$A$ coil of area $80\, cm^2$ and $50\, turns$ is rotating with $2000$ revolutions per minute about an axis perpendicular to a magnetic field of $0.05\, T$. The maximum value of the $emf$ developed in it is

If $y=a e^x+b e^{-x}+c$,where $a, b, c$ are parameters,then $y^{\prime \prime \prime}$ is equal to

The maximum energy in the thermal radiation from a hot source occurs at a wavelength of $11 \times 10^{-5} \text{ cm}$. According to Wien's law,the temperature of the source (on Kelvin scale) will be $n$ times the temperature of another source (on Kelvin scale) for which the wavelength at maximum energy is $5.5 \times 10^{-5} \text{ cm}$. The value of $n$ is

Which of the following cylindrical rods will conduct the most heat when their ends are maintained at the same steady temperatures?

The polymer having an amide linkage is

Solution $X$ contains $Na_2CO_3$ and $NaHCO_3$. $20 \ mL$ of $X$ when titrated using methyl orange indicator consumed $60 \ mL$ of $0.1 \ M$ $HCl$ solution. In another experiment,$20 \ mL$ of $X$ solution when titrated using phenolphthalein consumed $20 \ mL$ of $0.1 \ M$ $HCl$ solution. The concentrations (in $mol \ L^{-1}$) of $Na_2CO_3$ and $NaHCO_3$ in $X$ are respectively: