Let $\alpha, \beta ; \alpha>\beta$, be the roots of the equation $x^2-\sqrt{2} x-\sqrt{3}=0$. Let $P_n=\alpha^n-\beta^n, n \in N$. Then $(11 \sqrt{3}-10 \sqrt{2}) \mathrm{P}_{10}+(11 \sqrt{2}+10) \mathrm{P}_{11}-11 \mathrm{P}_{12}$ is equal to :
$10 \sqrt{2} \mathrm{P}_9$
$10 \sqrt{3} \mathrm{P}_9$
$11 \sqrt{2} \mathrm{P}_9$
$11 \sqrt{3} \mathrm{P}_9$
If $\alpha ,\beta ,\gamma$ are the roots of $x^3 - x - 2 = 0$, then the value of $\alpha^5 + \beta^5 + \gamma^5$ is-
The set of all real numbers $x$ for which ${x^2} - |x + 2| + x > 0,$ is
If $x$ is real and satisfies $x + 2 > \sqrt {x + 4} ,$ then
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?
Let $\alpha, \beta$ be two roots of the equation $x^{2}+(20)^{\frac{1}{4}} x+(5)^{\frac{1}{2}}=0$. Then $\alpha^{8}+\beta^{8}$ is equal to: