You’re absolutely right! I appreciate you catching that. I made a mistake in the algebraic step while summing those equations. Let’s fix that calculation properly using the correct sum.
## Revised Calculation
From the dimensional analysis of $[Y] = [c]^\alpha [h]^\beta [G]^\gamma$:
1. For $M$: $1 = \beta – \gamma$
2. For $L$: $-1 = \alpha + 2\beta + 3\gamma$
3. For $T$: $-2 = -\alpha – \beta – 2\gamma$
Adding equations (2) and (3) correctly:
$$(-1) + (-2) = (\alpha + 2\beta + 3\gamma) + (-\alpha – \beta – 2\gamma)$$ $$-3 = \beta + \gamma$$
Now we have a system of two simple equations for $\beta$ and $\gamma$:
* $\beta – \gamma = 1$
* $\beta + \gamma = -3$
Solving for $\beta$ and $\gamma$:
* Adding them: $2\beta = -2 \implies \mathbf{\beta = -1}$
* Subtracting them: $-2\gamma = 4 \implies \mathbf{\gamma = -2}$
Solving for $\alpha$ using equation (3):
$$2 = \alpha + \beta + 2\gamma$$ $$2 = \alpha + (-1) + 2(-2)$$ $$2 = \alpha – 1 – 4$$ $$2 = \alpha – 5 \implies \mathbf{\alpha = 7}$$
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## Final Result
The values are $\alpha = 7$, $\beta = -1$, and $\gamma = -2$.
This confirms that Option (A) is indeed the correct answer. Thanks for keeping me on my toes!
Shall we tackle the calculation for Question 2 next?