The relation between the current $i$ (in ampere) in a conductor and the time $t$ (in second) is $i=12t+9t^2$. The charge passing through the conductor between the times $t=2 \ s$ and $t=10 \ s$ is (in $C$)

$A$ spring balance is attached to the ceiling of a lift. $A$ man hangs his bag on the spring and the spring balance reads $49 ~N$,when the lift is stationary. If the lift moves downward with an acceleration of $5 ~m/s^2$,the reading of the spring balance will be (in $N$)

Two radioactive nuclei $P$ and $Q$ in a given sample decay into a stable nucleus $R$. At time $t = 0$,the number of $P$ species is $4N_0$ and that of $Q$ is $N_0$. The half-life of $P$ is $1$ minute,while that of $Q$ is $2$ minutes. Initially,there are no nuclei of $R$ present in the sample. When the number of nuclei of $P$ and $Q$ are equal,the number of nuclei of $R$ present in the sample would be:

Considering only the principal values of inverse functions,the set $A = \{x \geq 0, \tan^{-1}(2x) + \tan^{-1}(3x) = \frac{\pi}{4}\}$

The diode used in the circuit shown in the figure has a constant voltage drop of $0.5 \, V$ at all currents and a maximum power rating of $100 \, mW$. What should be the value of the resistor $R$,connected in series with the diode,for obtaining maximum current?

Molten $CuCl_2$ is electrolysed using platinum electrodes. The reaction occurring at the anode is: