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Assertion : If x=(a^(n))/(b^(m))" the "...

Assertion : If `x=(a^(n))/(b^(m))" the "(Deltax)/(x)=n((pmDeltaa)/(a))-m((pmDeltab)/(b))`
The change in `a` or `b` i.e., `DeltaaorDeltab` may be comparable to `a and b`.
Reason : The above relation is valid when `Deltaaltlta` and `Deltabltltb`.

A

If both Assertion and Reason are correct and Reason is the correct explaination of Assertion.

B

If both Assertion and Reason are correct but Reason is not the correct explaination of Assertion.

C

If Assertion is true but Reason is false.

D

If Assertion is false but Reason is true.

Text Solution

AI Generated Solution

The correct Answer is:
To solve the given problem, we need to analyze both the assertion and the reason provided in the question. ### Step 1: Analyze the Assertion The assertion states that if \( x = \frac{a^n}{b^m} \), then the relative change in \( x \) is given by: \[ \frac{\Delta x}{x} = n \left( \frac{\Delta a}{a} \right) - m \left( \frac{\Delta b}{b} \right) \] This formula is derived from the rules of propagation of uncertainties. The assertion claims that the changes \( \Delta a \) or \( \Delta b \) may be comparable to \( a \) and \( b \). **Conclusion for Assertion**: The assertion is **incorrect** because the formula should include a plus sign, indicating that the changes in \( a \) and \( b \) contribute positively to the uncertainty in \( x \). ### Step 2: Analyze the Reason The reason states that the above relation is valid when \( \Delta a \ll a \) and \( \Delta b \ll b \). This means that the changes in \( a \) and \( b \) must be much smaller than their respective values for the relation to hold true. **Conclusion for Reason**: The reason is **correct**. The relation is indeed valid under the condition that the changes in the measurements are significantly smaller than the original values. ### Step 3: Determine the Relationship Between Assertion and Reason Now we need to determine the relationship between the assertion and the reason: - The assertion is false. - The reason is true. ### Final Conclusion Based on the analysis: - The assertion is false, and the reason is true. Thus, the correct option is: **Assertion is false, but reason is true.**
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