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) $$\ne$$

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

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

If$${k_1},{k_2},...,{k_n} \geqslant 0$$ , then$$max\{ {k_i},1 \leqslant i \leqslant n,n \in N\} - - - 0.$$

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

$$\forall {x_1},{x_2},{x_3} \in \mathbb{R},if\,\left| {{x_1}} \right| + \left| {{x_2}\,} \right| + \left| {{x_3}\,} \right| = 0,then - - - .$$

• A) $${x_2} = 0$$
• B) $${x_3} = 0$$
• C) $${x_1} = {x_2} = {x_3} = 0$$
• D) $${x_1} = 0$$

$l^1 :$the set of all sequences whose corresponding series: -------- converges.

• A) $\sum\limits_{i = 1}^\infty {\frac{1}{{\left| {x_i } \right|}}}$
• B) $\sum\limits_{i = 1}^\infty {\frac{1}{{x_i }}}$
• C) $\sum\limits_{i = 1}^\infty {x_i }$
• D) $\sum\limits_{i = 1}^\infty {\left| {x_i } \right|}$

Which of the following is an example of unbounded function?

• A) $f(x) = \cos x$
• B) $f(x) = \sin x^2$
• C) $f(x) = \sin x\cos x$
• D) $f(x) = e^x$

$$\forall \,x,y \in \mathbb{R}, - - - - \sqrt {\left| x \right|} + \sqrt {\left| y \right|} \,\,.$$

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

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

• A) $$d( - x,y)$$
• B) $$d(\frac{1}{x},\frac{1}{y})$$
• C) $$d(y,x)$$
• D) $$d(x, - y)$$

For the distance function$$d(x,y) = \sqrt {\left| {x - y} \right|}$$ in (R,d), d(1,17)=--------.

• A) -4
• B) 16
• C) -16
• D) 4

A metric space$\left( {X,d} \right)$ is bounded if $\exists \,\,M > 0,$ such that $d\left( {x,y} \right) - - - - M,\forall x,y \in X.$

• A) $\ne$
• B) =
• C) $\ge$
• D) $\le$

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

• A) $x < a\,\,\,{\text{and}}\,\,\,x > - a$
• B) $x < a\,\,\,\,{\text{and}}\,\,\,x > a$
• C) $x < - a\,\,\,{\text{and}}\,\,\,x > a$
• D) $x < - a\,\,\,{\text{and}}\,\,\,x > - a$

$$\max \{ {x_1} + {y_1},{x_2} + {y_2}\} - - - \max \{ {x_1} + {x_2}\} + \max \{ {y_1} + {y_2}\}$$

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