A simple harmonic oscillator of angular frequency 2 rad s^-1 is acted upon by an external force F= sin⁡t N.If the oscillator is at rest in its…

It is given that oscillator at rest at $t=0$t=0 i.e. at $t=0$t=0, $v=0$v=0. So, in option, we can check by putting $v=\frac{dx}{dt}=0$v=dtdx​=0.

(1) $If x\propto \mathrm{sin}⁡t+\frac{1}{2}\mathrm{cos}⁡2t$Ifx∝sin⁡t+21​cos⁡2t

$$\Longrightarrow v\propto \mathrm{cos}⁡t+\frac{1}{2}\times 2 (-\mathrm{sin}⁡2t)$$⟹v∝cos⁡t+21​×2(−sin⁡2t)

$$\Longrightarrow at t=0, v\propto 1-0\neq 0$$⟹att=0,v∝1−0=0

(2) $If x \propto \mathrm{cos}⁡t-\frac{1}{2}\mathrm{sin}⁡t$Ifx∝cos⁡t−21​sin⁡t

$$\Longrightarrow v\propto -\mathrm{sin}⁡t-\frac{1}{2}\mathrm{cos}⁡t$$⟹v∝−sin⁡t−21​cos⁡t

$$\Longrightarrow at t=0, v\propto -\frac{1}{2}\neq 0$$⟹att=0,v∝−21​=0

(3) $If x\propto \mathrm{sin}⁡t-\frac{1}{2} sin 2t$Ifx∝sin⁡t−21​sin2t

$$then \upsilon \propto \mathrm{cos}⁡t-\frac{1}{2}\times 2\mathrm{cos}⁡2t$$thenυ∝cos⁡t−21​×2cos⁡2t

$$\Longrightarrow at t=0, v\propto 1-1=0$$⟹att=0,v∝1−1=0

(4) $If x\propto \mathrm{sin}⁡t+\frac{1}{2}\mathrm{sin}⁡2t$Ifx∝sin⁡t+21​sin⁡2t

$$\Longrightarrow v\propto \mathrm{cos}⁡t+\frac{1}{2}\times 2\mathrm{cos}⁡2t$$⟹v∝cos⁡t+21​×2cos⁡2t

$$\Longrightarrow at t=0, v\propto 1+1$$⟹att=0,v∝1+1

$$\Longrightarrow v\propto 2\neq 0$$⟹v∝2=0

$$\therefore in option (3) v=0 at t=0$$∴inoption(3)v=0att=0