A conducting loop of finite resistance lies on the x - y plane....

The magnetic flux passing through the loop is given by:
$\Phi = B \cdot A$

Given that the area varies with time as $A = A_0(1 + \sin t)$ and the magnetic field $B$ is constant, the flux is:
$\Phi = B A_0(1 + \sin t)$

According to Faraday's law of electromagnetic induction, the induced emf $\varepsilon$ is:
$\varepsilon = -\dfrac{d\Phi}{dt} = -B A_0 \dfrac{d}{dt}(1 + \sin t) = -B A_0 \cos t$

The power $P$ dissipated in the loop of resistance $R$ is:
$P = \dfrac{\varepsilon^2}{R} = \dfrac{(-B A_0 \cos t)^2}{R} = \dfrac{B^2 A_0^2}{R} \cos^2 t$

From this expression, we can see that the power $P$ varies as $\cos^2 t$.
At $t = 0$, $P = \dfrac{B^2 A_0^2}{R} \cos^2(0) = \dfrac{B^2 A_0^2}{R}$ (which is a maximum value).
At $t = \dfrac{\pi}{2}$, $P = 0$.

The graph of $P$ versus $t$ will start from a maximum positive value, decrease to zero, and then increase again, oscillating between $0$ and a maximum value. This behavior is correctly represented by the graph in option (3).

Answer: