If $a$ is a positive integer such that the roots of the equation $7x^2 - 13x + a = 0$ are rational numbers,then the smallest possible value of $a$ is

If $\alpha, \beta, \gamma, \delta, \varepsilon$ are the roots of the equation $x^5+x^4-13x^3-13x^2+36x+36=0$ and $\alpha < \beta < \gamma < \delta < \varepsilon$,then find the value of $\frac{\varepsilon}{\alpha}+\frac{\delta}{\beta}+\frac{1}{\gamma}$.

What are the real roots of the equation $x^2 + 5|x| + 4 = 0$?

The number of distinct real roots of the equation $|x||x+2|-5|x+1|-1=0$ is:

Let $a, b$ and $c$ be the sides of a scalene triangle. If $\lambda$ is a real number such that the roots of the equation $x^2+2(a+b+c)x+3\lambda(ab+bc+ca)=0$ are real,then the interval in which $\lambda$ lies is

If $x = 2 + \sqrt{3}$ and $xy = 1$,then $\frac{x}{\sqrt{2} + \sqrt{x}} + \frac{y}{\sqrt{2} - \sqrt{y}} = $