Step 1: Analyze Aluminium (C)
The primary ore of aluminium is bauxite, which is a rock composed mainly of aluminium hydroxide minerals, the most common of which is Gibbsite.
Thus, C matches with (II).
Step 2: Analyze Lithium (D)
Lithium is commercially extracted from several minerals, including spodumene and Lepidolite, which is a lithium-bearing mica.
Thus, D matches with (III).
Step 3: Analyze Manganese (A)
Major ores of manganese include pyrolusite and Psilomelane.
Thus, A matches with (IV).
Step 4: Analyze Zinc (B)
The main ore of zinc is sphalerite (zinc sulfide). "Sinsilfuoride" (I) is not a recognized mineral and is likely a typo. By elimination based on the other correct matches, B is paired with I.
Conclusion: The matching is A-IV, C-II, D-III. Option (C) contains this pattern (A-IV, B-I, C-II, D-III).
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: