Answer the following questions: Let X be a continuous random variable where `f(x) = kx (3 – x), 0 le x le 3`: Then obtain k.

Answer the following questions: If X is a discrete random variable, then define median of X.

Answer the following questions (Alternatives are to be noted) : If X be a continuous random variable with p.d.f. f(x) = k, ale xleb 0, otherwise Find E(x)

Answer the following questions (Alternatives are to be noted): If x is continuous random variable with p. d. f. {(1/2 - ax, ""0 le x le 4),(0,""otherwise):} where a is a constant, then find the value of a.

Answer the following questions : (Alternatives are to be noted): A Continuous random variable x has a density function given by f(x) = 1/2 - ax, 0lexle4=0 else where Find (i) The constant a (ii) p(2x+3>5).

The probability density function of a random variable X is f(x) = k(x-1)^2 , 1 le x le 2 . The value of the constant K is

Following is the graph of y = f' (x) , given that f(c) = 0. Analyse the graph and answer the following questions. (a) How many times the graph of y = f(x) will intersect the x - axis? (b) Discuss the type of roots of the equation f (x) = 0, a le x le b . (c) How many points of inflection the graph of y = f(x), a le x le b , has? ,(d) Find the points of local maxima/minima of y = f(x), a le x le b , , (e) f"(x)=0 has how many roots?

Answer the following questions (Alternatives are to be noted): If r be the correlation coefficient between two variables x and y then prove that -1 le r le1 .

Show that Rolle's theorem is not applicable to the following functions in the specified intervals: f(x)={(x^(2)+1" when " 0 le x le 1),(3-x" when "1 lt x le 2):}

Find the range of the function f(x) = abs(x - 1) + abs(x - 2) , -1 le x le 3

Find the number of distinct solutions of sec x + tan x = sqrt3 , where 0 le x le 3pi .