Match List-I with List-II, where t > 0 and, in (B), \(0 \lt t \lt \displaystyle\frac{\pi}{2}\)
| List-I | List-II | ||
|---|---|---|---|
| Parametric equations | \(\displaystyle\frac{dy}{dx}\) | ||
| (A) | \(x = t^{2},\ y = t^{3}\) | (I) | \(e^{t}\) |
| (B) | \(x = \sin t,\ y = \cos 2t\) | (II) | \(3t^{3}\) |
| (C) | \(x = 2e^{t},\ y = e^{2t}\) | (III) | \(\displaystyle\frac{3t}{2}\) |
| (D) | \(x = \log_e t,\ y = t^{3}\) | (IV) | \(-4\sin t\) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A)-(III), (B)-(IV), (C)-(I), (D)-(II)
In each case \(\frac{dy}{dx} = \frac{dy/dt}{dx/dt}\) (parametric form).
(A) \(x = t^{2},\ y = t^{3}\)
This matches (III) \(\frac{3t}{2}\):
\( \displaystyle \frac{dy}{dx} = \frac{3t^{2}}{2t} = \frac{3t}{2} \)
(B) \(x = \sin t,\ y = \cos 2t\)
This matches (IV) \(-4\sin t\). Using \(\sin 2t = 2\sin t\cos t\):
\( \displaystyle \frac{dy}{dx} = \frac{-2\sin 2t}{\cos t} = \frac{-4\sin t\cos t}{\cos t} = -4\sin t \)
(C) \(x = 2e^{t},\ y = e^{2t}\)
This matches (I) \(e^{t}\):
\( \displaystyle \frac{dy}{dx} = \frac{2e^{2t}}{2e^{t}} = e^{t} \)
(D) \(x = \log_e t,\ y = t^{3}\)
This matches (II) \(3t^{3}\):
\( \displaystyle \frac{dy}{dx} = \frac{3t^{2}}{1/t} = 3t^{3} \)