The horizontal asymptote for the function y = \frac{{2x - 8}}{{{x^2} - 16}} is . . . . . . . .

• A) y = 0
• B) y = 16
• C) y = 1
• D) y = 2

Let{\text{ f(x) = 3x - 5, }}and{\text{ }}\frac{{df(x)}}{{dx}} > 0,\,then

• A) f(x) must has a critical point
• B) f(x) is an increasing function
• C) None of the other
• D) f(x) is a decreasing function

Let P(r,\theta ) be a point on a straight line “L” and “p” is the length of the perpendicular drawn on the line from pole, making an angle \alpha with the initial line, then the equation in polar form of “L” is . . . . . . . . .

• A) p = r\cos (\alpha \theta )
• B) p = r\cos (\theta + \alpha )
• C) p = r\cos (\theta - \alpha )
• D) p = \alpha r\cos (\theta )

If{\text{ }}{\left( {{f_{xy}}} \right)^2} - {f_{xx}}{f_{yy}} > {\text{ }}0,{\text{ }}then{\text{ }}the{\text{ }}double{\text{ }}point{\text{ }}\left( {x,{\text{ }}y} \right){\text{ }}would{\text{ }}be{\text{ }}a

• A) Both (a) and (b)
• B) Cusp
• C) Node
• D) Isolated point

The singular points of {x^2}{(x - 3)^2}y'' + (x - 3)y' + 3{x^2}y = \,0 are

• A) None of the other
• B) 4,3
• C) 1,3
• D) 0,3

\[The{\text{ }}graph{\text{ }}of{\text{ }}f\left( x \right) = \left| x \right|{\text{ }}at{\text{ }}x = 0{\text{ }}has{\text{ }}a\]

• A) Node
• B) Derivative
• C) Cusp
• D) Both (a) and (b)

Let{\text{ f(x) = }}4{x^2} + 10x + 50,{\text{ and }}\frac{{df(x)}}{{dx}}\, < \,0,\,\;at\,R(x,y),then

• A) f(x) is a decreasing function at P(x,y)
• B) None of the other
• C) f(x) is an increasing function at P(x,y)
• D) f(x) must has a critical point at P(x,y)

If{\text{ }}\frac{{df}}{{dx}}\, < \,0,\,\;at\,R(x,y),then

• A) f(x) is a decreasing function at R(x,y)
• B) f(x) must has a critical point at R(x,y)
• C) f(x) is an increasing function at R(x,y)
• D) None of the other

\[In{\text{ }}curve{\text{ }}{y^2} = x{\left( {x - a} \right)^2},{\text{ }}the{\text{ }}singular{\text{ }}point{\text{ }}\left( {a,0} \right){\text{ }}is{\text{ }}a{\text{ }} \ldots \ldots \,when{\text{ }}{f_{xx}}\left( {a,0} \right) = {\text{ }} - 2a,{\text{ }}{f_{yy}}\left( {a,0} \right) = 2{\text{ }}and{\text{ }}{f_{xy}}\left( {a,0} \right) = 0.\]

• A) Conjugate point
• B) Isolated point
• C) Node
• D) Cusp

According{\text{ }}to{\text{ }}the{\text{ }}second{\text{ }}derivative{\text{ }}test{\text{ }}f{\text{ }}has{\text{ }}a{\text{ }}relative{\text{ }}maximum{\text{ }}value{\text{ }}at{\text{ }}c{\text{ }}if{\text{ }}f''\left( c \right)

• A) <0
• B) >0
• C) ≤0
• D) =0

For a rational function r(x) = \frac{{p(x)}}{{q(x)}} = \frac{{{a_n}{x^n} + {a_{n - 1}}{x^{n - 1}} + ... + {a_0}}}{{{b_m}{x^m} + {b_{m - 1}}{x^{m - 1}} + ... + {b_0}}} , if n < m, then . . . . . . . .

• A) The line y = \frac{{{a_n}}}{{{b_m}}} is the horizontal asymptote.
• B) The line y = 0 is the horizontal asymptote.
• C) The line y = {b_m} is the vertical asymptote.
• D) The line y = {a_n} is the horizontal asymptote.

The horizontal asymptote for the function y = \frac{{2x + 1}}{{x + 2}} is . . . . . . . . .

• A) y =2
• B) y = 4
• C) y = 0
• D) None

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• A) data:image/png;base64,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
• B) data:image/png;base64,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
• C) None of these.
• D) data:image/png;base64,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

\frac{{{x^2}}}{4} + \frac{{{y^2}}}{9} + 1 = \frac{{{z^2}}}{{16}}, is an equation of ______.

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

\frac{{{x^2}}}{{16}} + \frac{{{y^2}}}{{25}} - \frac{{{z^2}}}{{36}} = - 1, is an equation of ______.

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

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

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

\frac{{{x^2}}}{{{3^2}}} + \frac{{{y^2}}}{{{4^2}}} = 1,\,\,\,{\text{is }}\,{\text{the equation }}\,{\text{of _____}}\,\,{\text{cylinder}}{\text{.}}

• A) hyperbolic
• B) parabolic
• C) elliptic
• D) circular

\frac{{{x^2}}}{{64}} + \frac{{{y^2}}}{{81}} + 1 = \frac{{{z^2}}}{{100}}, is an equation of ______.

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

{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