Let $$\alpha,\beta$$ be the roots of the equation $$x^2-px+r=0$$ and $$\frac{\alpha}{2},2\beta$$ be the roots of the equation $$x^2-qx+r=0$$. Then the value of r is

Let $$f(x)$$ be differentiable on the interval (0, $$\infty$$) such that $$f(1)=1$$, and $$\mathop {\lim }\limits_{t \to x} {{{t^2}f(x) - {x^2}f(t)} \over {t - x}} = 1$$ for each $$x > 0$$. Then $$f(x)$$ is

One Indian and four American men and their wives are to be seated randomly around a circular table. Then the conditional probability that the Indian man is seated adjacent to his wife given that each American man is seated adjacent to his wife is

The tangent to the curve $$y=e^x$$ drawn at the point ($$c,e^c$$) intersects the line joining the points ($$c-1,e^{c-1}$$) and ($$c+1,e^{c+1}$$)