Symmetric property states that for all real numbers x and y --------- .

• A) $$d(x,y) \geqslant d(y,x)$$
• B) $$d(x,y) \leqslant d(y,x)$$
• C) $$d(x,y) = d(y,x)$$
• D) $$d(x,y) = 0$$

If d is a metric on R defined by$$d(x,y) = \left| x \right| + \left| y \right|$$ , then d(-2,0)=---------.

• A) -2
• B) 0
• C) -0.50
• D) 2

$${\text{Which of the following functions is usual or Eucledian metric on }}{R^2}{\text{ for points }}{P_1} = \left( {{x_1},{y_1}} \right){\text{ and }}{P_2} = \left( {{x_2},{y_2}} \right){\text{?}}$$

• A) $$d\left( {{P_1},{P_2}} \right) = \sqrt {\mid {x_1} - {x_2}{\mid ^2} + \mid {y_1} - {y_2}{\mid ^2}}$$
• B) $$d\left( {{P_1},{P_2}} \right) = \sqrt {{{({x_1} - {x_2})}^2} + {{({y_1} - {y_2})}^2}}$$
• C) $$d\left( {{P_1},{P_2}} \right) = \sqrt {\mid {x_1} - {x_2}{\mid ^2} - \mid {y_1} - {y_2}{\mid ^2}}$$
• D) $$d\left( {{P_1},{P_2}} \right) = \sqrt {\mid {x_1} - {x_2}\mid + \mid {y_1} - {y_2}\mid }$$

For the distance function $$d(x,y) = \sqrt {\left| {x - y} \right|}$$ in R ,$$d(1,\frac{1}{2})$$ =-----------.

• A) 2
• B) $$\frac{1}{{\sqrt 2 }}$$
• C) 1
• D) $$\sqrt 2$$

$${\text{In }}{{\text{R}}^4}{\text{, the taxicab distance between points }}\left( {1,2,3,4} \right){\text{ and}}\left( {4,1,5,6} \right){\text{ is:}}$$

• A) 7
• B) 5
• C) 8
• D) 6

$${\text{Which of the following points is at a taxicab distance of 4 from the point }}\left( {2,3} \right) \in {R^2}{\text{?}}$$

• A) $$\left( {{\text{2,7}}} \right)$$
• B) $$\left( {{\text{0,0}}} \right)$$
• C) $$\left( {{\text{5,5}}} \right)$$
• D) $$\left( {{\text{0,2}}} \right)$$

If d is a usual(natural) metric on R, then d(1,0.50)= ----------.

• A) 1
• B) 1.50
• C) -0.50
• D) 0.50

If $${d_0}$$ is a discrete metric on R then $${d_0}( - 2, - 2) = - - - - - .$$

• A) 0
• B) -2
• C) -1
• D) 1

$$\begin{gathered} {\text{Given the set }}X = \{ 1,2,3\} {\text{ with the discrete metric defined by }}d(a,b) = 1{\text{ if }}a \ne b{\text{ and }}d(a,a) = 0{\text{, what is the distance between 1 and 3?}} \hfill \\ \hfill \\ \end{gathered}$$

• A) 2
• B) 1
• C) 0
• D) 3

If d is a usual metric on$${\mathbb{R}^2}$$ , then d((0,0),(-1,-1)) =-----------.

• A) 0
• B) 2
• C) 1
• D) $$\sqrt 2$$

Dot product $\overrightarrow u \cdot \overrightarrow v$ of two vectors $\overrightarrow u$ and $\overrightarrow v$ is given by----.

• A) $\left| {\overrightarrow u } \right|\left| {\overrightarrow v } \right|\csc \theta$
• B) $\left| {\overrightarrow u } \right|\left| {\overrightarrow v } \right|\cos \theta$
• C) $\left| {\overrightarrow u } \right|\left| {\overrightarrow v } \right|\tan \theta$
• D) $\left| {\overrightarrow u } \right|\left| {\overrightarrow v } \right|\sin \theta$

Which of the following is an example of bounded function?

• A) $${x^2}$$
• B) $${\log _{10}}x$$
• C) Sin x
• D) $${e^x}$$

If$\forall x \in \mathbb{R},$$\left| x \right| < - a,$ where $a < 0,$ then ----.

• A) $- a > x > a$
• B) $- a < x < a$
• C) $a < x < - a$
• D) $- a > x < a$

$$\forall x,y \in \mathbb{R},if\,\,\min \{ \left| x \right|,\left| y \right|\} = 0,then - - - .$$

• A) x=0 and y=0
• B) x=0
• C) y=0
• D) either x=0 or y=0

If $x_1 ,x_2 , \cdots ,x_n ,y_1 ,y_2 , \cdots ,y_n \in R$ , then which of the following is Cauchy-Schwarz inequality?

• A) $$\left( {x_1 y_1 + x_2 y_2 + \cdots + x_n y_n } \right)^2 \le \left( {x_1^2 + x_2^2 + \cdots + x_n^2 } \right)\left( {y_1^2 + y_2^2 + \cdots + y_n^2 } \right)$$
• B) $$\sqrt {\left( {x_1 + y_1 } \right)^2 + \left( {x_2 + y_2 } \right)^2 + \cdots + \left( {x_n + y_n } \right)^2 } \le \sqrt {x_1^2 + x_2^2 + \cdots + x_n^2 } \sqrt {y_1^2 + y_2^2 + \cdots + y_n^2 }$$
• C)
• D)

Open set in R under usual metric space is ------.

• A) $\left[ {0,1} \right]$
• B) $\left\{ 1 \right\}$
• C) $\left\{ 0 \right\}$
• D) $\left] {0,1} \right[$

Which of the following is true about the bounded-ness of $f(x) = \cos x$ ?

• A) $- 1 \leqslant \cos x \leqslant 1$
• B) $- 1 \leqslant \cos x \leqslant 0$
• C) $0 \leqslant \cos x \leqslant 1$
• D) $- \frac{1}{2} \leqslant \cos x \leqslant \frac{1}{2}$

In$R^2$ space under the usual metric space defined as $d\left( {P_1 ,P_2 } \right) = \sqrt {\left( {x_1 - x_2 } \right)^2 + \left( {y_1 - y_2 } \right)^2 } ,\,\,\forall P_1 \left( {x_1 ,y_1 } \right),P_2 \left( {x_2 ,y_2 } \right) \in R^2 ,$ the open sphere is an open -----.

• A) disk
• B) interval
• C) circle
• D) square

$$\forall {a_k},{b_k} \in \mathbb{R},{\left( {\sum\limits_{k = 1}^n {{a_k}{b_k}} } \right)^2} - - - -$$

• A) $$\leqslant \left( {\sum\limits_{k = 1}^n {a_k^2} } \right)\left( {\sum\limits_{k = 1}^n {b_k^2} } \right)$$
• B) $$= \left( {\sum\limits_{k = 1}^n {a_k^2} } \right)\left( {\sum\limits_{k = 1}^n {b_k^2} } \right)$$
• C) $$\geqslant \left( {\sum\limits_{k = 1}^n {a_k^2} } \right)\left( {\sum\limits_{k = 1}^n {b_k^2} } \right)$$
• D) $$= \sum\limits_{k = 1}^n {a_k^2b_k^2}$$

If$$\max \{ \left| {{x_i} - {y_i}} \right|\} = 0\,\,\,$$ where $$1 \leqslant i \leqslant n,{x_i},{y_i} \in \mathbb{R},then\,\,{x_i} - - - {y_i}.$$

• A) $$\leqslant$$
• B) $$\geqslant$$
• C) =
• D) $$\ne$$