If $x+\frac{1}{x}=a, x^2+\frac{1}{x^3}=b$, then $x^3+\frac{1}{x^2}$ is
$a^3+a^2-3 a-2-b$
$a^3-a^2-3 a+4-b$
$a^3-a^2+3 a-6-b$
$a^3+a^2+3 a-16-b$
If $\alpha , \beta $ are the roots of the equation $x^2 - 2x + 4 = 0$ , then the value of $\alpha ^n +\beta ^n$ is
A real root of the equation ${\log _4}\{ {\log _2}(\sqrt {x + 8} - \sqrt x )\} = 0$ is
The number of non-negative integer solutions of the equations $6 x+4 y+z=200$ and $x+y+z=100$ is
If $\alpha $ and $\beta $ are the roots of the quadratic equation, $x^2 + x\, sin\,\theta -2sin\,\theta = 0$, $\theta \in \left( {0,\frac{\pi }{2}} \right)$ then $\frac{{{\alpha ^{12}} + {\beta ^{12}}}}{{\left( {{\alpha ^{ - 12}} + {\beta ^{ - 12}}} \right){{\left( {\alpha - \beta } \right)}^{24}}}}$ is equal to
The sum of all the solutions of the equation $(8)^{2 x}-16 \cdot(8)^x+48=0$ is :