A physical quantity Q is related to measurable quantities L,M and N as `Q = KL^(-2)M^(2)N^(3)`, where k is constant. If the % errors in the measurements of L, M and N are respectively 1%, 2% and 1%, then the % error in hc mcasurement of Q is

Options

A. `9%`
B. `4%`
C. `2%`
D. `1%`

Answer

To find the percentage error in the measurement of the physical quantity \( Q \), which is related to measurable quantities \( L \), \( M \), and \( N \) as given by the equation:

\[ Q = K L^{-2} M^{2} N^{3} \]

where \( K \) is a constant, we can follow these steps:

### Step 1: Understand the relationship and errors
The percentage errors in the measurements of \( L \), \( M \), and \( N \) are given as:
- Percentage error in \( L \) = 1%
- Percentage error in \( M \) = 2%
- Percentage error in \( N \) = 1%

### Step 2: Write the formula for percentage error in \( Q \)
The formula for the percentage error in \( Q \) based on the powers of \( L \), \( M \), and \( N \) is given by:

\[ \frac{\Delta Q}{Q} = 2 \cdot \frac{\Delta M}{M} + (-2) \cdot \frac{\Delta L}{L} + 3 \cdot \frac{\Delta N}{N} \]

Where:
- \( \Delta Q \) is the absolute error in \( Q \)
- \( \Delta L \), \( \Delta M \), and \( \Delta N \) are the absolute errors in \( L \), \( M \), and \( N \) respectively.

### Step 3: Substitute the percentage errors
Now, substituting the percentage errors into the formula:

\[ \frac{\Delta Q}{Q} = 2 \cdot \left(\frac{2}{100}\right) + (-2) \cdot \left(\frac{1}{100}\right) + 3 \cdot \left(\frac{1}{100}\right) \]

### Step 4: Calculate the contributions
Calculating each term:

1. Contribution from \( M \):
\[ 2 \cdot \frac{2}{100} = \frac{4}{100} \]

2. Contribution from \( L \):
\[ -2 \cdot \frac{1}{100} = -\frac{2}{100} \]

3. Contribution from \( N \):
\[ 3 \cdot \frac{1}{100} = \frac{3}{100} \]

### Step 5: Combine the contributions
Now, combine these contributions:

\[ \frac{\Delta Q}{Q} = \frac{4}{100} - \frac{2}{100} + \frac{3}{100} = \frac{5}{100} = 0.05 \]

### Step 6: Convert to percentage
To find the percentage error in \( Q \):

\[ \text{Percentage error in } Q = 0.05 \times 100 = 5\% \]

### Final Answer
The percentage error in the measurement of \( Q \) is **5%**.

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