Let $S=\left\{ x : x \in R \text { and }(\sqrt{3}+\sqrt{2})^{ x ^2-4}+(\sqrt{3}-\sqrt{2})^{ x ^2-4}=10\right\} \text {. }$ Then $n ( S )$ is equal to

The set of all real numbers $x$ for which ${x^2} - |x + 2| + x > 0,$ is

If $\alpha ,\beta ,\gamma$  are the roots of $x^3 - x - 2 = 0$, then the value of $\alpha^5 + \beta^5 + \gamma^5$ is-

Let $\alpha, \beta, \gamma$ be the three roots of the equation $x ^3+ bx + c =0$. If $\beta \gamma=1=-\alpha$, then $b^3+2 c^3-3 \alpha^3-6 \beta^3-8 \gamma^3$ is equal to $......$.

Let $P(x) = x^3 - ax^2 + bx + c$ where $a, b, c \in R$ has integral roots such that $P(6) = 3$, then $' a '$ cannot be equal to

Consider the cubic equation $x^3+c x^2+b x+c=0$ where $a, b, c$ are real numbers. Which of the following statements is correct?