Step 1: Analyze statement A.
Excessive nutrient inputs (eutrophication) cause massive algal blooms. When these algae die, they are decomposed by bacteria, a process that consumes large amounts of dissolved oxygen, leading to hypoxia or anoxia (low or no oxygen). Thus, the statement is incorrect.
Step 2: Analyze statement B.
Extensive algal blooms are a primary symptom of eutrophication, which is a state of poor ecological health for a lake, often leading to fish kills and loss of biodiversity. Thus, the statement is incorrect.
Step 3: Analyze statement C.
Eutrophic lakes are rich in nutrients and have high biological productivity. This leads to a large amount of organic matter (dead algae, etc.) that needs to be decomposed. This decomposition requires a lot of oxygen, meaning the lake has a high Biochemical Oxygen Demand (BOD). This statement is correct.
Step 4: Analyze statement D.
Oligotrophic lakes are nutrient-poor and have low biological productivity. There is little organic matter to decompose, so they have a very low BOD. The statement claims they have high BOD compared to eutrophic lakes, which is the exact opposite of the truth. Thus, the statement is incorrect.
If \(f(t)\) is the inverse Laplace transform of \( F(s) = \frac{s+1+s^{-2}}{s^2-1} \), then \(f(t)\) is
Match LIST-I with LIST-II
LIST-I (Differential Equation)
(A) \(\frac{dy}{dx} = 2x(y-x^2+1)\)
(B) \(x\frac{dy}{dx} + 2(x^2+1)y=6\)
(C) \((x^2+1)\frac{dy}{dx} + 2xy = x \sin x\)
(D) \(x^3\frac{dy}{dx} + 2xy = 2x^2e^{x^2}\)
LIST-II (Integrating Factor)
(I) \(x^2\)
(II) \(e^{-x^2}\)
(III) \(x^2e^x\)
(IV) \(1+x^2\)
Choose the correct answer from the options given below: