Step 1: Analyze Statement I
Ozonolysis is a reaction where an alkene undergoes cleavage in the presence of ozone (\( \text{O}_3 \)), forming an ozonide intermediate. This ozonide is then reduced by zinc (\( \text{Zn} \)) and water (\( \text{H}_2 \text{O} \)) to give two carbonyl compounds. For cis-2-butene, the reaction proceeds as follows: \[ \text{cis-2-butene} \xrightarrow{O_3} \text{ozonide intermediate} \xrightarrow{Zn, H_2O} \text{ethanal (acetaldehyde)}. \] When cis-2-butene undergoes ozonolysis, the double bond is cleaved symmetrically, producing two molecules of ethanal (acetaldehyde), which is a simple aldehyde. Therefore, Statement I is true
.
Step 2: Analyze Statement II
For \(\textit{3, 6-dimethyloct-4-ene}\), when ozonolysis occurs followed by treatment with \( \text{Zn} \) and \( \text{H}_2 \text{O} \), the reaction will lead to cleavage of the double bond, forming two products. The key point here is the symmetry of the molecule. 3, 6-dimethyloct-4-ene is a symmetric compound, and the cleavage of the double bond leads to the formation of two carbonyl groups. The product is symmetrical, and therefore, it will not have any chiral centers because both products are non-chiral and identical. In other words, the two products obtained are symmetrical and lack any chiral centers, making Statement II false.
The reaction proceeds as follows:
Step I: Starting with cis-2-butene: \[ \text{Cis-2-butene} \xrightarrow{(i) \, O_2 / Zn, (ii) \, H_2O} \text{Ethanol} \]
Step II: Starting with 3-methylpent-2-ene: \[ \text{3-methylpent-2-ene} \xrightarrow{(i) \, O_2 / Zn, (ii) \, H_2O} 2 \, \text{CH}_3\text{CH}_2\text{C}\left( \text{OH} \right)\text{CH}_3 \] Here, the reaction involves the oxidation of cis-2-butene and a subsequent hydration step to produce ethanol.



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: