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 $x$ is real, the expression $\frac{{x + 2}}{{2{x^2} + 3x + 6}}$ takes all value in the interval
The number of distinct real roots of the equation $x ^{7}-7 x -2=0$ is
If $x$ is real, then the value of $\frac{{{x^2} + 34x - 71}}{{{x^2} + 2x - 7}}$ does not lie between
The solutions of the quadratic equation ${(3|x| - 3)^2} = |x| + 7$ which belongs to the domain of definition of the function $y = \sqrt {x(x - 3)} $ are given by
$\alpha$, $\beta$ ,$\gamma$ are roots of equatiuon $x^3 -x -1 = 0$ then equation whose roots are $\frac{1}{{\beta + \gamma }},\frac{1}{{\gamma + \alpha }},\frac{1}{{\alpha + \beta }}$ is