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

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

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

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

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

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

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 m = n, then . . . . . . . .

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

The{\text{ }}functions{\text{ }}y = 4\sqrt x {\text{ }}and{\text{ }}y = - 4\sqrt x {\text{ }}are{\text{ }}two{\text{ }}branches{\text{ }}of{\text{ }}parabola

• A) {y^2} = 4x
• B) {y^2} = 16x
• C) y = 16x
• D) {y^2} =- 4x

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) All of them
• B) Isolated point
• C) Node
• D) Cusp

Which of the following is not true for the function f(y) = {y^2};\,\,\,1 \leqslant \,y\, \leqslant \,10

• A) None of the other
• B) It has a critical point at x = 0
• C) It has no critical point
• D) It has a critical point at x = 5

For the function y = f(x) , the line x = c is called the vertical asymptote if and only if . . . . . .

• A) \mathop {\lim }\limits_{x \to {c^ + }} f(x) = 0
• B) \mathop {\lim }\limits_{x \to {c^ + }} f(x) = \pm \infty
• C) \mathop {\lim }\limits_{x \to {c^ + }} f(x) = \pm c
• D) \mathop {\lim }\limits_{x \to \infty } f(x) = \pm c

\[The{\text{ }}functions{\text{ }}y = 4\sqrt x {\text{ }}and{\text{ }}y = - 4\sqrt x {\text{ }}are{\text{ }}two{\text{ }}branches{\text{ }}of{\text{ }}parabola\]

• A) \[{y^2} =- 4x\]
• B) \[{y^2} = 4x\]
• C) \[y = 16x\]
• D) \[{y^2} = 16x\]

For the function y = f(x) , the line y = b is called the horizontal asymptote iff as . . . . . .

• A) x \to 0, y \to \infty
• B) x \to 0, y \to b
• C) x \to \infty, y \to b
• D) x \to \infty, y \to 0

\[The{\text{ }}critical{\text{ }}points{\text{ }}of{\text{ }}the{\text{ }}polynomial{\text{ }}p\left( x \right) = {x^3} - 2{x^2}\,are\]

• A) 0,-2
• B) 0,4/3
• C) ±2
• D) 2

\[The{\text{ }}critical{\text{ }}points{\text{ }}of{\text{ }}the{\text{ }}polynomial{\text{ }}p\left( x \right) = {x^3} - 3x + 1{\text{ }}are\]

• A) 0
• B) ±1
• C) -1
• D) 1

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

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

\[For{\text{ }}the{\text{ }}curve{\text{ }}{x^3} + {y^3} - 3axy = 0,{\text{ }}the{\text{ }}tangents{\text{ }}at{\text{ }}the{\text{ }}origin{\text{ }}are{\text{ }}x = 0{\text{ }}and{\text{ }}y = 0,{\text{ }}then{\text{ }}the{\text{ }}origin{\text{ }}is{\text{ }}a\]

• A) Cusp
• B) Node
• C) None of these
• D) Isolated point

\[If{\text{ }}f'\left( x \right){\text{ }}has{\text{ }}the{\text{ }}same{\text{ }}sign{\text{ }}on{\text{ }}both{\text{ }}left{\text{ }}and{\text{ }}right{\text{ }}sides{\text{ }}of{\text{ }}{x_0}\,on{\text{ }}an{\text{ }}open{\text{ }}interval,{\text{ }}then{\text{ }}f.........\,relative{\text{ }}extremum/extrema{\text{ }}at{\text{ }}{x_0}.\]

• A) Does not have
• B) Three
• C) One
• D) Two

The{\text{ }}second{\text{ }}derivative{\text{ }}test{\text{ }}gives{\text{ }}no{\text{ }}information{\text{ }}if{\text{ }}f''\left( c \right),\,\left( {where{\text{ }}c{\text{ }}is{\text{ }}a{\text{ }}critical{\text{ }}point} \right)

• A) =0
• B) <0
• C) ≥0
• D) >0

A{\text{ }}point{\text{ }}on{\text{ }}the{\text{ }}curve{\text{ }}through{\text{ }}which{\text{ }}r{\text{ }}branches{\text{ }}of{\text{ }}the{\text{ }}curve{\text{ }}pass{\text{ }}is{\text{ }}called{\text{ }}Multiple{\text{ }}point{\text{ }}of

• A) sth order
• B) pth order
• C) rth order
• D) Multiple order

The{\text{ }}curve{\text{ }}\left( {{x^2} + {y^2}} \right)x - 2a{y^2} = 0{\text{ }}has{\text{ }}tangents{\text{ }}at{\text{ }}origin{\text{ }}as{\text{ }}{y^2} = 0,{\text{ }}then{\text{ }}the{\text{ }}origin{\text{ }}is{\text{ }}a

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

Let the straight line y = mx + c be an asymptote to the graph of y = f(x) , then which of the following is true.

• A) \mathop {\lim }\limits_{x \to 0} \left[ {f(x) - (mx + c)} \right] = \infty
• B) \mathop {\lim }\limits_{x \to \infty } \left[ {f(x) - (mx + c)} \right] = 0
• C) \mathop {\lim }\limits_{x \to \infty } \left[ {f(x) + (mx + c)} \right] = \infty
• D) \mathop {\lim }\limits_{x \to \infty } \left[ {f(x) \times (mx + c)} \right] = 0