The velocity of sound in a gas. in which two wavelengths $4.08\,m$ and $4.16\,m$ produce $40$ beats in $12\,s$, will be ..............$ms ^{-1}$
JEE MAIN 2022, Medium
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List $-I$ gives the above four strings while list $-II$ lists the magnitude of some quantity.
($1$) If the tension in each string is $T _0$, the correct match for the highest fundamental frequency in $f _0$ units will be,List $-I$ List $-II$ $(I)$ String $-1( \mu$ ) $(P) 1$ $(II)$ String $-2 (2 \mu)$ $(Q)$ $1 / 2$ $(III)$ String $-3 (3 \mu)$ $(R)$ $1 / \sqrt{2}$ $(IV)$ String $-4 (4 \mu)$ $(S)$ $1 / \sqrt{3}$ $(T)$ $3 / 16$ $(U)$ $1 / 16$
$(1)$ $I \rightarrow P , II \rightarrow R , III \rightarrow S , IV \rightarrow Q$
$(2)$ $I \rightarrow P , II \rightarrow Q , III \rightarrow T , IV \rightarrow S$
$(3)$ $I \rightarrow Q , II \rightarrow S , III \rightarrow R , IV \rightarrow P$
$(4)$ I $\rightarrow Q , II \rightarrow P , III \rightarrow R$, IV $\rightarrow T$
($2$) The length of the string $1,2,3$ and 4 are kept fixed at $L _0, \frac{3 L _0}{2}, \frac{5 L _0}{4}$ and $\frac{7 L _0}{4}$, respectively. Strings $1,2,3$ and 4 are vibrated at their $1^{\text {tt }}, 3^{\text {rd }}, 5^{\text {m }}$ and $14^{\star}$ harmonics, respectively such that all the strings have same frequency. The correct match for the tension in the four strings in the units of $T _0$ will be.
$(1)$ $I \rightarrow P , II \rightarrow Q , III \rightarrow T , IV \rightarrow U$
$(2)$ $I \rightarrow T , II \rightarrow Q , III \rightarrow R$, IV $\rightarrow U$
$(3)$ $I \rightarrow P , II \rightarrow Q , III \rightarrow R , IV \rightarrow T$
$(4)$ I $\rightarrow P , II \rightarrow R , III \rightarrow T , IV \rightarrow U$