If $1 - i$ is a root of the equation ${x^2} - ax + b = 0$,then $b = $

If $\alpha, \beta, \gamma, \delta$ are the roots of the equation $12x^4-56x^3+89x^2-56x+12=0$ such that $\alpha\beta=\gamma\delta=1$ and $\frac{\alpha+\beta}{\gamma+\delta}>1$,then $\frac{\alpha+\beta}{\gamma+\delta}=$

The number of pairs of consecutive positive even integers such that the sum of their squares is $290$ is

Let $p, q, r \in \mathbb{R}$ and $r > p > 0$. If the quadratic equation $px^2 + qx + r = 0$ has two complex roots $\alpha$ and $\beta$,then what is the value of $|\alpha| + |\beta|$?

For what values of $a \in \mathbb{Z}$,the quadratic expression $(x+a)(x+1991)+1$ can be factorised as $(x+b)(x+c)$,where $b, c \in \mathbb{Z}$?

If $mn=3$ and $\frac{1}{m}+\frac{1}{n}=\frac{4}{3}$,then the value of $0.1+0.1^{\frac{1}{m}}+0.1^{\frac{1}{n}}$ is