Let $$f\left( x \right) = {a_n}{x^n} + {a_{n - 1}}{x^{n - 1}} + ........... + {a_1}x = 0$$

The other given equation,

$$na{}_n{x^{n - 1}} + \left( {n - 1} \right){a_{n - 1}}{x^{n - 2}} + .... + {a_1} = 0 = f'\left( x \right)$$

Given $${a_1} \ne 0 \Rightarrow f\left( 0 \right) = 0$$

Again $$f(x)$$ has root $$\alpha ,\,\,\, \Rightarrow f\left( \alpha \right) = 0$$

$$\therefore$$ $$\,\,\,\,\,f\left( 0 \right) = f\left( \alpha \right)$$

$$\therefore$$ By Roll's theorem $$f'(x)=0$$ has root be-

tween $$\left( {0,\alpha } \right)$$

Hence $$f'(x)$$ has a positive root smaller than $$\alpha .$$

Given that

$${{dv} \over {dt}} = 50\,c{m^3}/\min $$

$$ \Rightarrow {d \over {dt}}\left( {{4 \over 3}\pi {r^3}} \right) = 50$$

$$ \Rightarrow 4\pi {r^2}{{dr} \over {dt}} = 50$$

$$ \Rightarrow {{dr} \over {dt}} = {{50} \over {4\pi {{\left( {15} \right)}^2}}} = {1 \over {18\pi }}\,\,cm/\min \,\,$$

(here $$r=10+5)$$

$$x = a\left( {\cos \theta + \theta \sin \theta } \right)$$

$$ \Rightarrow {{dx} \over {d\theta }} = a\left( { - \sin \theta + \sin \theta + \theta \cos \theta } \right)$$

$$ \Rightarrow {{dx} \over {d\theta }} = a\theta \cos \theta \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...\left( 1 \right)$$

$$y = a\left( {\sin \theta - \theta \cos \theta } \right)$$

$${{dy} \over {d\theta }} = a\left[ {\cos \theta - \cos \theta + \theta \sin \theta } \right]$$

$$ \Rightarrow {{dy} \over {d\theta }} = a\theta \sin \theta \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...\left( 2 \right)$$

From equations $$(1)$$ and $$(2),$$ we get

$${{dy} \over {dx}} = \tan \theta \Rightarrow $$ Slope of normal $$ = - \cot \,\theta $$

Equation of normal at

$$'\theta '$$ is $$y - a\left( {\sin \theta - \theta \cos \theta } \right)$$

$$ = - \cot \theta \left( {x - a\left( {\cos \theta + \theta \sin \theta } \right)} \right.$$

$$ \Rightarrow y\sin \theta - a{\sin ^2}\theta + a\,\,\theta \cos \theta \sin \theta $$

$$ = - x\cos \theta + a{\cos ^2}\theta + a\,\theta \sin \theta \cos \theta $$

$$ \Rightarrow x\cos \theta + y\sin \theta = a$$

Clearly this is an equation of straight line -

which is at a constant distance $$'a'$$ from origin.

Area of rectangle $$ABCD$$ $$ = 2a\,\cos \,\theta $$

$$\left( {2b\,\sin \,\theta } \right) = 2ab\,\sin \,2\theta $$



$$ \Rightarrow $$ Area of greatest rectangle is equal to $$2ab$$

When $$\sin \,2\theta = 1.$$