(a) Draw the graph of $$y = {1 \over {\sqrt 2 }}\left( {cinx + \cos x} \right)$$ from $$x = - {\pi \over 2}$$ to $$x = {\pi \over 2}$$.

(b) If $$\cos \left( {\alpha + \beta } \right) = {4 \over 5},\,\,\sin \,\left( {\alpha - \beta } \right) = \,{5 \over {13}},$$ and $$\alpha ,\,\beta $$ lies between 0 and $${\pi \over 4}$$, find tan2$$\alpha $$.

If $$\alpha ,\,\beta $$ are the roots of $${x^2} + px + q = 0$$ and $$\gamma ,\,\delta $$ are the roots of $${x^2} + rx + s = 0,$$ evaluate $$\left( {\alpha - \gamma } \right)\left( {\alpha - \delta } \right)\left( {\beta - \gamma } \right)$$ $$\left( {\beta - \delta } \right)$$ in terms of $$p,\,q,\,r$$ and $$s$$.

deduce the condition that the equations have a common root.

Let a > 0, b > 0 and c > 0. Then the roots of the equation $$a{x^2} + bx + c = 0$$

If $$\ell $$, m, n are real, $$\ell \ne m$$, then the roots by the equation :
$$(\ell - m)\,{x^2} - 5\,(\ell + m)\,x - 2\,(\ell - m) = 0$$ are