Consider the equation $(1+a+b)^2=3(1+a^2+b^2)$ where $a, b$ are real numbers. Then,

Consider the equation $x^2+4x-n=0$,where $n \in [20, 100]$ is a natural number. Then the number of all distinct values of $n$,for which the given equation has integral roots,is equal to

The number of rational roots of the equation $x^{2016} - x^{2015} + x^{1008} + x^{1003} + 1 = 0$ is equal to

Let $S$ be the set of all non-zero real numbers $\alpha$ such that the quadratic equation $\alpha x^2 - x + \alpha = 0$ has two distinct real roots $x_1$ and $x_2$ satisfying the inequality $|x_1 - x_2| < 1$. Which of the following intervals is(are) a subset$(s)$ of $S$?
$(A) \left(-\frac{1}{2}, -\frac{1}{\sqrt{5}}\right)$
$(B) \left(-\frac{1}{\sqrt{5}}, 0\right)$
$(C) \left(0, \frac{1}{\sqrt{5}}\right)$
$(D) \left(\frac{1}{\sqrt{5}}, \frac{1}{2}\right)$

The number which exceeds its positive square root by $12$ is

If $\alpha$ is a repeated root of multiplicity $2$ of the equation $18x^3-33x^2+20x-4=0$,then