For a twice differentiable function $$f(x),g(x)$$ is defined as $$4\sqrt {65} g\left( x \right) = \left( {f'{{\left( x \right)}^2} + f''\left( x \right)} \right)\,\,f\left( x \right)$$ on $$\,\,\,\left[ {a,\,\,\,e} \right].$$ If for $$a < b < c < d < e,\,f\left( a \right) = 0,f\left( b \right) = 2,f\left( c \right) = - 1,f\left( d \right) = 2,f\left( e \right) = 0$$ then find the minimum number of zeros of $$g(x)$$.

The value of $$5050{{\int\limits_0^1 {{{\left( {1 - {x^{50}}} \right)}^{100}}} dx} \over {\int\limits_0^1 {{{\left( {1 - {x^{50}}} \right)}^{101}}} dx}}$$ is.

A student performs an experiment for determination of $$\mathrm g\left(=\frac{4\mathrm\pi^2\mathcal l}{\mathrm T^2}\right)$$. The error in length $$\mathcal l$$ is $$\triangle\mathcal l$$ and in the time T is $$\triangle\mathrm T$$ and n is number of times the reading is taken.The reading of g is most accurate for

In a screw gauge, the zero of main scale coincides with the fifth division of circular scale in figure (i).The circular division of screw gauge is 50. It moves 0.5 mm on main scale in one rotation.The diameter of the ball in figure (ii) is