In order to double the frequency of the fundamental note emitted by a stretched string, the length is reduced to $\frac{3}{4}$ of the original length and the tension is changed. The factor by which the tension is to be changed is

$A$ body is suspended from a string of length $1 \ m$ and mass $2 \ g$. The mass of the body required to produce a fundamental mode of $100 \ Hz$ frequency in the string is (Acceleration due to gravity $= 10 \ m \ s^{-2}$)

If $n_{1}, n_{2}$ and $n_{3}$ are the fundamental frequencies of three segments into which a string is divided,then the original fundamental frequency $n$ of the string is given by

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.
$(1)$ $I \rightarrow P, II \rightarrow Q, III \rightarrow T, IV \rightarrow U$
$(2)$ $I \rightarrow T, II \rightarrow Q, III \rightarrow R, IV \rightarrow U$
$(3)$ $I \rightarrow P, II \rightarrow Q, III \rightarrow R, IV \rightarrow T$
$(4)$ $I \rightarrow P, II \rightarrow R, III \rightarrow T, IV \rightarrow U$

The fundamental frequency of a sonometer wire increases by $6\, Hz$ if its tension is increased by $44\%$,keeping the length constant. The frequency of the wire is ...... $Hz$.

$A$ steel wire is used to stretch a spring. An oscillating magnetic field drives the steel wire up and down. $A$ standing wave with three antinodes is created when the spring is stretched by $4.0\, cm$. What stretch of the spring produces a standing wave with two antinodes with the same frequency (in $, cm$)?