Q.When an air column in a pipe closed at one end vibrates such that three nodes are formed in it, the frequency of its vibrations is ....... times the fundamental frequency. (A) 2
(B) 3
(C) 4
(D) 5

[!TLDR] For a pipe closed at one end, the p-th overtone has frequency $n_p=(2p+1)n$ and (p+1) nodes; 3 nodes means p+1=3, i.e. p=2, giving $n_2=(2\times2+1)n=5n$. [!ANSWER] (D) 5

For a pipe closed at one end (section 6.7.2), the closed end is always a node and the open end is always an antinode; the fundamental mode has just 1 node (at the closed end), the first overtone (third harmonic) has 2 nodes, the second overtone (fifth harmonic) has 3 nodes, and in general the mode with (p+1) nodes is the p-th overtone, of frequency $n_p=(2p+1)n$ where n is the fundamental. Here we are told 3 nodes are formed, so $p+1=3$, i.e. $p=2$, giving frequency $n_2=(2\times2+1)n=5n$ -- five times the fundamental frequency. This is consistent with the general rule that a closed pipe supports only ODD harmonics: with 3 nodes the air column is vibrating in its fifth-harmonic mode. [!ANSWER] (D) 5

Use the closed-pipe overtone formula $n_p=(2p+1)n$, matching the number of nodes (p+1) to the given count to find p, then substitute.

Confusing the node COUNT with the overtone/harmonic NUMBER directly (assuming 3 nodes means the 3rd harmonic, i.e. 3n) instead of correctly working out that 3 nodes corresponds to p=2, i.e. the 5th harmonic.