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(B) For a particle in simple harmonic motion $(SHM)$:$(i)$ The velocity $v$ as a function of displacement $x$ is given by $v = \omega \sqrt{A^2 - x^2}

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$(ii)$ and $(iii)$

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(B) For a particle in simple harmonic motion $(SHM)$:$(i)$ The velocity $v$ as a function of displacement $x$ is given by $v = \omega \sqrt{A^2 - x^2}$. Squaring both sides,$v^2 = \omega^2(A^2 - x^2)$,which represents an ellipse,not a parabola. Thus,$(i)$ is incorrect.$(ii)$ The velocity as a function of time is $v(t) = A\omega \cos(\omega t)$,which is a sinusoidal function. Thus,$(ii)$ is correct.$(iii)$ The acceleration is $a(t) = -A\omega^2 \sin(\omega t)$. From $v = A\omega \cos(\omega t)$ and $a = -A\omega^2 \sin(\omega t)$,we have $(v/A\omega)^2 + (a/A\omega^2)^2 = \cos^2(\omega t) + \sin^2(\omega t) = 1$. This is the equation of an ellipse. Thus,$(iii)$ is correct.Therefore,statements $(ii)$ and $(iii)$ are correct.

$A$ particle is performing simple harmonic motion. Which of the following statements are correct?
$(i)$ Its velocity-displacement graph is parabolic in nature.
$(ii)$ Its velocity-time graph is sinusoidal in nature.
$(iii)$ Its velocity-acceleration graph is elliptical in nature.

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$A$ particle is performing simple harmonic motion. Which of the following statements are correct?$(i)$ Its velocity-displacement graph is parabolic in nature.$(ii)$ Its velocity-time graph is sinusoidal in nature.$(iii)$ Its velocity-acceleration graph is elliptical in nature.