If the transverse and conjugate axes of a hyperbola are equal,then its eccentricity is

$A$ bullet of mass $m$ strikes a block of mass $M$ connected to a light spring of stiffness $k,$ with a speed $v_0.$ If the bullet gets embedded in the block,then the maximum compression in the spring is

Water flows in a streamlined manner through a capillary tube of radius $a$,the pressure difference being $P$ and the rate of flow $Q$. If the radius is reduced to $a/2$ and the pressure is increased to $2P$,the rate of flow becomes:

Consider a reaction $A+R \rightarrow$ Product. The rate of this reaction is measured to be $k[A][R]$. At the start of the reaction, the concentration of $R,[R]_0$, is $10-$times the concentration of $A,[A]_0$. The reaction can be considered to be a pseudo first order reaction with assumption that $k[R]=k^{\prime}$ is constant. Due to this assumption, the relative error $(in \%)$ in the rate when this reaction is $40 \%$ complete, is $....$
$[ k$ and $k^{\prime}$ represent corresponding rate constants$]$

In the following compounds,the order of acidity is:
$(I)$ Phenol
$(II)$ $p$-Cresol ($4$-methylphenol)
$(III)$ $m$-Nitrophenol
$(IV)$ $p$-Nitrophenol

Match the thermodynamic processes taking place in a system with the correct conditions. In the table: $\Delta Q$ is the heat supplied,$\Delta W$ is the work done and $\Delta U$ is change in internal energy of the system.

Process	Condition
$(I)$ Adiabatic	$(A)$ $\Delta W = 0$
$(II)$ Isothermal	$(B)$ $\Delta Q = 0$
$(III)$ Isochoric	$(C)$ $\Delta U \neq 0, \Delta W \neq 0, \Delta Q \neq 0$
$(IV)$ Isobaric	$(D)$ $\Delta U = 0$