Let $r_1, r_2, r_3$ be the roots of the equation $x^3 - 2x^2 + 4x + 5074 = 0$. Then the value of $(r_1 + 2)(r_2 + 2)(r_3 + 2)$ is:

If $\alpha, \beta$ are the roots of the equation $a x^{2}+b x+b=0,$ then $\sqrt{\frac{\alpha}{\beta}}+\sqrt{\frac{\beta}{\alpha}}+\sqrt{\frac{b}{a}}=$

The roots of the equation $ax^{2} + (4a^{2} - 3b)x - 12ab = 0$ are

The real number $k$ for which the equation $2x^2 + 3x + k = 0$ has two distinct real roots in $[0, 1]$:

If $x = \sqrt{6 + \sqrt{6 + \sqrt{6 + \dots \infty}}}$,then

Let $p(x)$ be a quadratic polynomial such that $p(0) = 1$. If $p(x)$ leaves a remainder of $4$ when divided by $x - 1$ and a remainder of $6$ when divided by $x + 1$,then: