Let $${C_1}$$ and $${C_2}$$ be the graphs of the functions $$y = {x^2}$$ and $$y = 2x,$$ $$0 \le x \le 1$$ respectively. Let $${C_3}$$ be the graph of a function $$y=f(x),$$ $$0 \le x \le 1,$$ $$f(0)=0.$$ For a point $$P$$ on $${C_1},$$ let the lines through $$P,$$ parallel to the axes, meet $${C_2}$$ and $${C_3}$$ at $$Q$$ and $$R$$ respectively (see figure.) If for every position of $$P$$ (on $${C_1}$$ ), the areas of the shaded regions $$OPQ$$ and $$ORP$$ are equal, determine the function$$f(x).$$

Three players, $$A,B$$ and $$C,$$ toss a coin cyclically in that order (that is $$A, B, C, A, B, C, A, B,...$$) till a head shows. Let $$p$$ be the probability that the coin shows a head. Let $$\alpha ,\,\,\,\beta $$ and $$\gamma $$ be, respectively, the probabilities that $$A, B$$ and $$C$$ gets the first head. Prove that $$\beta = \left( {1 - p} \right)\alpha $$ Determine $$\alpha ,\beta $$ and $$\gamma $$ (in terms of $$p$$).

Seven white balls and three black balls are randomly placed in a row. The probability that no two black balls are placed adjacently equals

There are four machines and it is known that exactly two of them are faulty. They are tested, one by one, in a random order till both the faulty machines are identified. Then the probability that only two tests are needed is