Consider the following two binary relations on the set A = {a, b, c} : R1 = {(c, a), (b, b), (a, c), (c, c), (b, c), (a, a)} and R2 = {(a, b), (b, a), (c, c), (c, a), (a, a), (b, b), (a, c)}. Then :

\mathop {\lim }\limits_{x \to 0} \,\,{{{{\left( {27 + x} \right)}^{{1 \over 3}}} - 3} \over {9 - {{\left( {27 + x} \right)}^{{2 \over 3}}}}} equals.

If the function f defined as f\left( x \right) = {1 \over x} - {{k - 1} \over {{e^{2x}} - 1}},x \ne 0, is continuous at x = 0, then the ordered pair (k, f(0)) is equal to :

Let f(x) be a polynomial of degree $$4$$ having extreme values at $$x=1$$ and $$x=2.$$ If \mathop {lim}\limits_{x \to 0} \left( {{{f\left( x \right)} \over {{x^2}}} + 1} \right) = 3 then f($$-$$1) is equal to :

Let f(x) = \left\{ {\matrix{ {{{\left( {x - 1} \right)}^{{1 \over {2 - x}}}},} & {x > 1,x \ne 2} \cr {k\,\,\,\,\,\,\,\,\,\,\,\,\,\,} & {,x = 2} \cr } } \right. Thevaue of k for which f s continuous at x = 2 is :

\mathop {\lim }\limits_{x \to 0} {{x\tan 2x - 2x\tan x} \over {{{\left( {1 - \cos 2x} \right)}^2}}} equals :

Let S = {($$\lambda$$, $$\mu$$) $$\in$$ R $$\times$$ R : f(t) = (|$$\lambda$$| e|t| $$-$$ $$\mu$$). sin (2|t|), t $$\in$$ R, is a differentiable function}. Then S is a subset of :

Let S = { t $$\in R:f(x)=\left|x-\pi \right|.\left(e^{|x|}-1\right)$$$$\sin \left|x\right|$$ is not differentiable at t}, then the set S is equal to

For each t $$\in R$$, let [t] be the greatest integer less than or equal to t. Then \mathop {\lim }\limits_{x \to {0^ + }} x\left( {\left[ {{1 \over x}} \right] + \left[ {{2 \over x}} \right] + ..... + \left[ {{{15} \over x}} \right]} \right)

If sum of all the solutions of the equation 8\cos x.\left( {\cos \left( {{\pi \over 6} + x} \right).\cos \left( {{\pi \over 6} - x} \right) - {1 \over 2}} \right) = 1 in [0, $$\pi$$] is k$$\pi$$, then k is equal to

The number of solutions of sin3x = cos 2x, in the interval \left( {{\pi \over 2},\pi } \right) is :

The number of four letter words that can be formed using the letters of the word BARRACK is :

n$$-$$digit numbers are formed using only three digits 2, 5 and 7. The smallest value of n for which 900 such distinct numbers can be formed, is :

From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and arranged in a row on a shelf so that the dictionary is always in the middle. The number of such arrangements is :

The number of numbers between 2,000 and 5,000 that can be formed with the digits 0, 1, 2, 3, 4 (repetition of digits is not allowed) and are multiple of 3 is :

If tanA and tanB are the roots of the quadratic equation, 3x2 $$-$$ 10x $$-$$ 25 = 0, then the value of 3 sin2(A + B) $$-$$ 10 sin(A + B).cos(A + B) $$-$$ 25 cos2(A + B) is :

Let S = { $$x$$ $$\in$$ R : $$x$$ $$\ge$$ 0 and $$2\left|\sqrt{x}-3\right|+\sqrt{x}\left(\sqrt{x}-6\right)+6=0$$}. Then S

If an angle A of a $$\Delta$$ABC satiesfies 5 cosA + 3 = 0, then the roots of the quadratic equation, 9x2 + 27x + 20 = 0 are :

Let p, q and r be real numbers (p $$\ne$$ q, r $$\ne$$ 0), such that the roots of the equation {1 \over {x + p}} + {1 \over {x + q}} = {1 \over r} are equal in magnitude but opposite in sign, then the sum of squares of these roots is equal to :

If f(x) is a quadratic expression such that f (1) + f (2) = 0, and $$-$$ 1 is a root of f (x) = 0, then the other root of f(x) = 0 is :