A physical quanity is represented by `X=[M^(a)L^(b)T^(-c)]`. If percentage error in the measurement of M, L and T are `alpha%,beta%andgamma%` respectively, then maximum percentage error in measurement of X should be (Given that `alpha,betaandgamma` are very small)

Options

A. `(alphaa-betab+gammac)%`
B. `(alphaa+betab+gammac)%`
C. `(alphaa-betab-gammac)%`
D. `(alphaa+betab-gammac)%`

Answer

To solve the problem, we need to determine the maximum percentage error in the measurement of the physical quantity \( X \) represented by the equation:

\[ X = [M^a L^b T^{-c}] \]

where \( M \), \( L \), and \( T \) are the physical quantities with their respective percentage errors given as \( \alpha\% \), \( \beta\% \), and \( \gamma\% \).

### Step-by-Step Solution:

1. **Understanding the Relationship**:
The physical quantity \( X \) depends on the quantities \( M \), \( L \), and \( T \) raised to certain powers. The percentage error in \( X \) can be derived from the percentage errors in \( M \), \( L \), and \( T \).

2. **Using the Formula for Percentage Error**:
The formula for the percentage error in a quantity that is a product of powers of other quantities is given by:
\[ \frac{\Delta X}{X} = a \frac{\Delta M}{M} + b \frac{\Delta L}{L} - c \frac{\Delta T}{T} \]
where \( \Delta M \), \( \Delta L \), and \( \Delta T \) are the absolute errors in \( M \), \( L \), and \( T \), respectively.

3. **Substituting the Percentage Errors**:
We know that:
- \( \frac{\Delta M}{M} = \frac{\alpha}{100} \)
- \( \frac{\Delta L}{L} = \frac{\beta}{100} \)
- \( \frac{\Delta T}{T} = \frac{\gamma}{100} \)

Substituting these into the formula gives:
\[ \frac{\Delta X}{X} = a \left(\frac{\alpha}{100}\right) + b \left(\frac{\beta}{100}\right) - c \left(\frac{\gamma}{100}\right) \]

4. **Calculating the Maximum Percentage Error**:
To find the maximum percentage error in \( X \), we need to consider the absolute values of the errors. Therefore, we take the absolute value of the terms:
\[ \text{Maximum Percentage Error in } X = |a| \alpha + |b| \beta + |c| \gamma \]

Since \( c \) is negative in the original expression, we can rewrite the equation as:
\[ \text{Maximum Percentage Error in } X = a \alpha + b \beta + c \gamma \]

5. **Final Expression**:
Thus, the maximum percentage error in the measurement of \( X \) is given by:
\[ \text{Maximum Percentage Error in } X = a \alpha + b \beta + c \gamma \]

### Conclusion:
The maximum percentage error in the measurement of \( X \) is \( a \alpha + b \beta + c \gamma \).