1. Given data

• Full scale current of galvanometer: $I_g = 10^{-4}\,\text{A}$
• Series resistance required to make voltmeter of range $0$ to $5\,\text{V}$: $R_s = 2\,\text{M}\Omega = 2\times 10^6\,\Omega$

We first find the resistance of the galvanometer, $G$.

2. Find galvanometer resistance

For a voltmeter made from a galvanometer: $V = I_g(G + R_s)$

Substitute the values: $5 = 10^{-4}(G + 2\times 10^6)$

So, $G + 2\times 10^6 = \frac{5}{10^{-4}} = 5\times 10^4$

Thus, $G = 5\times 10^4 - 2\times 10^6 = -1.95\times 10^6\,\Omega$

This is physically impossible because resistance cannot be negative.

So the given data are inconsistent.

3. Check the ammeter conversion condition

Desired ammeter range: $I = 0.10\,\text{mA} = 10^{-4}\,\text{A}$

But this is exactly equal to the galvanometer full scale current: $I = I_g$

Hence, to convert the galvanometer into an ammeter of range $0.10\,\text{mA}$, no shunt is needed at all.

That means: $R_{sh} = \infty$

because the galvanometer itself already measures up to $0.10\,\text{mA}$.

4. Evaluate options

• A: $200\,\Omega$ — incorrect
• B: $500\,\Omega$ — incorrect
• C: $100\,\Omega$ — incorrect
• D: None of the options are correct — correct

5. Final answer

The required shunt resistance is not any finite value from the options; effectively no shunt is required.

So the correct option is: $\boxed{\text{D}}$