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

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

The order of (4 5) is ----------

• A) 1
• B) 2
• C) 4
• D) 6

The order of (1 2 3) (4 5 6) is _________.

• A) 4
• B) 2
• C) 5
• D) 3

The product of two even permutations is an even permutation.

• A) True
• B) False
• C)
• D)

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

• A) 7
• B) 6
• C) 8
• D) 5

The order of (1 4 5) (2 3) is __________.

• A) 4
• B) 3
• C) 6
• D) 5

The permutation (1 3)(1 2)is an even permutation.

• A) True
• B) False
• C)
• D)

The order of the cyclic permutation (1 2 3) is ----------

• A) 3
• B) 1
• C) 4
• D) 2

The permutation (1 2)(1 3)(1 4) is an even permutation.

• A) True
• B) False
• C)
• D)

\[{{\text{What is the taxicab distance between the points }}\left( {{\text{1,2}}} \right){\text{ and }}\left( {4,6} \right){\text{ in }}{R^2}{\text{?}}}\]

• A) 8
• B) 5
• C) 6
• D) 7

{{\text{For points }}\left( {{x_1},{x_2}, \ldots ,{x_n}} \right){\text{ and }}\left( {{y_1},{y_2}, \ldots ,{y_n}} \right){\text{ in }}{R^n}{\text{, the taxicab distance is given by:}}}

• A) {max_{i = 1}^n\mid {x_i} - {y_i}}
• B) \sum\limits_{i = 1}^n {{{({x_i} - {y_i})}^2}}
• C) {\sum\limits_{i = 1}^n {({x_i} - {y_i})} }
• D) {\sum\limits_{i = 1}^n {\mid {x_i} - {y_i}} \mid }