The horizontal asymptote for the function $y = \frac{{{x^2} - 25}}{{{x^2} - x - 2}}$ is . . . . . . . .

• A) y = 1
• B) y= 0
• C) y = 5/4
• D) None

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

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

A function f is said to have no absolute maximum when . . . . . . . . . . . . . . .

• A) As $x \to 0$ , then $f(x) \to 0$
• B) As $x \to \infty$ ,then $f(x) \to \infty$
• C) As $x \to 0$ , then $f(x) \to \infty$
• D) As $x \to \infty$ , then $f(x) \to 0$

Which of the following is true for the function$f(x) = 3{x^2} - 6x - 10{\text{ }};x \in \mathbb{R}$

• A) None of the other
• B) It has a critical point at x = 1
• C) It has no critical point
• D) It has a critical number at x= -1

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 $y = {b_m}$ is the vertical asymptote.
• B) The line $y = {a_n}$ is the horizontal asymptote.
• C) The line $y = \frac{{{a_n}}}{{{b_m}}}$ is the horizontal asymptote.
• D) The rational function r(x) becomes unbounded for large values of x.

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

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

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

If for a rational function $r(x) = \frac{{p(x)}}{{q(x)}}$ , degree of p(x) is strictly one great than the degree of q(x) then r(x) will have a/an . . . . . . .

• A) Horizontal asymptote
• B) Vertical asymptote
• C) y = 1 line as asymptote
• D) Oblique asymptote

If two real branches of a curve passing through the double point are real and tangents to them are distinct then the double point is a

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

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

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

• A) <0
• B) ≥0
• C) =0
• D) >0

$$If{\text{ }}f\left( x \right) = {\left( {x - 1} \right)^{2/3}} - 3\left( {x - 1} \right){\text{ }}and{\text{ }}f'\left( x \right) = \frac{2}{{3{{\left( {x - 1} \right)}^{1/3}}}} - 3{\text{ }}then{\text{ }}the{\text{ }}singular{\text{ }}point{\text{ }}of{\text{ }}f\left( x \right){\text{ }}is$$

• A) 0
• B) 1
• C) -1
• D) 2

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

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

Which of the following cannot be an asymptote to the graph of a function $y = f(x)$.

• A) Curved Asymptote
• B) Oblique Asymptote
• C) Horizontal Asymptote
• D) Vertical Asymptote

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

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

A Point on the curve through which more than one branch of the curve pass is called

• A) Singular point
• B) Critical point
• C) None of the other
• D) Multiple Point

A function f has an absolute maximum at a point “c” if ......

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

In polar coordinates, asymptote to a curve can be defined with the help of parameter(s). . . . . . .

• A) The distance $r$ and the initial line.
• B) The distance $r$ and the angle $\theta$.
• C) The distance $r$ only.
• D) The angle $\theta$ only.

If two real branches of a curve passing through the double point are real and tangents to them are coincident then the double point is a

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

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

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