The sum of all the real roots of the equation $\left( e ^{2 x }-4\right)\left(6 e ^{2 x }-5 e ^{ x }+1\right)=0$ is

The number of solutions of $\frac{{\log 5 + \log ({x^2} + 1)}}{{\log (x - 2)}} = 2$ is

The number of solution$(s)$ of the equation $2^x = x^2$ is

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

Let $\alpha$ and $\beta$ be the roots of the equation $5 x^{2}+6 x-2=0 .$ If $S_{n}=\alpha^{n}+\beta^{n}, n=1,2,3 \ldots$ then :

The number of real solutions of the equation $3\left(x^2+\frac{1}{x^2}\right)-2\left(x+\frac{1}{x}\right)+5=0$, is