A function f has an absolute minimum at a point “d” if . . . . .

• A) $f(d)f(x)\, = 0\,\,\,\,\,\,\forall \,\,\,x \in D$
• B) $f(d)f(x)\, \geqslant 0\,\,\,\,\,\,\forall \,\,\,x \in D$
• C) $f(d) \geqslant f(x)\,\,\,\,\,\,\,\forall \,\,\,x \in D$
• D) $f(d) < f(x)\,\,\,\,\,\,\,\forall \,\,\,x \in D$

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

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

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

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

A singular point of a curve is called ----------.

• A) None of the other
• B) Cusp
• C) Node
• D) Both Node and Cusp

A point on the curve through which two branches of the curve does not pass is called

• A) Double point
• B) Singular point
• C) None of the other
• D) Triple point

A function f has a local maximum at a point “c” if ……………….. in some open interval containing “c”.

• A) $f(c) \leqslant f(x)\,\,\,\,\,\,$
• B) $f(c) \geqslant f(x)\,\,\,\,$
• C) $f(c) + f(x) \leqslant 0\,$
• D) $f(c)f(x)\, \geqslant 0$

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

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

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

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

Which of the following is true for the function $f(x) = 4{x^2} - 20x - 1000{\text{ }};x \in \mathbb{R}$

• A) It has a critical point at x= 1000
• B) It has no critical point
• C) None of the other
• D) It has a critical number at x = 5/2

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

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

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

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

For a function “f” if at some point “c”, f'(c) = 0 or does not exist at it, then “c” is called . . . . . . . . . .

• A) Critical point
• B) Inflation point
• C) Unique point
• D) Singular point

If two tangents at the origin are imaginary then the origin is a

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

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

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

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

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

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

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

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

For the graph of a function $y = f(x)$ , asymptote is a line that as $x \to \infty$, . . . . . . . .

• A) Gets arbitrarily close to the curve but never touches it.
• B) Crosses the curve after constant intervals.
• C) Remains at a constant distance from the curve.
• D) Gets arbitrarily away from the curve.

For the graph of the function $y = f(x)$ , a line $y = b$ is called an asymptote if for the distance $d = \left| {f({x_i}) - b} \right|$

• A) $$d \to b$$
• B) $$d \to 1$$
• C) $$d \to \infty$$
• D) $$d \to 0$$

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