If the roots of ${x^2} + x + a = 0$exceed $a$, then
$2 < a < 3$
$a > 3$
$ - 3 < a < 3$
$a < - 2$
If $\alpha, \beta$ are roots of the equation $x^{2}+5 \sqrt{2} x+10=0, \alpha\,>\,\beta$ and $P_{n}=\alpha^{n}-\beta^{n}$ for each positive integer $\mathrm{n}$, then the value of $\left(\frac{P_{17} P_{20}+5 \sqrt{2} P_{11} P_{19}}{P_{18} P_{19}+5 \sqrt{2} P_{18}^{2}}\right)$ is equal to $....$
If $x$ is real and satisfies $x + 2 > \sqrt {x + 4} ,$ then
If$\frac{{2x}}{{2{x^2} + 5x + 2}} > \frac{1}{{x + 1}}$, then
In the equation ${x^3} + 3Hx + G = 0$, if $G$ and $H$ are real and ${G^2} + 4{H^3} > 0,$ then the roots are
Let $a, b, c, d$ be real numbers between $-5$ and $5$ such that $|a|=\sqrt{4-\sqrt{5-a}},|b|=\sqrt{4+\sqrt{5-b}},|c|=\sqrt{4-\sqrt{5+c}}$ $|d|=\sqrt{4+\sqrt{5+d}}$ Then, the product $a b c d$ is