Match the list-I with list-II 
Step 1: Understanding the reagents and their uses.
We need to match the reagents from List-I with the corresponding chemical reactions or reagents used in List-II. Let's analyze each reagent and its application:
Step 2: Reagent analysis.
- (I) Tollen's reagent: Tollen's reagent is used for the detection of aldehydes, particularly in the silver mirror test. It contains \(\text{Cu}^{2+}\) and \(\text{OH}^-\) ions. Hence, it matches with \((P)\).
- (II) Fehling's reagent: Fehling's reagent is used to test for the presence of reducing sugars and aldehydes, where it reacts with \([Ag(NH_3)_2]^+\) and \(\text{OH}^-\) ions. It corresponds with \((S)\).
- (III) Williamson method: The Williamson ether synthesis method uses an alkoxide (\(RO^-\)) reacting with alkyl halides (R-X) to form ethers, which corresponds with \((R)\).
- (IV) Bayer's reagent: Bayer's reagent is used for the oxidation of alkenes to diols using diluted potassium permanganate (\(KMnO_4\)), hence it corresponds with \((R)\).
Step 3: Conclusion.
The correct matching is (B) because each reagent corresponds with the correct chemical reagent or reaction as described. (B) (I)-(P), (II)-(S), (III)-(R), (IV)-(R).
Let the lines $L_1 : \vec r = \hat i + 2\hat j + 3\hat k + \lambda(2\hat i + 3\hat j + 4\hat k)$, $\lambda \in \mathbb{R}$ and $L_2 : \vec r = (4\hat i + \hat j) + \mu(5\hat i + + 2\hat j + \hat k)$, $\mu \in \mathbb{R}$ intersect at the point $R$. Let $P$ and $Q$ be the points lying on lines $L_1$ and $L_2$, respectively, such that $|PR|=\sqrt{29}$ and $|PQ|=\sqrt{\frac{47}{3}}$. If the point $P$ lies in the first octant, then $27(QR)^2$ is equal to}
As shown in the figure, the ratio of \(T_1\) and \(T_2\) is 