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

Unit-I (Classical Algebra)

[Marks: 22]
1. Answer any one question :
(a) Find the geomtric image of the complex number $z$ satisfying $|z-i|\le3$.
(b) If $x+\frac{1}{x}=2~cos\frac{\pi}{7}$, prove that $x^{7}+x^{-7}=-2$.
(c) Use Descarte's rule of sign to show that the equation $x^{8}+x^{4}+1=0$ has no real root.

2. Answer any two questions :
(a) If $n$ be a positive integer, then prove that $(1+i)^{n}+(1-i)^{n}=2^{\frac{n}{2}+1}cos\frac{n\pi}{4}$.
(b) Solve the equation $3x^{4}+20x^{3}-70x^{2}-60x+27=0$ given that the roots are in geometric progression.
(c) State and prove the Cauchy-Schwarz inequality.

3. Answer any one question :
(a) (i) If $\alpha, \beta, \gamma$ be the roots of the equation $x^{3}-px^{2}+qx-r=0$, form an equation whose roots are $\beta\gamma+\frac{1}{\alpha}$, $\gamma\alpha+\frac{1}{\beta}$, $\alpha\beta+\frac{1}{\gamma}$.
(ii) Prove that $sin(log~i^{i})=-1$.
(iii) If $x, y, z$ are positive real numbers such that $xy+yz+zx=8$, then find the greatest value of $xyz$.
(b) (i) If $s_{n}=1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n}$ prove that $s_{n}>\frac{2n}{n+1}$ if $n>1$.
(ii) Prove that the roots of the equation $\frac{1}{x+a_{1}}+\frac{1}{x+a_{2}}+\cdot\cdot\cdot+\frac{1}{x+a_{n}}=\frac{1}{x}$ are all real, where $a_{1}, a_{2},...,a_{n}$ are all positive real numbers.
(iii) If $a, b, c$ be positive real numbers, prove that $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\ge3$.

Unit-II (Sets and Integers)

[Marks: 15]
4. Answer any five questions :
(a) Prove that $1^{n}-3^{n}-6^{n}+8^{n}$ is divisible by 10 $\forall n\in N$.
(b) When a function is invertible. Find the inverse of the function $f: R^{-}\rightarrow R^{+}$ defined by $f(x)=x^{2}$.
(c) Use mathematical induction to establish the following: $\sum_{i=1}^{n}(i+1)2^{i}=n.2^{n+1}$.
(d) Prove that the intersection of two symmetric relations is a symmetric relation.
(e) Let $P=\{n\in Z:0\le n\le5\}$, $Q=\{n\in Z:-5\le n\le0\}$ be two sets. Prove that the cardinality of two sets are equal.
(f) If two mappings $f:R\rightarrow R$ and $g:R\rightarrow R$ be defined by $f(x)=x^{2}$ and $g(x)=x-2$, respectively, then show that $f\circ g\ne g\circ f$.
(g) If $a$ is prime to $b$, prove that $a+b$ is prime to $ab$.
(h) Examine whether the mapping $f: Z\rightarrow Z$ defined by $f(x)=|x|\forall x\in z$ is injective.

5. Answer any one question :
(a) State Euclidean Algorithm for computation of $gcd(a, b)$. Hence find $gcd(1575, 231)$.
(b) (i) State the division algorithm on the set of integers.
(ii) Show that the product of any three consecutive integers is divisible by 6.

Unit-III (System of Linear Equations)

[Marks: 9]
6. Answer any two questions :
(a) Find the condition(s) for which the system $a_{1}x+b_{1}y=c_{1}, a_{2}x+b_{2}y=c_{2}$ has many solution and no solution.
(b) For what values of $K$ the system of equations $2x+Ky=0, 5x+2y=0$ has a non-trivial solution.
(c) Determine $K$ so that the set $s=\{(K,1,1), (1, K, 1), (1, 1, K)\}$ is linearly independent in $R^{3}$.

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) For what values of $K$ the planes $x+y+z=2, 3x+y-2z=K$ and $2x+4y+7z=K+2$ intersect in a line?
(ii) Find a row-reduced echelon matrix which is row equivalent to $$\begin{pmatrix} 0&0&2&2&0 \\ 1&3&2&4&1 \\ 2&6&2&6&2 \end{pmatrix}$$

Unit-IV (Linear Transformations & Eigen Values)

[Marks: 14]
8. Answer any two questions :
(a) Find the rank of the matrix $$\begin{pmatrix} 1&2&3 \\ 3&4&5 \\ 2&1&1 \end{pmatrix}$$
(b) If $\lambda$ be an eigen value of an $n\times n$ matrix $A$, then show that $\lambda$ is also an eigen value of its transpose matrix $A^{t}$.
(c) Let $P_{1}$ be the vector space of polynomials in $t$ of degree 1 over the field of real numbers $R$. If $T:P_{1}\rightarrow P_{1}$ is a linear transformation such that $T(1+t)=t$, $T(1-t)=1$, find $T(2-3t)$.

9. Answer any one question :
(a) (i) State Cayley-Hamilton theorem. Verify Cayley-Hamilton theorem for the matrix $$A = \begin{pmatrix} 1&0&0 \\ 1&0&1 \\ 0&1&0 \end{pmatrix}$$ Hence find $A^{-1}$ and $A^{100}$.
(ii) Find the eigen values of $\begin{pmatrix} 1&-i \\ i&1 \end{pmatrix}$.
(b) (i) Define rank and nullity of a linear transformation. Find the matrix of the linear transformation $T:R^{3}\rightarrow R^{3}$ defined by $T(a, b, c) = (a + b, a - b, 2c)$ with respect to the ordered basis $B=\{(0,1,1), (1, 0, 1), (1, 1, 0)\}$.
(ii) Let $V=\{(x,y,z)|x,y,z\in R\}$, where $R$ is a field of real numbers. Show that $W=\{(x,y,z)|x-3y+4z=0\}$ is a sub-space of $V$ over $R$. Find the dimension of $W$.