Question

If $x=a\text{(}\cos \theta +\theta \sin \theta ),y=a\text{(}\sin \theta -\theta \cos \theta ),$ then $\frac{d^{2}x}{d{\theta }^2}=a\text{(}\cos \theta -\theta \sin \theta ),\frac{d^{2}y}{d{\theta }^2}=a\text{(}\sin \theta +\theta \cos \theta )$

• A. True
• B. False

Answer 1

The correct option is A True

Here, $x=a(\cos \theta +\theta \sin \theta )$

$\frac{dx}{d\theta }=a(-\sin \theta +\sin \theta +\theta \cos \theta )=a\left(\theta \cos \theta \right)$

$\frac{d^{2}x}{d{\theta }^2}=a(\cos \theta -\theta \sin \theta )$

$y=a(\sin \theta -\theta \cos \theta )$

$\frac{dy}{d\theta }=a(\cos \theta -\cos \theta +\theta \sin \theta )$

$\frac{d^{2}x}{d{\theta }^2}=a(\sin \theta +\theta \cos \theta )$

Hence above statement is true.