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

• 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) $$\geqslant$$

If $x_1 ,x_2 , \cdots ,x_n ,y_1 ,y_2 , \cdots ,y_n \in R$ , then which of the following is Minkoski’s inequality?

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

$\left] {0,1} \right[$ in$\mathbb{R}$ under usual metric space is an example of -------.

• A) open interval
• B) open set
• C) open sphere
• D) all choices are equivalent

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) $- \infty < x < a$
• D) $- \infty < x < \infty$

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) None of these

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) 0
• B) 1
• C) -1
• D) 2

Which of the following is true about an open sphere $S_r \left( {x_0 } \right)$ with center at $x_0$ in a metric space$\left( {X,d} \right)$ ?

• A) $S_r \left( {x_0 } \right) \subset \varphi$
• B) $S_r \left( {x_0 } \right) \ne \varphi$
• C) $S_r \left( {x_0 } \right) = \varphi$
• D) $X \subset S_r \left( {x_0 } \right)$

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

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

If $\overrightarrow u = \left( \begin{array}{l} u_1 \\\\ u_2 \\\\ \end{array} \right)$ and $\overrightarrow v = \left( \begin{array}{l} v_1 \\\\ v_2 \\\\ \end{array} \right)$ , then their dot product is given by;

• A) $u_1 v_1 + u_2 v_2$
• B) $\sqrt {u_1 ^2 + u_2 ^2 } \sqrt {v_1 ^2 + v_2 ^2 }$
• C) $u_1 v_2 - u_2 v_1$
• D) $\sqrt {u_1 ^2 - u_2 ^2 } \sqrt {v_1 ^2 - v_2 ^2 }$

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

• A) -a
• B) 0
• C) $$\pm a$$
• D) a