If $\alpha, \beta, \gamma, \delta$ are the roots of the equation $x^4+x^2+1=0$,then $\frac{\alpha^3+\beta^3+\gamma^3+\delta^3}{\alpha^6+\beta^6+\gamma^6+\delta^6}=$

Let $[r]$ denote the greatest integer not exceeding $r$. The roots of the equation $3 x^2 + 6 x + 5 + \alpha (x^2 + 2 x + 2) = 0$ are complex numbers whenever $\alpha > L$ or $\alpha < M$. If $(L - M)$ is minimum,then the greatest value of $[r]$ such that $L y^2 + M y + r < 0$ for all $y \in R$ is:

The sum of the squares of all the roots of the equation $x^2+|2x-3|-4=0$ is:

Let $a, b, c$ be non-zero real numbers such that $a+b+c=0$. Let $q=a^2+b^2+c^2$ and $r=a^4+b^4+c^4$. Then,

If $x$ is real,the function $f(x) = \frac{(x - a)(x - b)}{(x - c)}$ will assume all real values,provided

Let $\alpha$ and $\beta$ be the roots of $x^{2}-3x+p=0$ and $\gamma$ and $\delta$ be the roots of $x^{2}-6x+q=0$. If $\alpha, \beta, \gamma, \delta$ form a geometric progression,then the ratio $(2q+p):(2q-p)$ is: