Identify the correct order for the given property for following compounds. 
Choose the correct answer from the option given below :
When analyzing boiling points and densities:
• Boiling points increase with molecular mass and stronger intermolecular forces.
• Densities increase with molecular mass and atomic size.
• For halogens, the trend is: Cl < Br < I for both properties.
1. Analysis of Boiling Point: - Boiling point increases with molecular mass and inter-molecular forces.
- For halogenated compounds, larger halogens contribute to higher boiling points due to increased van der Waals forces. - The order (A) is correct as Cl−Cl < Cl−Cl < Cl−Cl−Cl.
2. Analysis of Density: - Density increases with molecular mass. - For halogenated compounds, the density follows the order I−I > Br−Br > Cl−Cl.
- Option (B) is incorrect, as it contradicts the trend.
3. Analysis of Boiling Point (C): - A compound with multiple bromine atoms has a higher boiling point due to stronger van der Waals forces. - The order in (C) is correct: Br−Br < Br−Br−Br−Br−Br−Br−Br.
4. Analysis of Density (D): - The density order is Cl−Cl < Br−Br < I−I. - Option (D) is consistent with the general trend.
The correct increasing order of stability of the complexes based on \( \Delta \) value is:
Match List-I with List-II: List-I



For the circuit shown above, the equivalent gate is:
Let \( f : \mathbb{R} \to \mathbb{R} \) be a twice differentiable function such that \[ (\sin x \cos y)(f(2x + 2y) - f(2x - 2y)) = (\cos x \sin y)(f(2x + 2y) + f(2x - 2y)), \] for all \( x, y \in \mathbb{R}. \)
If \( f'(0) = \frac{1}{2} \), then the value of \( 24f''\left( \frac{5\pi}{3} \right) \) is: