{x^2} + \frac{{{y^2}}}{{16}} - \frac{{{z^2}}}{{25}} = 1, is an equation of ______.

• A) Hyperboloid of one sheet
• B) Ellipsoid
• C) Paraboloid
• D) Hyperboloid of two sheets

\frac{{{x^2}}}{{36}} + \frac{{{y^2}}}{{49}} = z, is an equation of ______.

• A) Hyperboloid of two sheets
• B) Hyperbolic paraboloid
• C) Hyperboloid of one sheet
• D) Elliptic paraboloid

\frac{{{x^2}}}{{16}} + \frac{{{y^2}}}{{25}} = \frac{{{z^2}}}{{36}},\,\,\,{\text{is }}\,{\text{the equation }}\,{\text{of _____}}{\text{.}}

• A) paraboloid
• B) cylinder
• C) cone
• D) sphere

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• A) Only the x-axis
• B) None
• C) Only the xy-plane
• D) All coordinate planes and the origin

{\text{Trace}}\,\,{\text{of}}\,\,{\text{a cone}}\,\,\,{x^2} + \,\,\frac{{{y^2}}}{9} = {z^2},\,\,\,{\text{in}}\,\,xz - {\text{plane}}\,\,\,{\text{is }}\_\_\_\_\_.

• A) y = \pm \,3\,z
• B) x = \pm \,\,\frac{y}{3}
• C) no trace
• D) x = \pm \,\,z

\frac{{{x^2}}}{9} + \frac{{{y^2}}}{{16}} + \frac{{{z^2}}}{{16}} = 1,\,\,\,{\text{is }}\,{\text{the equation }}\,{\text{of _____}}{\text{.}}

• A) ellipsoid
• B) cone
• C) paraboloid
• D) hyperboloid

{\text{When}}\,\,a = b = c,\,\,\,{\text{the}}\,\,{\text{ellipsoid}}\,\,\,\,\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} + \frac{{{z^2}}}{{{c^2}}} = 1,\,\,\,{\text{becomes}}\,\,{\text{the}}\,\_\_\_\_\_.

• A) cone
• B) paraboloid
• C) hyperboloid
• D) sphere

\[\frac{{{x^2}}}{9} + \frac{{{y^2}}}{{16}} + \frac{{{z^2}}}{{16}} = 1,\,\,\,{\text{is }}\,{\text{the equation }}\,{\text{of _____}}{\text{.}}\]

• A) ellipsoid
• B) hyperboloid
• C) cone
• D) paraboloid

\frac{{{x^2}}}{{16}} + \frac{{{y^2}}}{{25}} = z, is an equation of ______.

• A) Elliptic paraboloid
• B) Hyperboloid of two sheets
• C) Hyperboloid of one sheet
• D) Hyperbolic paraboloid

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

• A) 0
• B) 1
• C) 3
• D) 2

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

• A) 2
• B) 1
• C) 0.5
• D) 0

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

• A) 2
• B) 0
• C) 1
• D) -1

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

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

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

• A) -1
• B) 2
• C) 1
• D) -2

$${\text{Determine if }}d\left( {x,y} \right) = \frac{{\mid x - y\mid }}{{1 + \mid x - y\mid }}{\text{ is a metric on R?}}$$

• A) $${\text{No}}$$
• B) $${\text{Yes}}$$
• C)
• D)

$${\text{If }}d\left( {a,b} \right) = \mid a - b\mid {\text{defines a metric on R, which of the following points are at distance 3 from the point 2?}}$$

• A) $${\text{ - 1 and 4}}$$
• B) $${\text{1 and - 4}}$$
• C) $${\text{2 and 5}}$$
• D) $${\text{ - 4 and 4}}$$

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

• A) 6
• B) 7
• C) 8
• D) 5

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

• A) 7
• B) 8
• C) 6
• D) 5

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

• A) 8
• B) 7
• C) 6
• D) 5

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

• A) 1
• B) 2
• C) 0
• D) -1