b
(b) Let initial velocity of the bullet \(= u\)

After penetrating \(3\, cm\) its velocity becomes \(\frac{u}{2}\)

From \({v^2} = {u^2} - 2as\)

\({\left( {\frac{u}{2}} \right)^2} = {u^2} - 2a\,(3)\)

\(⇒\) \(6a = \frac{{3{u^2}}}{4}\) \(⇒\)  \(a = \frac{{{u^2}}}{8}\)

Let further it will penetrate through distance \(x\) and stops at point \(C\).

For distance \(BC\), \(v = 0,\,u = u/2,\,s = x,\,a = {u^2}/8\)

From \({v^2} = {u^2} - 2as\) \(⇒\) \(0 = {\left( {\frac{u}{2}} \right)^2} - 2\left( {\frac{{{u^2}}}{8}} \right)\,.\,x\) \(⇒\) \(x = 1\,cm\).