For rate-concentration graphs:
• Linear regions indicate first-order behavior.
• A constant rate indicates zero-order behavior.
• Non-linear regions suggest fractional orders of reaction.
1. Analysis of the Graph:
- In region-I, the rate of reaction increases linearly with the concentration, which is characteristic of a first-order reaction.
- In region-II, the graph shows non-linear behavior, indicating that the reaction order is fractional (in the range of 0.1 to 0.9).
- In region-III, the rate becomes constant, indicating a zero-order reaction.
2. Verification of Statements:
- (A) Incorrect. The overall order cannot be determined directly from the graph as it changes across regions.
- (B) Incorrect. The order can be inferred for specific regions.
- (C) Correct. Region-I corresponds to first-order behavior, and region-III corresponds to zero-order behavior.
- (D) Incorrect. In region-II, the reaction is not of first order.
- (E) Correct. In region-II, the reaction order lies between 0.1 and 0.9.
3. Conclusion:
- The correct statements are (C) and (E).
Let one focus of the hyperbola $ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 $ be at $ (\sqrt{10}, 0) $, and the corresponding directrix be $ x = \frac{\sqrt{10}}{2} $. If $ e $ and $ l $ are the eccentricity and the latus rectum respectively, then $ 9(e^2 + l) $ is equal to:
The largest $ n \in \mathbb{N} $ such that $ 3^n $ divides 50! is: