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 $......$.
$21$
$\frac{169}{8}$
$19$
$\frac{155}{8}$
Let $\mathrm{S}=\left\{x \in R:(\sqrt{3}+\sqrt{2})^x+(\sqrt{3}-\sqrt{2})^x=10\right\}$. Then the number of elements in $\mathrm{S}$ is :
In the real number system, the equation $\sqrt{x+3-4 \sqrt{x-1}}+\sqrt{x+8-6 \sqrt{x-1}}=1$ has
The maximum value $M$ of $3^x+5^x-9^x+15^x-25^x$, as $x$ varies over reals, satisfies
If $x,\;y,\;z$ are real and distinct, then $u = {x^2} + 4{y^2} + 9{z^2} - 6yz - 3zx - zxy$ is always
The sum of all integral values of $\mathrm{k}(\mathrm{k} \neq 0$ ) for which the equation $\frac{2}{x-1}-\frac{1}{x-2}=\frac{2}{k}$ in $x$ has no real roots, is ..... .