The function f(x), that satisfies the condition
$$f(x) = x + \int\limits_0^{\pi /2} {\sin x.\cos y\,f(y)\,dy} $$, is :

Let $$\theta$$ be the acute angle between the tangents to the ellipse $${{{x^2}} \over 9} + {{{y^2}} \over 1} = 1$$ and the circle $${x^2} + {y^2} = 3$$ at their point of intersection in the first quadrant. Then tan$$\theta$$ is equal to :

Let X be a random variable with distribution.

x	$$ - $$2	$$ - $$1	3	4	6
P(X = x)	$${1 \over 5}$$	a	$${1 \over 3}$$	$${1 \over 5}$$	b

If the mean of X is 2.3 and variance of X is $$\sigma$$², then 100 $$\sigma$$² is equal to :

Let $$f(x) = {x^6} + 2{x^4} + {x^3} + 2x + 3$$, x $$\in$$ R. Then the natural number n for which $$\mathop {\lim }\limits_{x \to 1} {{{x^n}f(1) - f(x)} \over {x - 1}} = 44$$ is __________.