The maximum possible number of real roots of equation ${x^5} - 6{x^2} - 4x + 5 = 0$ is

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}} = $

Let $f(x)=x^4+a x^3+b x^2+c$ be a polynomial with real coefficients such that $f(1)=-9$. Suppose that $i \sqrt{3}$ is a root of the equation $4 x^3+3 a x^2+2 b x=0$, where $i=\sqrt{-1}$. If $\alpha_1, \alpha_2, \alpha_3$, and $\alpha_4$ are all the roots of the equation $f(x)=0$, then $\left|\alpha_1\right|^2+\left|\alpha_2\right|^2+\left|\alpha_3\right|^2+\left|\alpha_4\right|^2$ is equal to. . . . . .

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 real solution of equation $(\frac{3}{2})^x =  -x^2 + 5x-10$ :-

If the graph of $y = ax^3 + bx^2 + cx + d$ is symmetric about the line $x = k$ then