A physical quantity `X` is represented by `X = (M^(x) L^(-y) T^(-z)`. The maximum percantage errors in the measurement of `M , L , and T`, respectively , are `a% , b% and c%`. The maximum percentage error in the measurement of `X` will be

Options

A. `( ax + by - cz)%`
B. `( ax - by - cz)%`
C. `( ax + by + cz)%`
D. `( ax - by + cz)%`

Explanation

To find the maximum percentage error in the measurement of the physical quantity \( X \) represented by the equation \( X = M^{x} L^{-y} T^{-z} \), we can follow these steps:

### Step 1: Understand the relationship of errors
The percentage error in a product or quotient of quantities can be determined by the formula:
\[ \text{Percentage Error in } X = \left| \frac{\partial X}{X} \right| \times 100 \]
For a quantity expressed in terms of powers of other quantities, the total percentage error can be calculated by summing the individual percentage errors, each multiplied by their respective power.

### Step 2: Write the expression for \( X \)
Given:
\[ X = M^{x} L^{-y} T^{-z} \]
We can express the total percentage error in \( X \) in terms of the percentage errors in \( M \), \( L \), and \( T \).

### Step 3: Apply the error propagation formula
The maximum percentage error in \( X \) can be calculated using the formula:
\[ \text{Percentage Error in } X = |x| \cdot a + |-y| \cdot b + |-z| \cdot c \]
Where:
- \( a \) is the maximum percentage error in \( M \)
- \( b \) is the maximum percentage error in \( L \)
- \( c \) is the maximum percentage error in \( T \)

### Step 4: Substitute the values
Substituting the values into the formula, we get:
\[ \text{Percentage Error in } X = |x| \cdot a + |y| \cdot b + |z| \cdot c \]

### Step 5: Conclusion
Thus, the maximum percentage error in the measurement of \( X \) is given by:
\[ \text{Maximum Percentage Error in } X = x \cdot a + y \cdot b + z \cdot c \]

### Final Answer
The maximum percentage error in the measurement of \( X \) is:
\[ x \cdot a + y \cdot b + z \cdot c \]