Match List I with List - II
| List - I (Set Laws) | List - II (Form) |
|---|---|
| A. Idempotent law | III. \(A \cup A = A\) |
| B. Commutative law | I. \(A \cup B = B \cup A\) |
| C. Associative law | II. \(A \cap (B \cap C) = (A \cap B) \cap C\) |
| D. De-Morgan law | IV. \((A \cap B)' = A' \cup B'\) |
Let \( y = f(x) \) be the solution of the differential equation\[\frac{dy}{dx} + \frac{xy}{x^2 - 1} = \frac{x^6 + 4x}{\sqrt{1 - x^2}}, \quad -1 < x < 1\] such that \( f(0) = 0 \). If \[6 \int_{-1/2}^{1/2} f(x)dx = 2\pi - \alpha\] then \( \alpha^2 \) is equal to ______.
If \[ \frac{dy}{dx} + 2y \sec^2 x = 2 \sec^2 x + 3 \tan x \cdot \sec^2 x \] and
and \( f(0) = \frac{5}{4} \), then the value of \[ 12 \left( y \left( \frac{\pi}{4} \right) - \frac{1}{e^2} \right) \] equals to: