$A$ $100 \ W$ bulb is rated at $200 \ V$ and a $200 \ W$ bulb is rated at $100 \ V$. Then:

If $E = at + bt^2$,what is the temperature of inversion?

$A$ thermocouple develops $200\,\mu V$ between $0\,^{\circ}C$ and $100\,^{\circ}C$. If it develops $64\,\mu V$ and $76\,\mu V$ respectively between $(0\,^{\circ}C - 32\,^{\circ}C)$ and $(32\,^{\circ}C - 70\,^{\circ}C)$,then what will be the thermo-emf it develops between $70\,^{\circ}C$ and $100\,^{\circ}C$ in $\mu V$?

At time $t = 0$,terminal $A$ in the circuit shown in the figure is connected to $B$ by a key and alternating current $I(t) = I_0 \cos(\omega t)$,with $I_0 = 1 \text{ A}$ and $\omega = 500 \text{ rad s}^{-1}$ starts flowing in it with the initial direction shown in the figure.
At $t = \frac{7\pi}{6\omega}$,the key is switched from $B$ to $D$. Now onwards only $A$ and $D$ are connected. $A$ total charge $Q$ flows from the battery to charge the capacitor fully. If $C = 20 \mu\text{F}$,$R = 10 \Omega$ and the battery is ideal with emf of $50 \text{ V}$,identify the correct statement$(s)$.
$(A)$ Magnitude of the maximum charge on the capacitor before $t = \frac{7\pi}{6\omega}$ is $1 \times 10^{-3} \text{ C}$.
$(B)$ The current in the left part of the circuit just before $t = \frac{7\pi}{6\omega}$ is clockwise.
$(C)$ Immediately after $A$ is connected to $D$,the current in $R$ is $10 \text{ A}$.
$(D)$ $Q = 2 \times 10^{-3} \text{ C}$.

$A$ conductor is made of an isotropic material and has the shape of a truncated cone. $A$ battery of constant emf is connected across it and its left end is earthed as shown in the figure. If at a section distant $x$ from the left end, electric field intensity, potential, and the rate of generation of heat per unit length are $E, V$, and $H$ respectively, which of the following graphs is/are correct?

$A$ cell of emf $\varepsilon$ and internal resistance $r$ is connected across a variable load resistance $R$. The graph drawn between its terminal voltage $V$ and resistance $R$ is