The number of distinct real roots of $x^4-4 x^3+12 x^2+x-1=0$ 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

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

The number of real solutions of the equation $|x{|^2}$-$3|x| + 2 = 0$ are

If $x$ is real and $k = \frac{{{x^2} - x + 1}}{{{x^2} + x + 1}},$ 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 ..... .