AC basics and RMS values

A resistance of $40\ \Omega$ is connected to a source of alternating current rated 220 V, 50 Hz. Find the time taken by the current to change from its maximum value to the rms value :

The current flowing through an ac circuit is given by
$$I=5\sin(120\pi t)\ \mathrm A$$
How long will the current take to reach the peak value starting from zero ?

A 220 V, 50 Hz AC source is connected to a 25 V, 5 W lamp and an additional resistance R in series (as shown in figure) to run the lamp at its peak brightness, then the value of R (in ohm) will be ______.

A direct current of 4 A and an alternating current of peak value 4 A flow through resistance of 3 Ω and 2 Ω respectively. The ratio of heat produced in the two resistances in same interval of time will be :

An alternating emf $E=440\sin100\pi t$ is applied to a circuit containing an inductance of $\dfrac{\sqrt2}\pi$ H. If an a.c. ammeter is connected in the circuit, its reading will be :

The instantaneous voltages at three terminals marked $X$, $Y$ and $Z$ are given by

$$V_X = V_0\sin\omega t,\quad V_Y = V_0\sin\left(\omega t + \frac{2\pi}{3}\right)\ \text{and}\ V_Z = V_0\sin\left(\omega t + \frac{4\pi}{3}\right).$$

An ideal voltmeter is configured to read rms value of the potential difference between its terminals. It is connected between points $X$ and $Y$ and then between $Y$ and $Z$. The reading(s) of the voltmeter will be

The AC voltage across a resistance can be measured using a :

At time $t=0$, terminal A in the circuit shown in the figure is connected to B by a key and an alternating current $I(t)=I_0\cos(\omega t)$, with $I_0=1$ A and $\omega=500\ \mathrm{rad\,s^{-1}}$ starts flowing in it with the initial direction shown in the figure. At $t=\dfrac{7\pi}{6\omega}$, 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 $C=20\ \mu$F, $R=10\ \Omega$ and the battery is ideal with emf of 50 V, identify the correct statement(s).