Match List-I with List-II

Choose the correct answer from the options given below:

(A)-(III), (B)-(IV), (C)-(I), (D)-(II)

The correct answer is Option (4) → (A)-(III), (B)-(IV), (C)-(I), (D)-(II)

(A) $x=\frac{2}{t},\; y=2t$

$\frac{dx}{dt}=-\frac{2}{t^2},\;\frac{dy}{dt}=2$

$\frac{dy}{dx}=\frac{2}{-2/t^2}=-t^2 \;\;\;\Rightarrow\;$ (III)

(B) $x=t^3,\; y=3t+2$

$\frac{dx}{dt}=3t^2,\;\frac{dy}{dt}=3$

$\frac{dy}{dx}=\frac{3}{3t^2}=\frac{1}{t^2} \;\;\;\Rightarrow\;$ (IV)

(C) $x=\log t,\; y=2t^2$

$\frac{dx}{dt}=\frac{1}{t},\;\frac{dy}{dt}=4t$

$\frac{dy}{dx}=\frac{4t}{1/t}=4t^2 \;\;\;\Rightarrow\;$ (I)

(D) $x=e^t,\; y=2te^t$

$\frac{dx}{dt}=e^t,\;\frac{dy}{dt}=2e^t+2te^t=2(t+1)e^t$

$\frac{dy}{dx}=\frac{2(t+1)e^t}{e^t}=2(t+1) \;\;\;\Rightarrow\;$ (II)