




Step 1: Definition of acidic hydrogen
An acidic hydrogen is one that can be easily removed as a proton (\( \text{H}^+ \)), resulting in a stable conjugate base. The more stable the conjugate base, the more acidic the hydrogen.
Step 2: Analyze conjugate base stability
- In Option (B), the removal of the hydrogen leads to a carbanion which is resonance stabilized by conjugation with two adjacent double bonds. This forms an allylic anion extended over a conjugated cyclic system. - In Options (A), (C), and (D), although allylic positions are present, the resonance stabilization is less extensive compared to (B). Especially, (A) and (C) are sterically hindered or less effectively delocalized.
Step 3: Resonance in Option (B)
Removing the proton from the carbon adjacent to two double bonds gives a conjugate base that can delocalize the negative charge over a five-carbon conjugated system, similar to a cyclopentadienyl-like system — which is known for its aromatic stabilization when fully conjugated.
Conclusion: Option (B) has the most stable conjugate base, hence the most acidic hydrogen.
Final Answer: \( \boxed{\text{B}} \)
As shown in the figures, a uniform rod $ OO' $ of length $ l $ is hinged at the point $ O $ and held in place vertically between two walls using two massless springs of the same spring constant. The springs are connected at the midpoint and at the top-end $ (O') $ of the rod, as shown in Fig. 1, and the rod is made to oscillate by a small angular displacement. The frequency of oscillation of the rod is $ f_1 $. On the other hand, if both the springs are connected at the midpoint of the rod, as shown in Fig. 2, and the rod is made to oscillate by a small angular displacement, then the frequency of oscillation is $ f_2 $. Ignoring gravity and assuming motion only in the plane of the diagram, the value of $\frac{f_1}{f_2}$ is:
The reaction sequence given below is carried out with 16 moles of X. The yield of the major product in each step is given below the product in parentheses. The amount (in grams) of S produced is ____. 
Use: Atomic mass (in amu): H = 1, C = 12, O = 16, Br = 80
Let $ a_0, a_1, ..., a_{23} $ be real numbers such that $$ \left(1 + \frac{2}{5}x \right)^{23} = \sum_{i=0}^{23} a_i x^i $$ for every real number $ x $. Let $ a_r $ be the largest among the numbers $ a_j $ for $ 0 \leq j \leq 23 $. Then the value of $ r $ is ________.
Let $ \mathbb{R} $ denote the set of all real numbers. Then the area of the region $$ \left\{ (x, y) \in \mathbb{R} \times \mathbb{R} : x > 0, y > \frac{1}{x},\ 5x - 4y - 1 > 0,\ 4x + 4y - 17 < 0 \right\} $$ is