The function $f(x) = \left\{ {\begin{array}{*{20}{l}} {x,}&{0 \leqslant x < 1,} \\\ {2,}&{1 \leqslant x \leqslant 2,} \end{array}} \right.$ is

• A) is nonincreasing on $$I = \left[ {0,2} \right].$$
• B) is undefined on $$I = \left[ {0,2} \right].$$
• C) is having negative values on $$I = \left[ {0,2} \right].$$
• D) is nondecreasing on $$I = \left[ {0,2} \right].$$

$\begin{array}{l} The\,series\,\sum\limits_{}^{} {a_n b_n } \,\,\,converges\,\,if\,\,a_{n + 1} \le \,a_{n\,} \,for\,\,n\, \ge \,k,\,\,\mathop {\lim }\limits_{x \to \infty } \,\,a_n = - - - - - ,\,\,and\, \\\ |b_k + \,b_{k + 1} + ... + b_n |\,\, \le \,\,M,\,\,for\,\,some\,\,constant\,M. \\\ \end{array}$

• A) $0\,$
• B) $None\,of\,these$
• C) $1$
• D) $- 1$

$Investigate\,\,the\,\,\lim it\,\,of\,\,\,\mathop {\lim }\limits_{x \to 0} \,\,x\sin \frac{{1\,\,}}{x} = - - - - .$

• A) $1$
• B) $x$
• C) $none\,\,of\,\,these$
• D) $0$

The inverse of the given function $f(x) = 2x + 4,\quad 0 \leqslant x \leqslant 2,$ is

• A) ${f^{ - 1}}(y) = \frac{{y - 4}}{4}$
• B) ${f^{ - 1}}(y) = \frac{{y - 2}}{4}.$
• C) ${f^{ - 1}}(y) = \frac{{x - 2}}{4}.$
• D) ${f^{ - 1}}(y) = \frac{{y - 4}}{2}.$

$Investigate\,\,the\,\,\lim it\,\,of\,\,\,\mathop {\lim }\limits_{x \to 0} \,\,\frac{{\sin x\,\,}}{x} = - - - - .$

• A) $x\,$
• B) $0$
• C) $1\,$
• D) $none\,\,of\,\,these$

If a function $f$ is continuous on the closed interval then $f$ attains its

• A) extreme values in the closed interval.
• B) may or may not attain extreme values in the closed interval.
• C) derivative in the closed interval.
• D) extreme values at one point in the closed interval.

$The\,series\,\sum\limits_{}^{} {( - 1)^n a_n } \,\,\,converges\,\,\,\,if\,\,0 \le a_{n + 1} \le \,a_{n\,} \,and\,\mathop {\lim }\limits_{x \to \infty } \,\,a_n = \, - - - - - .$

• A) $- 1$
• B) $0\,\,$
• C) $None\,\,of\,\,these\,\,$
• D) $1$

If $f$ is differentiable at a local extreme point $x_0\in D_{f}^{0},$ then

• A) $f'(x_0)=~0.$
• B) $f'(x_0) \neq~0.$
• C) $x_0=~0.$
• D) $|f'(x_0)|<~0.$

$Investigate\,\,the\,\,\lim it\,\,of\,\,\,\mathop {\lim }\limits_{x \to 0^ - } \,\frac{1}{x} = - - - - .$

• A) $0$
• B) $\infty \,$
• C) $none\,\,of\,\,these$
• D) $- \infty$

$Investigate\,\,the\,\,\lim it\,\,of\,\,\,\mathop {\lim }\limits_{x \to 0} \,\,2x\sin \sqrt x = - - - - .$

• A) $x$
• B) $0$
• C) $1\,$
• D) $none\,\,of\,\,these$

$$The\,\,\,function\,\,g\left( x \right) = - x^3 \,\,is\,\,decrea\sin g\,\,on\,\, - - - - - .\,$$

• A) $$\left( { - \infty \,,0} \right)$$
• B) $$none\,\,of\,\,these\,$$
• C) $$\left( { - \infty ,\infty } \right)$$
• D) $$[0,\infty )\,$$

$Investigate\,\,the\,\,\lim it\,\,of\,\,\,\mathop {\lim }\limits_{x \to 0^ + } \,\,\left( {\frac{{\left| x \right|}}{x} + x} \right) = - - - - .$

• A) $none\,\,of\,\,these$
• B) $- 1$
• C) $1\,$
• D) $0$

lf $f:I \to \mathbb{R}$ has a derivative at $c \in \,I$ , then $f$ is

• A) having one sided limit.
• B) discontinuous at c.
• C) undefined at c.
• D) continuous at c.

Let f: A → B and g : B → C be one - one functions. Then, gof: A → C is

• A) onto
• B) one-one
• C) not onto
• D) does not exist

Which function has $0/0$ form

• A) $f(x) = \frac{e^{-x}}{x}$ at $x=2$
• B) $f(x) = 2\ln x$ at $x=2$.
• C) $f(x) = \frac{\sin \pi x}{x-1}$ at $x=1$.
• D) $f(x) = \cos x + x$ at $x =0$.

The inverse of the function $f(x) = {x^2},$ is

• A) ${f^{ - 1}}(y) = \sqrt y .$
• B) ${f^{ - 1}}(y) = {y^2}$
• C) ${f^{ - 1}}(y) = \frac{1}{{{y^2}}}$
• D) ${f^{ - 1}}(y) = \frac{1}{{{x^2}}}$

$The\,series\,\sum\limits_{}^{} {( - 1)^n a_n } \,\,\, - - - - - - \,\,\,if\,\,0 \le a_{n + 1} \le \,a_{n\,} \,and\,\mathop {\lim }\limits_{x \to \infty } \,\,a_n = 0$

• A) $converges$
• B) $diverges$
• C)
• D)

$\begin{array}{l} The\,\,radius\,\,\,of\,\,\,convergence\,\,\,of\,\,\,\,\sum\limits_{}^{} {a_n \,(x - x_n )^n \,\,\,is\,\,\,given\,\,\,by\,\, - - - - = \mathop {\lim }\limits_{n \to \infty } } |\frac{{a_{n + 1} }}{{a_n }}|\, \\\ if\,the\,limit\,exists\,in\,the\,extended\,real\,system. \\\ \end{array}$

• A) $R$
• B) $\frac{1}{R}$
• C) $n$
• D) $none\,of\,these$

$\begin{array}{l} The\,series\,\sum\limits_{}^{} {a_n b_n } \,\,\,converges\,\,if\,\,a_{n + 1} \le \,a_{n\,} \,for\,\,n\, - - - k,\,\,\mathop {\lim }\limits_{x \to \infty } \,\,a_n = 0,\,\,and\, \\\ |b_k + \,b_{k + 1} + ... + b_n | \le M,\,\,for\,\,some\,\,constant\,M. \\\ \end{array}$

• A) $\ge \,$
• B) $\, \le$
• C) $>$
• D) $<$

$$The\,\,\,function\,\,g\left( x \right) = x^2 \,\,is\,\,inrea\sin g\,\,on\,\, - - - - - .\,$$

• A) $$none\,\,of\,\,these\,\,\,$$
• B) $$[0,\infty )$$
• C) $$\left( { - \infty , - 1} \right)\,$$
• D) $$\left( { - \infty \,,0} \right)$$