If the roots of the quadratic equation $${x^2} + px + q = 0\,\,{\text{are }}\tan {30^ \circ }\,{\text{and tan1}}{{\text{5}}^ \circ },$$        respectively, then the value of $$2 + q - p$$   is

Solution :
$${x^2} + px + q = 0\,$$
Sum of roots $$ = \,\tan {30^ \circ } + \tan {15^ \circ } = - p$$
Product of roots $$ = \,\tan {30^ \circ }.\tan {15^ \circ } = q$$
$$\eqalign{ & \tan {45^ \circ } = \frac{{\tan {{30}^ \circ } + \tan {{15}^ \circ }}}{{1 - \tan {{30}^ \circ }.\tan {{15}^ \circ }}} = \frac{{ - p}}{{1 - q}} = 1 \cr & \Rightarrow \,\, - p = 1 - q \cr & \Rightarrow \,\,\,q - p = 1 \cr & \therefore \,\,2 + q - p = 3 \cr} $$