If $x=\sqrt{t}, y=\sin t$ and $\frac{d^2 y}{d x^2}=a \cos x^2+b x^2 \sin x^2$ then $a^2+b^2$ is equal to

20

$x=\sqrt{t}, y=\sin t$

$t = x^2  ~~\Rightarrow y = \sin x^2$

$\frac{d y}{d x}=2 x \cos x^2$

$\Rightarrow \frac{d^2 y}{d x^2}=2 \cos x^2-4 x^2 \sin x^2=a \cos x^2+b x^2 \sin x^2$

⇒  a = 2  ,  b = -4

$a^2 + b^2 = 4 + 16 = 20$

Option: C