The equation $\sqrt {3 {x^2} + x + 5} = x - 3$ , where $x$ is real, has

Let $a, b$ be non-zero real numbers. Which of the following statements about the quadratic equation $a x^2+(a+b) x+b=0$ is necessarily true?

$I$. It has at least one negative root.

$II$. It has at least one positive root.

$III$. Both its roots are real.

What is the sum of all natural numbers $n$ such that the product of the digits of $n$ (in base $10$ ) is equal to $n^2-10 n-36 ?$

If $3$ distinct real number $a$,$b$,$c$ satisfy $a^2(a + p) = b^2 (b + p) = c^2 (c + p)$ where $p \in R$, then value of $bc + ca + ab$ is

If $x$ is real, the expression $\frac{{x + 2}}{{2{x^2} + 3x + 6}}$ takes all value in the interval

Let $\alpha$ and $\beta$ be the roots of the equation $\mathrm{x}^{2}-\mathrm{x}-1=0 .$ If $\mathrm{p}_{\mathrm{k}}=(\alpha)^{\mathrm{k}}+(\beta)^{\mathrm{k}}, \mathrm{k} \geq 1,$ then which one of the following statements is not true?