If $\alpha , \beta $ are the roots of the equation $x^2 - 2x + 4 = 0$ , then the value of $\alpha ^n +\beta ^n$ is
${2^n}\cos \left( {\frac{{n\pi }}{3}} \right)$
${2^{n + 1}}\cos \left( {\frac{{n\pi }}{3}} \right)$
${2^n}\sin \left( {\frac{{n\pi }}{3}} \right)$
${2^{n + 1}}\sin \left( {\frac{{n\pi }}{3}} \right)$
If $\alpha ,\beta,\gamma$ are the roots of equation $x^3 + 2x -5 = 0$ and if equation $x^3 + bx^2 + cx + d = 0$ has roots $2 \alpha + 1, 2 \beta + 1, 2 \gamma + 1$ , then value of $|b + c + d|$ is (where $b,c,d$ are coprime)-
The polynomial equation $x^3-3 a x^2+\left(27 a^2+9\right) x+2016=0$ has
Let $a, b, c$ be non-zero real roots of the equation $x^3+a x^2+b x+c=0$. Then,
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 ..... .
The equation $e^{4 x}+8 e^{3 x}+13 e^{2 x}-8 e^x+1=0, x \in R$ has: