1. Assume the required forms

We need to form:

• a unit of length $L$ using $h, c, G$
• a unit of mass $M$ using $h, c, G$

Let $L \propto h^a c^b G^d$

and $M \propto h^p c^q G^r$

2. Write dimensions of the constants

The dimensions are: $[h] = M L^2 T^{-1}$ $[c] = L T^{-1}$ $[G] = M^{-1} L^3 T^{-2}$

3. Find the unit of length

We want $[L] = [h^a c^b G^d]$

So, $[h^a c^b G^d] = M^{a-d} L^{2a+b+3d} T^{-a-b-2d}$

For length, compare powers with $M^0L^1T^0$:

1. Mass exponent: $a-d=0 \Rightarrow a=d$

2. Time exponent: $-a-b-2d=0$ Using $a=d$: $-d-b-2d=0 \Rightarrow b=-3d$

3. Length exponent: $2a+b+3d=1$ Using $a=d, b=-3d$: $2d-3d+3d=1 \Rightarrow 2d=1 \Rightarrow d=\frac12$

Hence, $a=\frac12, \qquad b=-\frac32$

Therefore, $L \propto h^{1/2} c^{-3/2} G^{1/2}$

So:

• $L \propto \sqrt h$ ✅
• $L \propto \sqrt G$ ✅

Thus options C and D are correct.

4. Find the unit of mass

We want $[M] = [h^p c^q G^r]$

So, $[h^p c^q G^r] = M^{p-r} L^{2p+q+3r} T^{-p-q-2r}$

Compare with $M^1L^0T^0$:

1. Mass exponent: $p-r=1$

2. Length exponent: $2p+q+3r=0$

3. Time exponent: $-p-q-2r=0 \Rightarrow p+q+2r=0$

From the third equation, $q=-p-2r$

Substitute into the second: $2p+(-p-2r)+3r=0$ $p+r=0 \Rightarrow p=-r$

Now use $p-r=1$: $-r-r=1 \Rightarrow -2r=1 \Rightarrow r=-\frac12$

Thus, $p=\frac12$

and $q=-p-2r=-\frac12+1=\frac12$

Therefore, $M \propto h^{1/2} c^{1/2} G^{-1/2}$

So:

• $M \propto \sqrt c$ ✅
• $M \propto \sqrt G$ ❌, since actually $M \propto G^{-1/2}$

Thus option A is correct, B is incorrect.

5. Final answer

The correct options are: $\boxed{A,\ C,\ D}$

6. Comparison with stored answer

Stored correct answer: $D, A, C$

This matches exactly (order does not matter).