The solution set of the equation $pq{x^2} - {(p + q)^2}x + {(p + q)^2} = 0$ is
$\left\{ {\frac{p}{q},\,\frac{q}{p}} \right\}$
$\left\{ {pq,\,\frac{p}{q}} \right\}$
$\left\{ {\frac{q}{p},\,pq} \right\}$
$\left\{ {\frac{{p + q}}{p},\,\frac{{p + q}}{q}} \right\}$
The roots of $|x - 2{|^2} + |x - 2| - 6 = 0$are
If $|{x^2} - x - 6| = x + 2$, then the values of $x$ are
The number of solution$(s)$ of the equation $ln(lnx)$ = $log_xe$ is -
Number of natural solutions of the equation $x_1 + x_2 = 100$ , such that $x_1$ and $x_2$ are not multiple of $5$
Let $\alpha $ and $\beta $ be the roots of the quadratic equation ${x^2}\,\sin \,\theta - x\,\left( {\sin \,\theta \cos \,\,\theta + 1} \right) + \cos \,\theta = 0\,\left( {0 < \theta < {{45}^o}} \right)$ , and $\alpha < \beta $. Then $\sum\limits_{n = 0}^\infty {\left( {{\alpha ^n} + \frac{{{{\left( { - 1} \right)}^n}}}{{{\beta ^n}}}} \right)} $ is equal to