At $27^{\circ} C$,$500 \ mL$ of helium diffuses in $30 \ minutes$. What is the time (in hours) taken for $1000 \ mL$ of $SO_2$ to diffuse under the same experimental conditions?

For any integer $n \geq 2$,let $I_n = \int \tan^n x \, dx$. If $I_n = \frac{1}{a} \tan^{n-1} x - b I_{n-2}$ for $n \geq 2$,then the ordered pair $(a, b)$ equals to

$A$ diode having a potential difference of $0.5\,V$ across its junction,which does not depend on the current,is connected in series with a resistance of $20\,\Omega$ across a source. If $0.1\,A$ current passes through the resistance,what is the voltage of the source (in $,V$)?

$A$ paramagnetic sample shows a net magnetization of $8 \ Am^{-1}$ when placed in an external magnetic field of $0.6 \ T$ at a temperature of $4 \ K$. When the same sample is placed in an external magnetic field of $0.2 \ T$ at a temperature of $16 \ K$,the magnetization will be

If $y = \frac{e^x \log x}{x^2}$,then $\frac{dy}{dx} = $

When $100 \, V$ $DC$ is applied across a solenoid,a current of $1 \, A$ flows in it. When $100 \, V$ $AC$ is applied across the same coil,the current drops to $0.5 \, A$. If the frequency of the $AC$ source is $50 \, Hz$,the impedance and inductance of the solenoid are: