Question

A particle starts at the origin and moves along the $x$-axis in such a way that its velocity at the point $(x, 0)$ is given by the formula $\frac{d x}{d t}=\cos ^{2} \pi x$. Then the particle never reaches the point on

• A. $x=\frac{1}{4}$
• B. $x=\frac{3}{4}$
• C. $x=\frac{1}{2}$
• D. $x=1$

Answer 1

Answer: (C)

Given: $\frac{d x}{d t}=\cos ^{2} \pi x$. Differentiate with respect to $t$,
$\frac{d^{2} x}{d t^{2}}=-2 \pi \sin 2 \pi x=- ve$
$\because \frac{d^{2} x}{d t^{2}}=0$
$\Rightarrow 2 \pi \sin 2 \pi x=0 $
$\Rightarrow \sin 2 \pi x=\sin \pi$
$\Rightarrow 2 \pi x=\pi \Rightarrow x=\frac{1}{2}$