The nuclear transmutation of beryllium (Be) is given as:

$$ _4^9 \mathrm{Be} + X \to _4^8 \mathrm{Be} + Y $$

We examine the possible reactions to determine the values of X and Y:

(A) If $ X = _0^0\gamma $ and $ Y = _0^1n $

The reaction is:

$$ _4^9 \mathrm{Be} + _0^0\gamma \longrightarrow _4^8 \mathrm{Be} + _0^1n $$

Using the conditions of:

(i) Mass balance:

Mass of reactants $ = 9 + 0 = 9 $

Mass of products $ = 8 + 1 = 9 $

(ii) Charge balance:

Charge of reactants $ = 4 + 0 = 4 $

Charge of products $ = 4 + 0 = 4 $

Both mass and charge are balanced. Therefore, Option (A) is correct.

(B) If $ X = p = _1^1H $ and $ Y = D = _1^2H $

The reaction is:

$$ _4^9 \mathrm{Be} + _1^1H \longrightarrow _4^8 \mathrm{Be} + _1^2H $$

Using the conditions of:

(i) Mass balance:

Mass of reactants: $ 9 + 1 = 10 $

Mass of products $ = 8 + 2 = 10 $

(ii) Charge balance:

Charge of reactants $ = 4 + 1 = 5 $

Charge on products $ = 4 + 1 = 5 $

Both mass and charge are balanced. Therefore, Option (B) is correct.

(C) If $ X = _0^1n $ and $ Y = D = _1^2H $

The reaction is:

$$ _4^9 \mathrm{Be} + _0^1n \longrightarrow _4^8 \mathrm{Be} + _1^2H $$

Using the conditions of:

(i) Mass balance:

Mass of reactants $ = 9 + 1 = 10 $

Mass of products $ = 8 + 2 = 10 $

(ii) Charge balance:

Charge on reactants $ = 4 + 0 = 4 $

Charge on products $ = 4 + 1 = 5 $

Charge on reactants $ \neq $ Charge on products. Since, charge is not balanced, Option (C) is incorrect.

(D) If $ X = _0^0\gamma $ and $ Y = p = _1^1H $

The reaction is:

$$ _4^9 \mathrm{Be} + _0^0\gamma \longrightarrow _4^8 \mathrm{Be} + _1^1H $$

Using the conditions of:

(i) Mass balance:

Mass of reactants: $ 9 + 0 = 9 $

Mass of products: $ 8 + 1 = 9 $

(ii) Charge balance:

Charge on products: $ 4 + 1 = 5 $

Charge on reactants: $ 4 + 0 = 4 $

Charge on reactants $ \neq $ Charge on products. Since the charge is not balanced, Option (D) is incorrect.