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)

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

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

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

If \left| {{a^2} - {b^2}} \right| = 0 , then b=------.

• A) \pm a
• B) a
• C) 0
• D) -a

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

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

If f(x)=1 and g(x)=2 , then the distance as defined by \[d(f,g) = \int\limits_0^1 {\left| {f(x) - g(x)} \right|} dx = - - - - .\]

• A) -1
• B) 1
• C) 2
• D) 0

Supremum of open interval(0,1)in R is-----.

• A) Not defined
• B) 0
• C) + \infty
• D) 1

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) \[ \ne \]
• B) \[ \leqslant \]
• 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) \[{x_1} = 0\]
• B) \[{x_2} = 0\]
• C) \[{x_3} = 0\]
• D) \[{x_1} = {x_2} = {x_3} = 0\]

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

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

Supremum of open interval\[(0,1)\]in R is-----.

• A) 0
• B) Not defined
• C) \[ + \infty \]
• D) 1