If $$\alpha $$ be a repeated root of a quadratic equation $$f(x)=0$$ and $$A(x), B(x)$$ and $$C(x)$$ be polynomials of degree $$3$$, $$4$$ and $$5$$ respectively,
then show that $$\left| {\matrix{ {A\left( x \right)} & {B\left( x \right)} & {C\left( x \right)} \cr {A\left( \alpha \right)} & {B\left( \alpha \right)} & {C\left( \alpha \right)} \cr {A'\left( \alpha \right)} & {B'\left( \alpha \right)} & {C'\left( \alpha \right)} \cr } } \right|$$ is
divisible by $$f(x)$$, where prime denotes the derivatives.

With usual notation, if in a triangle $$ABC$$;
$${{b + c} \over {11}} = {{c + a} \over {12}} = {{a + b} \over {13}}$$ then prove that $${{\cos A} \over 7} = {{\cos B} \over {19}} = {{\cos C} \over {25}}$$.

For a triangle $$ABC$$ it is given that $$\cos A + \cos B + \cos C = {3 \over 2}$$. Prove that the triangle is equilateral.

The numerical value of $$\tan \left\{ {2{{\tan }^{ - 1}}\left( {{1 \over 5}} \right) - {\pi \over 4}} \right\}$$ is equal to __________