Step 1: Analyze statement A.
Pyrite (FeS\(_2\)) is a sulfide mineral. When exposed to air and water, it oxidizes to form sulfuric acid (H\(_2\)SO\(_4\)), a strong acid. This process, known as acid mine drainage, drastically increases soil and water acidity. Statement A is a cause of acidity.
Step 2: Analyze statement B.
The primary process for extracting aluminum (Bayer process) involves digesting bauxite ore in a hot solution of sodium hydroxide (a strong base). The waste product, red mud, is highly alkaline, not acidic. Statement B is not a cause of acidity.
Step 3: Analyze statement C.
Ammonium-based nitrogen fertilizers can cause soil acidity through nitrification. Soil bacteria convert ammonium (NH\(_4^+\)) to nitrate (NO\(_3^-\)), releasing hydrogen ions (H\(^+\)) in the process, which lowers the soil pH. Statement C is a cause of acidity.
Step 4: Analyze statement D.
Limestone is calcium carbonate (CaCO\(_3\)). It is an alkaline material used to neutralize soil acidity by reacting with H\(^+\) ions. It is a treatment for acidity, not a cause. Statement D is not a cause of acidity.
Conclusion: A and C are causes of soil acidity.
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: