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. . . . . .

If the equation $\frac{{{x^2} + 5}}{2} = x – 2\cos \left( {ax + b} \right)$ has atleast one solution, then $(b + a)$ can be equal to

The number of solutions of the equation $\log _{(x+1)}\left(2 x^{2}+7 x+5\right)+\log _{(2 x+5)}(x+1)^{2}-4=0, x\,>\,0$, is $….$

Suppose $a$ is a positive real number such that $a^5-a^3+a=2$. Then,

If the expression $\left( {mx – 1 + \frac{1}{x}} \right)$ is always non-negative, then the minimum value of m must be