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

Solve the given system of three linear equations and select the correct relationship between $x, y,$ and $z$ from the given options.
$I. 7x + 6y + 4z = 122$
$II. 4x + 5y + 3z = 88$
$III. 9x + 2y + z = 78$

The roots of the equation $3 a^{2} x^{2}-a b x-2 b^{2}=0$ are

The largest possible right circular cylinder is cut out from a wooden cube of edge $7 \, cm$.
$Quantity \, I$: Volume of the cube left over after cutting out the cylinder.
$Quantity \, II$: Surface area of the cube remaining after cutting out the cylinder.
$Note$: Compare the magnitudes of both quantities.

If $\alpha_1 < \alpha_2 < \alpha_3 < \alpha_4 < \alpha_5 < \alpha_6$,then the equation $(x - \alpha_1)(x - \alpha_3)(x - \alpha_5) + 3(x - \alpha_2)(x - \alpha_4)(x - \alpha_6) = 0$ has :-

If $x^{2}+y^{2}+z^{2}=2(x+z-1)$,then find the value of $x^{3}+y^{3}+z^{3}$.