The number of pairs of reals $(x, y)$ such that $x=x^2+y^2$ and $y=2 x y$ is

Suppose that $x$ and $y$ are positive number with $xy = \frac{1}{9};\,x\left( {y + 1} \right) = \frac{7}{9};\,y\left( {x + 1} \right) = \frac{5}{{18}}$ . The value of $(x + 1) (y + 1)$ equals

The number of solutions of $\sin ^2 \mathrm{x}+\left(2+2 \mathrm{x}-\mathrm{x}^2\right) \sin \mathrm{x}-3(\mathrm{x}-1)^2=0$, where $-\pi \leq \mathrm{x} \leq \pi$, is....................

The number of integral values of $m$ for which the quadratic expression, $(1 + 2m)x^2 -2(1+ 3m)x + 4(1 + m),$ $x\in R,$ is always positive, is

The number of integers $a$ in the interval $[1,2014]$ for which the system of equations $x+y=a$, $\frac{x^2}{x-1}+\frac{y^2}{y-1}=4$ has finitely many solutions is

If $\alpha , \beta $ are the roots of the equation $x^2 - 2x + 4 = 0$ , then the value of $\alpha ^n +\beta ^n$ is