An urn contains nine balls of which three are red, four are blue and two are green. Three balls are drawn at random without replacement from the urn. The probability that the three balls have different colours is :

Four numbers are chosen at random (without replacement) from the set $$\left\{ {1,2,3,....20} \right\}.$$

Statement - 1: The probability that the chosen numbers when arranged in some order will form an AP is $${1 \over {85}}.$$

Statement - 2: If the four chosen numbers form an AP, then the set of all possible values of common difference is $$\left( { \pm 1, \pm 2, \pm 3, \pm 4, \pm 5} \right).$$

Solution of the differential equation

$$\cos x\,dy = y\left( {\sin x - y} \right)dx,\,\,0 < x <{\pi \over 2}$$ is :

Let $$p(x)$$ be a function defined on $$R$$ such that $$p'(x)=p'(1-x),$$ for all $$x \in \left[ {0,1} \right],p\left( 0 \right) = 1$$ and $$p(1)=41.$$ Then $$\int\limits_0^1 {p\left( x \right)dx} $$ equals :