{\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) 8
• B) 7
• C) 5
• D) 6

The permutation (a b)(a c)(a d) is an even permutation.

• A) True
• B) False
• C)
• D)

The order of (1 2)(3 4) is -----------------

• A) 6
• B) 2
• C) 4
• D) 3

The order of the cyclic permutation (1 2 3 4 5) is ----------

• A) 2
• B) 6
• C) 3
• D) 5

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

{\text{What is the distance between the points }}\left( {{\text{1,2}}} \right){\text{ and }}\left( {4,6} \right){\text{ under usual or Eucledian metric on }}{R^2}{\text{ ?}}

• A) 6
• B) 5
• C) 7
• D) 8

\[{{\text{The points }}\left( {{x_1},{x_2}, \ldots ,{x_n}} \right),\left( {{y_1},{y_2}, \ldots ,{y_n}} \right) \in {R^n}{\text{ have a taxicab distance of 0 if:}}}\]

• A) \[{x_i} = y{_i}{\text{ }}for{\text{ }}all{\text{ }}i\]
• B) \[{{x_i} = y{_i}{\text{ }}for{\text{ }}some{\text{ }}i}\]
• C)
• D)

{\text{For points }}\left( {{x_1},{y_1}} \right){\text{and }}\left( {{x_2},{y_2}} \right){\text{, the taxicab distance on }}{R^2}{\text{ is given by:}}

• A) \mid {x_1} + {x_2}\mid + \mid {y_1} + {y_2}\mid
• B) \mid {x_1} + {x_2}\mid - \mid {y_1} + {y_2}\mid
• C) \mid {x_1} - {x_2}\mid - \mid {y_1} - {y_2}\mid
• D) \mid {x_1} - {x_2}\mid + \mid {y_1} - {y_2}\mid

In{\text{ }}a{\text{ }}metric{\text{ }}space{\text{ }}(X,d),{\text{ }}the{\text{ }}triangle{\text{ }}inequality{\text{ }}states{\text{ }}that{\text{ }}for{\text{ }}any{\text{ }}points{\text{ }}a,b,c \in X:

• A) d(a,c) \leqslant d(a,b) + d(b,c)
• B) d(a,c) = d(a,b) + d(b,c)
• C) d(a,b) \geqslant d(b,c)
• D) d(a,b) \leqslant d(b,c)

\[{\text{What is the distance between the points }}\left( {{\text{1,2}}} \right){\text{ and }}\left( {4,6} \right){\text{ under usual or Eucledian metric on }}{R^2}{\text{ ?}}\]

• A) 6
• B) 7
• C) 5
• D) 8

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

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

Let \alpha be a permutation such that {\alpha ^7}\,\, = \,\,I , then order of \alpha is ------------

• A) 5
• B) 6
• C) 8
• D) 7

{\text{What is the distance between the points }}\left( {{\text{1,3}}} \right){\text{ and }}\left( {5,6} \right){\text{ under usual or Eucledian metric on }}{R^2}{\text{ ?}}

• A) 7
• B) 6
• C) 5
• D) 8

\[{\text{In }}{{\text{R}}^5}{\text{, the taxicab distance between points }}\left( {1,2,0,3,4} \right){\text{ and}}\left( {0,4,1,5,6} \right){\text{ is:}}\]

• A) 8
• B) 5
• C) 6
• D) 7

For the distance function d(x,z) = \left| {x - z} \right|in (R,d), d(x,z) \geqslant 0

• A) True
• B) False
• C)
• D)

\[{\text{For points }}\left( {{x_1},{y_1}} \right){\text{and }}\left( {{x_2},{y_2}} \right){\text{, the taxicab distance on }}{R^2}{\text{ is given by:}}\]

• A) \[\mid {x_1} - {x_2}\mid - \mid {y_1} - {y_2}\mid \]
• B) \[\mid {x_1} + {x_2}\mid - \mid {y_1} + {y_2}\mid \]
• C) \[\mid {x_1} + {x_2}\mid + \mid {y_1} + {y_2}\mid \]
• D) \[\mid {x_1} - {x_2}\mid + \mid {y_1} - {y_2}\mid \]

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

• A) x=0 and y=0
• B) y=0
• C) x=0
• D) either x=0 or y=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) \sqrt {u_1 ^2 - u_2 ^2 } \sqrt {v_1 ^2 - v_2 ^2 }
• B) u_1 v_1 + u_2 v_2
• C) \sqrt {u_1 ^2 + u_2 ^2 } \sqrt {v_1 ^2 + v_2 ^2 }
• D) u_1 v_2 - u_2 v_1

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) \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)

\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_1} = {x_2} = {x_3} = 0
• C) {x_3} = 0
• D) {x_1} = 0