a) i) Let $S = \{ (x,y,z,w) \in \mathbb{R}^{4} : x+2y-z=0, 2x+y+w=0 \}$. Prove that S is a subspace of the real vector space $\mathbb{R}^{4}$. Also find the basis of S and the dimension of S.
ii) A is a $3 \times 3$ real matrix having the eigen values 2, 3, 1. If $\begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix}, \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}$ are the eigen vectors of A corresponding to the eigen values 2, 3, 1 respectively. Find the matrix A.
(b) i) Prove that the eigen values of a real skew-symmetric matrix are purely imaginary or zero.
ii) Let V be a vector space over a field F and let $\alpha, \beta \in V$. Then prove that the set $W = \{ c\alpha+d\beta : c \in F, d \in F \}$ forms a subspace of V. If $\alpha=(1,2,3), \beta=(3,1,0)$ and $\gamma=(2,-1,3)$ then examine for $\gamma \in W$ or not.