Step 1: Recall the properties of equivalence relations: a relation \( R \) is an equivalence relation if it is reflexive, symmetric, and transitive.
Step 2: - Reflexive: For every \( x \) in the given interval, \( x R x \) must hold. That is, we check if \( \sec^2 x - \tan^2 x = 1 \). This is true for all \( x \) in the interval \( \left[ 0, \frac{\pi}{2} \right] \), so the relation is reflexive.
- Symmetric: For the relation to be symmetric, if \( x R y \), then \( y R x \) must also hold. Since the equation involves both \( x \) and \( y \) in a symmetric manner, the relation is symmetric.
- Transitive: For transitivity, if \( x R y \) and \( y R z \), then \( x R z \) must hold. This property holds as well, meaning the relation is transitive. Thus, \( R \) is reflexive, symmetric, and transitive, so it is an equivalence relation.
Let a line passing through the point $ (4,1,0) $ intersect the line $ L_1: \frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4} $ at the point $ A(\alpha, \beta, \gamma) $ and the line $ L_2: x - 6 = y = -z + 4 $ at the point $ B(a, b, c) $. Then $ \begin{vmatrix} 1 & 0 & 1 \\ \alpha & \beta & \gamma \\ a & b & c \end{vmatrix} \text{ is equal to} $
Resonance in X$_2$Y can be represented as
The enthalpy of formation of X$_2$Y is 80 kJ mol$^{-1}$, and the magnitude of resonance energy of X$_2$Y is: