Euclidean metric on R is defined by …………

• A) $$d(x,y) = \left| x \right| - \left| y \right|$$
• B) $$d(x,y) = \sqrt {\left| {x - y} \right|}$$
• C) $$d(x,y) = \left| {x - y} \right|$$
• D) $$d(x,y) = \left| x \right| + \left| y \right|$$

Triangular inequality states that length of one side of a triangle is…………… than the sum of the length of other two sides.

• A) less
• B) greater
• C)
• D)

If d is a metric on R defined by$$d(x,y) = \left| x \right| + \left| y \right|$$ and d(x,y)=1, then ---------.

• A) x=y=1
• B) No conclusion
• C) x=0 and y=1
• D) x=1 and y=0

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

• 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) $$\mid {x_1} - {x_2}\mid + \mid {y_1} - {y_2}\mid$$

$$Which{\text{ }}of{\text{ }}the{\text{ }}following{\text{ }}is{\text{ }}an{\text{ }}example{\text{ }}of{\text{ }}a{\text{ }}metric{\text{ }}space?$$

• A) $$The{\text{ }}set{\text{ }}of{\text{ }}all{\text{ }}integers{\text{ }}with{\text{ }}the{\text{ }}usual{\text{ }}addition.$$
• B) $$The{\text{ }}set{\text{ }}of{\text{ }}all{\text{ }}real{\text{ }}numbers{\text{ }}with{\text{ }}the{\text{ }}Euclidean{\text{ }}distance{\text{ }}d\left( {a,b} \right) = \mid a - b\mid .$$
• C) $$The{\text{ }}set{\text{ }}of{\text{ }}all{\text{ }}complex{\text{ }}numbers{\text{ }}with{\text{ }}the{\text{ }}operation{\text{ }}of{\text{ }}multiplication.$$
• D) $$The{\text{ }}set{\text{ }}of{\text{ }}all{\text{ }}positive{\text{ }}integers{\text{ }}with{\text{ }}the{\text{ }}function{\text{ }}d\left( {a,b} \right) = a + b.$$

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

• A) False
• B) True
• C)
• D)

Set of real numbers with distance function d(x,y)=|x-y| satisfy the reflexive property.

• A) True
• B) False
• C)
• D)

$${\text{Which of the following functions is a metric on R?}}$$

• A) $$d\left( {x,y} \right) = \mid x + y{\mid ^2}$$
• B) $$d\left( {x,y} \right) = \mid x + y\mid$$
• C) $$d\left( {x,y} \right) = \mid x\left| + \right|y\mid$$
• D) $$d\left( {x,y} \right) = \sqrt {\mid x - y\mid }$$

Set of real numbers with distance function d(x,y)=|x-y| satisfy the symmetric property.

• A) False
• B) True
• C)
• D)

Symmetric property states that for all real numbers x and y --------- .

• A) $$d(x,y) \geqslant d(y,x)$$
• B) $$d(x,y) \leqslant d(y,x)$$
• C) $$d(x,y) = d(y,x)$$
• D) $$d(x,y) = 0$$

If d is a metric on R defined by$$d(x,y) = \left| x \right| + \left| y \right|$$ , then d(-2,0)=---------.

• A) -2
• B) 0
• C) -0.50
• D) 2

$${\text{Which of the following functions is usual or Eucledian metric on }}{R^2}{\text{ for points }}{P_1} = \left( {{x_1},{y_1}} \right){\text{ and }}{P_2} = \left( {{x_2},{y_2}} \right){\text{?}}$$

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

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

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

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

• A) 7
• B) 5
• C) 8
• D) 6

$${\text{Which of the following points is at a taxicab distance of 4 from the point }}\left( {2,3} \right) \in {R^2}{\text{?}}$$

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

If d is a usual(natural) metric on R, then d(1,0.50)= ----------.

• A) 1
• B) 1.50
• C) -0.50
• D) 0.50

If $${d_0}$$ is a discrete metric on R then $${d_0}( - 2, - 2) = - - - - - .$$

• A) 0
• B) -2
• C) -1
• D) 1

$$\begin{gathered} {\text{Given the set }}X = \{ 1,2,3\} {\text{ with the discrete metric defined by }}d(a,b) = 1{\text{ if }}a \ne b{\text{ and }}d(a,a) = 0{\text{, what is the distance between 1 and 3?}} \hfill \\ \hfill \\ \end{gathered}$$

• A) 2
• B) 1
• C) 0
• D) 3

If d is a usual metric on$${\mathbb{R}^2}$$ , then d((0,0),(-1,-1)) =-----------.

• A) 0
• B) 2
• C) 1
• D) $$\sqrt 2$$

Dot product $\overrightarrow u \cdot \overrightarrow v$ of two vectors $\overrightarrow u$ and $\overrightarrow v$ is given by----.

• A) $\left| {\overrightarrow u } \right|\left| {\overrightarrow v } \right|\csc \theta$
• B) $\left| {\overrightarrow u } \right|\left| {\overrightarrow v } \right|\cos \theta$
• C) $\left| {\overrightarrow u } \right|\left| {\overrightarrow v } \right|\tan \theta$
• D) $\left| {\overrightarrow u } \right|\left| {\overrightarrow v } \right|\sin \theta$