$A$ sonometer wire has a length of $114 ~cm$ between two fixed ends. Where should two bridges be placed so as to divide the wire into three segments (in $cm$) whose fundamental frequencies are in the ratio $1:3:4$?

$A$ metal wire $PQ$ slides on parallel metallic rails having a separation of $0.25\ m$,each having negligible resistance. There is a $2\,\Omega$ resistor and a $10\ V$ battery as shown in the figure. $A$ uniform magnetic field of magnitude $0.5\ T$ is directed into the plane of the paper. $A$ force of $0.5\ N$ to the left is required to keep the wire $PQ$ moving with a constant speed to the right. With what speed is the wire $PQ$ moving? (Neglect self-inductance of the loop)

Which of the following is concerned with eubacteria?
$(1)$ Methanogens $(2)$ Anabaena $(3)$ $N_2$ fixation $(4)$ Thermoacidophiles $(5)$ Eukaryotes $(6)$ Diatoms $(7)$ Gelatinous sheath $(8)$ Heterocyst

What is the output $Y$ in the following circuit,when all the three inputs $A, B, C$ are first $0$ then $1$?

If $y=y(x)$ is the solution of the differential equation $\left(\frac{2+\sin x}{y+1}\right) \frac{d y}{d x}+\cos x=0$ with $y(0)=1$,then $y\left(\frac{\pi}{2}\right)$ is equal to

Match List-$I$ with List-$II$ for the composition of substances and select the correct answer using the code given below the lists.

List-$I$ (Substances)	List-$II$ (Composition)
$A$. Plaster of Paris	$i$. $CaSO_4 \cdot 1/2H_2O$
$B$. Epsomite	$ii$. $MgSO_4 \cdot 7H_2O$
$C$. Kieserite	$iii$. $MgSO_4 \cdot H_2O$
$D$. Gypsum	$iv$. $CaSO_4 \cdot 2H_2O$
	$v$. $CaSO_4$