\[\begin {array} {1 1} (a)\;\frac{3v}{4a} & \quad (b)\;\frac{3v}{2a} \\ (c)\;\frac{\sqrt 3 v}{\sqrt 2 a} & \quad (d)\;zero \end {array}\]

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When the block hits the ridges, the impulse of the impact force produces a torque about the centre of mass of the cube.

Angular momentum conservation can be applied about 0.

$Mv \times \large\frac{a}{2}$$= I _{about0} . w$

I for a about an axis through centre of mass= $\large\frac{Ma^2}{b}$

$OC= \large\frac{a}{\sqrt 2}$

I about an axis through o and $\perp$ to the plane of paper $=\large\frac{Ma^2}{6}$$+M\bigg(a / \sqrt 2\bigg)^2$

$\qquad=\large\frac{4Ma^2}{6}$

$\qquad=\large\frac{2}{3}$$Ma^2$

$\therefore mV\large\frac{a}{2}=\frac{2}{3}$$ Ma^2 w$

$\Rightarrow w = \large\frac{3v}{4a}$. Therefore (A) is the answer.

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