Flux $\phi$ (in weber) in a closed circuit of resistance $10\,\Omega$ varies with time $t$ (in sec) according to the equation $\phi = 6t^2 - 5t + 1$. What is the magnitude of the induced current at $t = 0.25\,s$ (in $,A$)?

The voltage of an $ac$ source varies with time according to the equation $V = 100 \sin(100 \pi t) \cos(100 \pi t)$,where $t$ is in seconds and $V$ is in volts. Then:

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:

The solution of the differential equation $\frac{dy}{dx} + \frac{3x^2}{1+x^3}y = \frac{\sin^2 x}{1+x^3}$ is

If $\cos x + \cos^2 x = 1$,then the value of $\sin^2 x + \sin^4 x$ is

$A$ rod of glass $(\mu = 1.5)$ with a square cross-section is bent into the shape shown in the figure. $A$ parallel beam of light falls on the plane flat surface $A$ as shown. If $d$ is the width of a side and $R$ is the radius of the inner circular arc, then for what maximum value of $\frac{d}{R}$ will light entering the glass slab through surface $A$ emerge from the glass through surface $B$?