Let f : R $$\to$$ R be defined as f(x) = \left\{ {\matrix{ {{{\lambda \left| {{x^2} - 5x + 6} \right|} \over {\mu (5x - {x^2} - 6)}},} & {x 2} \cr {\mu ,} & {x = 2} \cr } } \right. where [x] is the greatest integer is than or equal to x. If f is continuous at x = 2, then $$\lambda$$ + $$\mu$$ is equal to :

The value of \mathop {\lim }\limits_{x \to 0} \left( {{x \over {\root 8 \of {1 - \sin x} - \root 8 \of {1 + \sin x} }}} \right) is equal to :

Let $$f:[0,\infty )\to [0,3]$$ be a function defined by f(x) = \left\{ {\matrix{ {\max \{ \sin t:0 \le t \le x\} ,} & {0 \le x \le \pi } \cr {2 + \cos x,} & {x > \pi } \cr } } \right. Then which of the following is true?

Let f:\left( { - {\pi \over 4},{\pi \over 4}} \right) \to R be defined as f(x) = \left\{ {\matrix{ {{{(1 + |\sin x|)}^{{{3a} \over {|\sin x|}}}}} & , & { - {\pi \over 4} < x < 0} \cr b & , & {x = 0} \cr {{e^{\cot 4x/\cot 2x}}} & , & {0 < x < {\pi \over 4}} \cr } } \right. If f is continuous at x = 0, then the value of 6a + b2 is equal to :

Let f : R $$\to$$ R be a function such that f(2) = 4 and f'(2) = 1. Then, the value of \mathop {\lim }\limits_{x \to 2} {{{x^2}f(2) - 4f(x)} \over {x - 2}} is equal to :

Let [t] denote the greatest integer less than or equal to t. Let f(x) = x $$-$$ [x], g(x) = 1 $$-$$ x + [x], and h(x) = min{f(x), g(x)}, x $$\in$$ [$$-$$2, 2]. Then h is :

\mathop {\lim }\limits_{x \to 2} \left( {\sum\limits_{n = 1}^9 {{x \over {n(n + 1){x^2} + 2(2n + 1)x + 4}}} } \right) is equal to :

If $$\alpha$$, $$\beta$$ are the distinct roots of x2 + bx + c = 0, then \mathop {\lim }\limits_{x \to \beta } {{{e^{2({x^2} + bx + c)}} - 1 - 2({x^2} + bx + c)} \over {{{(x - \beta )}^2}}} is equal to :

If $$\underset{x\to \infty }{\lim }\left(\sqrt{x^2-x+1}-ax\right)=b$$, then the ordered pair (a, b) is :

The function $$f(x)=\left|x^2-2x-3\right|.e^{|9x^2-12x+4|}$$ is not differentiable at exactly :

If the function f(x) = \left\{ {\matrix{ {{1 \over x}{{\log }_e}\left( {{{1 + {x \over a}} \over {1 - {x \over b}}}} \right)} & , & {x 0} \cr } } \right. is continuous at x = 0, then {1 \over a} + {1 \over b} + {4 \over k} is equal to :

\mathop {\lim }\limits_{x \to 0} {{{{\sin }^2}\left( {\pi {{\cos }^4}x} \right)} \over {{x^4}}} is equal to :

If \alpha = \mathop {\lim }\limits_{x \to {\pi \over 4}} {{{{\tan }^3}x - \tan x} \over {\cos \left( {x + {\pi \over 4}} \right)}} and $$\beta =\underset{x\to 0}{\lim }(\cos x{)}^{\cot x}$$ are the roots of the equation, ax2 + bx $$-$$ 4 = 0, then the ordered pair (a, b) is :

Let f be any continuous function on [0, 2] and twice differentiable on (0, 2). If f(0) = 0, f(1) = 1 and f(2) = 2, then

A possible value of \tan \left( {{1 \over 4}{{\sin }^{ - 1}}{{\sqrt {63} } \over 8}} \right) is :

cosec\left[ {2{{\cot }^{ - 1}}(5) + {{\cos }^{ - 1}}\left( {{4 \over 5}} \right)} \right] is equal to :

If {{{{\sin }^1}x} \over a} = {{{{\cos }^{ - 1}}x} \over b} = {{{{\tan }^{ - 1}}y} \over c}; $$0<x<1$$, then the value of \cos \left( {{{\pi c} \over {a + b}}} \right) is :

If 0 < a, b < 1, and tan$$-$$1a + tan$$-$$1b = {\pi \over 4}, then the value of (a + b) - \left( {{{{a^2} + {b^2}} \over 2}} \right) + \left( {{{{a^3} + {b^3}} \over 3}} \right) - \left( {{{{a^4} + {b^4}} \over 4}} \right) + ..... is :

Given that the inverse trigonometric functions take principal values only. Then, the number of real values of x which satisfy {\sin ^{ - 1}}\left( {{{3x} \over 5}} \right) + {\sin ^{ - 1}}\left( {{{4x} \over 5}} \right) = {\sin ^{ - 1}}x is equal to :

The sum of possible values of x for tan$$-$$1(x + 1) + cot$$-$$1\left( {{1 \over {x - 1}}} \right) = tan$$-$$1\left( {{8 \over {31}}} \right) is :