The roots of $|x – 2{|^2} + |x – 2| – 6 = 0$are

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 number of ordered pairs $(x, y)$ of real numbers that satisfy the simultaneous equations $x+y^2=x^2+y=12$ is

Let $p(x)=a_0+a_1 x+\ldots+a_n x^n$ be a non-zero polynomial with integer coefficients. If $p(\sqrt{2}+\sqrt{3}+\sqrt{6})=0$, then the smallest possible value of $n$ is

If $\alpha ,\beta$ are the roots of $x^2 -ax + b = 0$ and if $\alpha^n + \beta^n = V_n$, then –