The fundamental frequency of a wire stretched by $2 \ kgwt$ is $100 \ Hz$. The weight required to produce its octave is (in $kgwt$)

If the length of a stretched string is reduced by $40 \%$ and the tension is increased by $44 \%$,then the ratio of the final to the initial frequencies of the stretched string is:

Answer the following by appropriately matching the lists based on the information given in the paragraph.
$A$ 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 $L_0$ and $2L_0$. It is found that in string-$1$ $(\mu)$ at free length $L_0$ and tension $T_0$ the fundamental mode frequency is $f_0$.
$List-I$ gives the above four strings while $List-II$ lists the magnitude of some quantity.

$List-I$	$List-II$
$(I)$ String-$1$ $(\mu)$	$(P) 1$
$(II)$ String-$2$ $(2\mu)$	$(Q) 1/2$
$(III)$ String-$3$ $(3\mu)$	$(R) 1/\sqrt{2}$
$(IV)$ String-$4$ $(4\mu)$	$(S) 1/\sqrt{3}$
	$(T) 3/16$
	$(U) 1/16$

$(1)$ If the tension in each string is $T_0$,the correct match for the fundamental frequency in $f_0$ units will be,
$(1)$ $I \rightarrow P, II \rightarrow R, III \rightarrow S, IV \rightarrow Q$
$(2)$ $I \rightarrow P, II \rightarrow Q, III \rightarrow T, IV \rightarrow S$
$(3)$ $I \rightarrow Q, II \rightarrow S, III \rightarrow R, IV \rightarrow P$
$(4)$ $I \rightarrow Q, II \rightarrow P, III \rightarrow R, IV \rightarrow T$
$(2)$ The lengths of the strings $1, 2, 3$ and $4$ are kept fixed at $L_0, 3L_0/2, 5L_0/4$ and $7L_0/4$,respectively. Strings $1, 2, 3$ and $4$ are vibrated at their $1^{st}, 3^{rd}, 5^{th}$ and $14^{th}$ harmonics,respectively,such that all the strings have the same frequency. The correct match for the tension in the four strings in the units of $T_0$ will be.
$(1)$ $I \rightarrow P, II \rightarrow Q, III \rightarrow T, IV \rightarrow U$
$(2)$ $I \rightarrow T, II \rightarrow Q, III \rightarrow R, IV \rightarrow U$
$(3)$ $I \rightarrow P, II \rightarrow Q, III \rightarrow R, IV \rightarrow T$
$(4)$ $I \rightarrow P, II \rightarrow R, III \rightarrow T, IV \rightarrow U$

$A$ sonometer wire of length $114\, cm$ is fixed at both the ends. Where should the two bridges be placed so as to divide the wire into three segments whose fundamental frequencies are in the ratio $1 : 3 : 4$?

The correct graph between the frequency $n$ and the square root of density $(\rho)$ of a wire,keeping its length,radius,and tension constant,is:

$A$ thin wire of length $99 \ cm$ is fixed at both ends as shown in the figure. The wire is kept under a tension and is divided into three segments of lengths $l_1, l_2$ and $l_3$ as shown in the figure. When the wire is made to vibrate,the segments vibrate respectively with their fundamental frequencies in the ratio $1: 2: 3$. Then,the lengths $l_1, l_2$ and $l_3$ of the segments respectively are (in $cm$):