$\lim_{x \to \frac{\pi}{2}} \dfrac{\cot x - \cos x}{(\pi - 2x)^3}$ equals
(1) $\dfrac{1}{24}$
(2) $\dfrac{1}{16}$
(3) $\dfrac{1}{8}$
(4) $\dfrac{1}{4}$

$\lim _ { x \rightarrow 0 } \frac { ( 27 + x ) ^ { \frac { 1 } { 3 } } - 3 } { 9 - ( 27 + x ) ^ { \frac { 2 } { 3 } } }$ equals
(1) $- \frac { 1 } { 6 }$
(2) $\frac { 1 } { 6 }$
(3) $\frac { 1 } { 3 }$
(4) $- \frac { 1 } { 3 }$

$\lim _ { x \rightarrow 1 ^ { - } } \frac { \sqrt { \pi } - \sqrt { 2 \sin ^ { - 1 } x } } { \sqrt { 1 - x } }$ is equal to
(1) $\sqrt { \pi }$
(2) $\sqrt { \frac { 2 } { \pi } }$
(3) $\frac { 1 } { \sqrt { 2 \pi } }$
(4) $\sqrt { \frac { \pi } { 2 } }$

$\lim_{x \rightarrow 0} \frac{\sin^2 x}{\sqrt{2} - \sqrt{1 + \cos x}}$ equals
(1) $4\sqrt{2}$
(2) $2\sqrt{2}$
(3) $\sqrt{2}$
(4) 4

$\lim _ { x \rightarrow a } \frac { ( a + 2 x ) ^ { \frac { 1 } { 3 } } - ( 3 x ) ^ { \frac { 1 } { 3 } } } { ( 3 a + x ) ^ { \frac { 1 } { 3 } } - ( 4 x ) ^ { \frac { 1 } { 3 } } } ( a \neq 0 )$ is equal to:
(1) $\left( \frac { 2 } { 9 } \right) \left( \frac { 2 } { 3 } \right) ^ { \frac { 1 } { 3 } }$
(2) $\left( \frac { 2 } { 3 } \right) ^ { \frac { 4 } { 3 } }$
(3) $\left( \frac { 2 } { 9 } \right) ^ { \frac { 4 } { 3 } }$
(4) $\left( \frac { 2 } { 3 } \right) \left( \frac { 2 } { 9 } \right) ^ { \frac { 1 } { 3 } }$

If $\alpha$ is the positive root of the equation, $p ( x ) = x ^ { 2 } - x - 2 = 0$, then $\lim _ { x \rightarrow \alpha ^ { + } } \frac { \sqrt { 1 - \cos p ( x ) } } { x + \alpha - 4 }$ is equal to
(1) $\frac { 3 } { 2 }$
(2) $\frac { 3 } { \sqrt { 2 } }$
(3) $\frac { 1 } { \sqrt { 2 } }$
(4) $\frac { 1 } { 2 }$

$\lim_{x\rightarrow 0} \frac{x\left(e^{\left(\sqrt{1+x^2+x^4}-1\right)/x}-1\right)}{\sqrt{1+x^2+x^4}-1}$
(1) is equal to $\sqrt{e}$
(2) is equal to 1
(3) is equal to 0
(4) does not exist

If $\lim _ { x \rightarrow 0 } \frac { a x - \left( e ^ { 4 x } - 1 \right) } { a x \left( e ^ { 4 x } - 1 \right) }$ exists and is equal to $b$, then the value of $a - 2 b$ is $\underline{\hspace{1cm}}$.

If $\lim _ { x \rightarrow 0 } \frac { a e ^ { x } - b \cos x + c e ^ { - x } } { x \sin x } = 2$, then $a + b + c$ is equal to $\_\_\_\_$.

If $\lim _ { x \rightarrow 0 } \left[ \frac { \alpha x e ^ { x } - \beta \log _ { e } ( 1 + x ) + \gamma x ^ { 2 } e ^ { - x } } { x \sin ^ { 2 } x } \right] = 10 , \alpha , \beta , \gamma \in R$, then the value of $\alpha + \beta + \gamma$ is $\underline{\hspace{1cm}}$.

$\lim _ { x \rightarrow 0 } \frac { \sin ^ { 2 } \left( \pi \cos ^ { 4 } x \right) } { x ^ { 4 } }$ is equal to :
(1) $2 \pi ^ { 2 }$
(2) $\pi ^ { 2 }$
(3) $4 \pi ^ { 2 }$
(4) $4 \pi$

$\lim _ { x \rightarrow 0 } \frac { \cos ( \sin x ) - \cos x } { x ^ { 4 } }$ is equal to
(1) $\frac { 1 } { 3 }$
(2) $\frac { 1 } { 6 }$
(3) $\frac { 1 } { 4 }$
(4) $\frac { 1 } { 12 }$

$\lim_{x \to \frac{\pi}{2}} \tan^2 x \left[(2\sin^2 x + 3\sin x + 4)^{\frac{1}{2}} - (\sin^2 x + 6\sin x + 2)^{\frac{1}{2}}\right]$ is equal to
(1) $\frac{1}{12}$
(2) $-\frac{1}{18}$
(3) $-\frac{1}{12}$
(4) $\frac{1}{6}$

$\lim_{x \rightarrow \frac{\pi}{4}} \frac{8\sqrt{2} - (\cos x + \sin x)^7}{\sqrt{2} - \sqrt{2}\sin 2x}$ is equal to
(1) 14
(2) 7
(3) $14\sqrt{2}$
(4) $7\sqrt{2}$

Let $x = 2$ be a root of the equation $x ^ { 2 } + p x + q = 0$ and $f ( x ) = \left\{ \begin{array} { c l } \frac { 1 - \cos \left( x ^ { 2 } - 4 p x + q ^ { 2 } + 8 q + 16 \right) } { ( x - 2 p ) ^ { 4 } } , & x \neq 2 p \\ 0 , & x = 2 p \end{array} \right.$. Then $\lim _ { x \rightarrow 2 p ^ { + } } [ f ( x ) ]$ where $[ \cdot ]$ denotes greatest integer function, is
(1) 2
(2) 1
(3) 0
(4) $- 1$

$\lim _ { x \rightarrow 0 } \left( \left( \frac { 1 - \cos ^ { 2 } ( 3 x ) } { \cos ^ { 3 } ( 4 x ) } \right) \left( \frac { \sin ^ { 3 } ( 4 x ) } { \left( \log _ { e } ( 2 x + 1 ) \right) ^ { 5 } } \right) \right)$ is equal to
(1) 15
(2) 9
(3) 18
(4) 24

$\lim _ { x \rightarrow 0 } \frac { e ^ { 2 \sin x } - 2 \sin x - 1 } { x ^ { 2 } }$
(1) is equal to $-1$
(2) does not exist
(3) is equal to 1
(4) is equal to 2

If $a = \lim _ { x \rightarrow 0 } \frac { \sqrt { 1 + \sqrt { 1 + x ^ { 4 } } } - \sqrt { 2 } } { x ^ { 4 } }$ and $b = \lim _ { x \rightarrow 0 } \frac { \sin ^ { 2 } x } { \sqrt { 2 } - \sqrt { 1 + \cos x } }$, then the value of $a b ^ { 3 }$ is:
(1) 36
(2) 32
(3) 25
(4) 30

If $\lim _ { x \rightarrow 0 } \frac { 3 + \alpha \sin x + \beta \cos x + \log _ { e } ( 1 - x ) } { 3 \tan ^ { 2 } x } = \frac { 1 } { 3 }$, then $2 \alpha - \beta$ is equal to :
(1) 2
(2) 7
(3) 5
(4) 1

Let $\mathrm { a } > 0$ be a root of the equation $2 x ^ { 2 } + x - 2 = 0$. If $\lim _ { x \rightarrow \frac { 1 } { \mathrm { a } } } \frac { 16 \left( 1 - \cos \left( 2 + x - 2 x ^ { 2 } \right) \right) } { ( 1 - \mathrm { a } x ) ^ { 2 } } = \alpha + \beta \sqrt { 17 }$, where $\alpha , \beta \in Z$, then $\alpha + \beta$ is equal to $\_\_\_\_$

The value of $\lim _ { x \rightarrow 0 } 2 \left( \frac { 1 - \cos x \sqrt { \cos 2 x } \sqrt [ 3 ] { \cos 3 x } \ldots \ldots \sqrt [ 10 ] { \cos 10 x } } { x ^ { 2 } } \right)$ is $\_\_\_\_$

Q85. Let $\mathrm { a } > 0$ be a root of the equation $2 x ^ { 2 } + x - 2 = 0$. If $\lim _ { x \rightarrow \frac { 1 } { \mathrm { a } } } \frac { 16 \left( 1 - \cos \left( 2 + x - 2 x ^ { 2 } \right) \right) } { ( 1 - \mathrm { a } x ) ^ { 2 } } = \alpha + \beta \sqrt { 17 }$, where $\alpha , \beta \in Z$, then $\alpha + \beta$ is equal to $\_\_\_\_$ Q86. Let the mean and the standard deviation of the probability distribution

X	$\alpha$	1	0	- 3
$\mathrm { P } ( \mathrm { X } )$	$\frac { 1 } { 3 }$	K	$\frac { 1 } { 6 }$	$\frac { 1 } { 4 }$

be $\mu$ and $\sigma$, respectively. If $\sigma - \mu = 2$, then $\sigma + \mu$ is equal to $\_\_\_\_$

Q85. The value of $\lim _ { x \rightarrow 0 } 2 \left( \frac { 1 - \cos x \sqrt { \cos 2 x } \sqrt [ 3 ] { \cos 3 x } \ldots \ldots \sqrt [ 10 ] { \cos 10 x } } { x ^ { 2 } } \right)$ is

$$\lim _ { x \rightarrow 0 } \frac { \ln \left( \sec ( e x ) \sec \left( e ^ { 2 } x \right) \sec \left( e ^ { 3 } x \right) \ldots \sec \left( e ^ { 10 } x \right) \right) } { \left( e ^ { 2 } - e ^ { 2 } \cos x \right) e ^ { 2 } / ( 1 - G x ) }$$ (A) $\frac { \mathrm { e } ^ { 18 } - 1 } { \mathrm { e } ^ { 2 } - 1 }$ (B) $\frac { \mathrm { e } ^ { 20 } - 1 } { \mathrm { e } ^ { 2 } - 1 }$ (C) $\frac { e ^ { 2 } - 1 } { e ^ { 2 } - 1 }$ (D) $\frac { \mathrm { e } ^ { 22 } - 1 } { \mathrm { e } ^ { 2 } - 1 }$

$$\lim _ { x \rightarrow 0 } \frac { x + \arcsin x } { \sin 2 x }$$
What is the value of this limit?
A) 0
B) 1
C) $\frac { 2 } { 3 }$
D) $\frac { 4 } { 3 }$
E) $\frac { 1 } { 6 }$