All the points $(x, y)$ in the plane satisfying the equation $x^2+2 x \sin (x y)+1=0$ lie on

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

Let $p$ and $q$ be two real numbers such that $p+q=$ 3 and $p^{4}+q^{4}=369$. Then $\left(\frac{1}{p}+\frac{1}{q}\right)^{-2}$ is equal to

If $\alpha ,\,\beta ,\,\gamma $ are the roots of the equation ${x^3} + 4x + 1 = 0,$ then ${(\alpha + \beta )^{ – 1}} + {(\beta + \gamma )^{ – 1}} + {(\gamma + \alpha )^{ – 1}} = $

Sum of the solutions of the equation $\left[ {{x^2}} \right] – 2x + 1 = 0$ is (where $[.]$ denotes greatest integer function)