JEE Advanced · 2019 · 27 May 2019 · Paper 2 · Waves & Sound

A musical instrument is made using four different metal strings, 1,2,3 and 4 with mass per unit length $\mu ,2\mu ,3\mu$ and $4\mu$ respectively. The instrument is played by vibrating the strings by varying the free length in between the range ${\mathrm{L}}_0$ and $2{\mathrm{L}}_0$. It is found that in string-1 $(\mu )$ at free length ${\mathrm{L}}_0$ and tension ${\mathrm{T}}_0$ the fundamental mode frequency is ${\mathrm{f}}_0$.

List-I gives the above four strings while list-II lists the magnitude of some quantity.

List-I
(I) String-1 $(\mu )$
(II) String-2 $(2\mu )$
(III) String-3 $(3\mu )$
(IV) String-4 $(4\mu )$

List-II
(P) 1
(Q) 1/2
(R) $1/\sqrt{2}$
(S) $1/\sqrt{3}$
(T) 3/16
(U) 1/16

The length of the string 1,2,3 and 4 are kept fixed at ${\mathrm{L}}_0,\frac{3{\mathrm{L}}_0}{2},\frac{5{\mathrm{L}}_0}{4}$ and $\frac{7{\mathrm{L}}_0}{4}$, respectively. Strings 1,2,3 and 4 are vibrated at their $1^{\text{st }},3^{\text{rd }},5^{\text{th }}$ and $14^{\text{th }}$ harmonics, respectively such that all the strings have same frequency. The correct match for the tension in the four strings in the units of ${\mathrm{T}}_0$ will be.