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 $\frac{dx}{dt}={\cos }^2\pi x$. 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

The correct option is C $x=\frac{1}{2}$

Given: $\frac{dx}{dt}={\cos }^2\pi x$.

Differentiate w.r.t $t$,

$\frac{d^{2}x}{{dt}^2}=-2\pi \sin 2\pi x=-ve$

$\because \frac{d^{2}x}{{dt}^2}=0\Rightarrow -2\pi \sin 2\pi x=0\Rightarrow \sin 2\pi x=\sin \pi$

$\Rightarrow 2\pi x=\pi \Rightarrow x=\frac{1}{2}.$