\begin{array}{l} If\,\,\,\sum\limits_{n = 1}^\infty {b_n } \,\,\,is\,\,rearrangement\,\,of\,\,an\,absolutely\,convergernt\,series \\\ \sum\limits_{n = 1}^\infty {a_n } ,\,\,then\,\,\sum\limits_{n = 1}^\infty {b_n } \,\,also\,\, - - - - - - \,\,absolutely,\,and\,\,to\,the\,\,same\,sum. \\\ \end{array}

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

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

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

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 nondecreasing on I = \left[ {0,2} \right].
• B) is having negative values on I = \left[ {0,2} \right].
• C) is nonincreasing on I = \left[ {0,2} \right].
• D) is undefined 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 = 0,\,\,and\, \\\ |b_k + \,b_{k + 1} + ... + b_n | - - - - M,\,\,for\,\,some\,\,constant\,M. \\\ \end{array}

• A) \, \le
• B) None\,of\,these
• C) \ge \,
• D) <

Generalized Mean Value Theorem says, If f and g are continuous on the closed interval [a,b] and differentiable on the open interval (a,b), then

• A) [g(b)-g(a)]=[f(b)-f(a)]g'(c) for some c in (a,b).
• B) g'(x)=f'(x) for all x in (a,b)
• C) [g(b)-g(a)]f'(c)=[f(b)-f(a)]g'(c) for some c in (a,b).
• D) g'(x)=0 for all x in (a,b)

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

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

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

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

\[ The\,\,\,function\,\,g\left( x \right) = x^2 \,\,is\, - - - - - \,on\,\,[0,\infty )\,.\, \]

• A) \[ inrea\sin g \]
• B) \[ decrea\sin g\,\, \]
• C)
• D)

Identify an unbounded function on [0,1]

• A) g(x) = \ln x
• B) h(x) = x^3
• C) f(x) =x^2+1
• D) v(x) = \sin x

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

• A) extreme values in the closed interval.
• B) derivative in the closed interval.
• C) extreme values at one point in the closed interval.
• D) may or may not attain extreme values 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) None\,\,of\,\,these\,\,
• B) - 1
• C) 1
• D) 0\,\,

\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) \, \le
• B) <
• C) >
• D) \ge \,

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

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

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

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

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

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

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

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

The Mean Value Theorem says, suppose that f is differentiable on [a,b], f'(a)\ne f'(b), and \mu is between f'(a) and f'(b).

• A) Then f'(c)=0 for some c in (a,b).
• B) Then f'(c)=\mu for some c in (a,b).
• C) Then f'(c) may be 0 or \mu for some c in (a,b).
• D) Then f'(c)\neq\mu for some c in (a,b).

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) 0
• D) - 1

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

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

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