Let $f(x)={{x}^{2}}-x+k-2,k\in R$ then the complete set of values of $k$ for which $y=\left| f\left( \left| x \right| \right) \right|$ is non-derivable at $5$ distinict points is 

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

The roots of the equation ${x^4} - 4{x^3} + 6{x^2} - 4x + 1 = 0$ are

Let $x, y, z$ be non-zero real numbers such that $\frac{x}{y}+\frac{y}{z}+\frac{z}{x}=7$ and $\frac{y}{x}+\frac{z}{y}+\frac{x}{z}=9$, then $\frac{x^3}{y^3}+\frac{y^3}{z^3}+\frac{z^3}{x^3}-3$ is equal to

Let $\alpha $ and $\beta $ be the roots of the quadratic equation ${x^2}\,\sin \,\theta  - x\,\left( {\sin \,\theta \cos \,\,\theta  + 1} \right) + \cos \,\theta  = 0\,\left( {0 < \theta  < {{45}^o}} \right)$ , and $\alpha  < \beta $.  Then $\sum\limits_{n = 0}^\infty  {\left( {{\alpha ^n} + \frac{{{{\left( { - 1} \right)}^n}}}{{{\beta ^n}}}} \right)} $ is equal to

The sum of the roots of the equation, ${x^2}\, + \,\left| {2x - 3} \right|\, - \,4\, = \,0,$ is