0.001 mol of `Cr(NH_(3))_(5)(NO_(3))(SO_(4))` was passed through a cation exchanger the acid coming out of it reguired `20mL` of 0.1M NaoH for netralisation Hence the complex is
`[Cr(NH_(3))_(5)SO_(4)]NO_(3)`
(b) `[Cr(NH_(3))_(5)NO_(3)]SO_(4)`
(c ) `[Cr(NH_(3))_(5)(SO_(4))](NO_(3)`
(d) `[Co(NH_(3))_(5)Br]SO_(4)` .

Explanation

To solve the problem, we need to analyze the information given about the complex `Cr(NH₃)₅(NO₃)(SO₄)` and the neutralization reaction that occurs when it is passed through a cation exchanger.

### Step-by-Step Solution:

1. **Determine the amount of acid produced:**
- We know that 20 mL of 0.1 M NaOH was required for neutralization.
- Calculate the moles of NaOH used:
\[ \text{Moles of NaOH} = \text{Volume (L)} \times \text{Molarity (mol/L)} = 0.020 \, \text{L} \times 0.1 \, \text{mol/L} = 0.002 \, \text{mol} \]

2. **Relate moles of acid to moles of complex:**
- The moles of acid produced (0.002 mol) are equivalent to the moles of H⁺ ions that were released from the complex.
- Since 1 mole of the complex releases 2 moles of H⁺ ions, we can find the moles of the complex:
\[ \text{Moles of complex} = \frac{\text{Moles of acid}}{2} = \frac{0.002 \, \text{mol}}{2} = 0.001 \, \text{mol} \]

3. **Determine the oxidation state of chromium in the complex:**
- The complex has a total charge of +2, which means the oxidation state of chromium must be +2.
- The ligands NH₃ are neutral, NO₃⁻ has a charge of -1, and SO₄²⁻ has a charge of -2.
- The overall charge balance can be calculated as follows:
\[ \text{Charge from ligands} = -1 \, (\text{from NO}_3) + -2 \, (\text{from SO}_4) = -3 \]
- Therefore, to balance the charge:
\[ \text{Oxidation state of Cr} + (-3) = +2 \implies \text{Oxidation state of Cr} = +5 \]

4. **Identify the correct complex:**
- The possible complexes given are:
- (a) `[Cr(NH₃)₅SO₄]NO₃`
- (b) `[Cr(NH₃)₅NO₃]SO₄`
- (c) `[Cr(NH₃)₅(SO₄)](NO₃)`
- (d) `[Co(NH₃)₅Br]SO₄`
- Since we have established that the oxidation state of chromium is +5, we can conclude that the complex must be one that reflects this oxidation state and the presence of the ligands.

5. **Final conclusion:**
- The correct complex that matches the given information and has chromium in the +3 oxidation state is option (b) `[Cr(NH₃)₅NO₃]SO₄`.

### Final Answer:
The complex is `[Cr(NH₃)₅NO₃]SO₄`.