Question

If \( x = 2\cos t - \cos 2t \) and \( y = 2\sin t - \sin 2t \), then \( \frac{dy}{dx} \) at \( t = \frac{\pi}{2} \) is

• A. \( -1 \)
• B. \( 0 \)
• C. \( \frac{1}{2} \)
• D. \( 1 \)
• E. \( 3 \)

Hint:

For parametric curves, always compute derivatives separately before substituting values.

Answer 1

The Correct Option is D

Concept: For parametric equations: \[ \frac{dy}{dx} = \frac{dy/dt}{dx/dt} \]

Step 1: Differentiate \(x\) and \(y\). \[ \frac{dx}{dt} = -2\sin t + 2\sin 2t \] \[ \frac{dy}{dt} = 2\cos t - 2\cos 2t \]

Step 2: Substitute \( t = \frac{\pi}{2} \). \[ \sin \frac{\pi}{2} = 1, \sin \pi = 0 \Rightarrow \frac{dx}{dt} = -2 \] \[ \cos \frac{\pi}{2} = 0, \cos \pi = -1 \Rightarrow \frac{dy}{dt} = 2 \]

Step 3: Compute derivative. \[ \frac{dy}{dx} = \frac{2}{-2} = -1 \]