An object of mass M moving with velocity V breaks into two pieces with mass ratio 1:5. If the velocity of the smaller piece is v1, find the velocity of the larger piece v2.

Let the initial mass of the object be $M$.
The initial velocity of the object is $\vec{V} = (20\hat{i} + 25\hat{j} - 12\hat{k})$.
The object breaks into two pieces with a mass ratio of $1:5$.
Let the mass of the smaller piece be $m_1$ and the mass of the larger piece be $m_2$.
So, $m_1 = \frac{1}{1+5}M = \frac{1}{6}M$.
And $m_2 = \frac{5}{1+5}M = \frac{5}{6}M$.

The velocity of the smaller piece is $\vec{v_1} = (100\hat{i} + 35\hat{j} + 8\hat{k})$.
We need to find the velocity of the larger piece, $\vec{v_2}$.

In the absence of external forces, the total momentum of the system before the explosion is equal to the total momentum after the explosion.

$$P_{initial} = P_{final}$$ $$M\vec{V} = m_1\vec{v_1} + m_2\vec{v_2}$$

$$6(20\hat{i} + 25\hat{j} - 12\hat{k}) = (100\hat{i} + 35\hat{j} + 8\hat{k}) + 5\vec{v_2}$$ $$(120\hat{i} + 150\hat{j} - 72\hat{k}) = (100\hat{i} + 35\hat{j} + 8\hat{k}) + 5\vec{v_2}$$

$$5\vec{v_2} = (120\hat{i} + 150\hat{j} - 72\hat{k}) - (100\hat{i} + 35\hat{j} + 8\hat{k})$$ $$5\vec{v_2} = (120 - 100)\hat{i} + (150 - 35)\hat{j} + (-72 - 8)\hat{k}$$ $$5\vec{v_2} = 20\hat{i} + 115\hat{j} - 80\hat{k}$$

$$\vec{v_2} = \frac{1}{5}(20\hat{i} + 115\hat{j} - 80\hat{k})$$ $$\vec{v_2} = \left(\frac{20}{5}\right)\hat{i} + \left(\frac{115}{5}\right)\hat{j} - \left(\frac{80}{5}\right)\hat{k}$$ $$\vec{v_2} = 4\hat{i} + 23\hat{j} - 16\hat{k}$$

The calculated velocity $\vec{v_2} = 4\hat{i} + 23\hat{j} - 16\hat{k}$ matches option (a).

The final answer is $\text{(a)}$.