Question

The number of solutions of the equation $\frac{x}{100}=\sin x$ is-

• A. $63$
• B. $32$
• C. $33$
• D. $1$

Answer 1

The correct option is A $63$

The number of solutions of the equation $\frac{x}{100}=\sin x$ is equal to the number of points of intersection of the graph $f\left(x\right)=\frac{x}{100}$ and $g\left(x\right)=\sin x$.
The period of $g\left(x\right)=\sin x$ is $2\pi$. The graph of $f\left(x\right)=\frac{x}{100}$ will intersect that of $\sin x$ twice in every interval of $2\pi$ (as shown in figure).
For all points of intersection, $-1<f\left(x\right)<1$ ( since $-1\le \sin x\le 1$)
$\Rightarrow -100<x<100$
In terms of $\pi$, $x$ is between $-32\pi$ and $32\pi$ (since $100<32\pi$).
So, we get total $16$ cycles of graph of $\sin x$ in the first quadrant. ( $\because$ the period is $2\pi$).
So, the total number of points of intersection in the $1^{st}$ quadrant is $16\times 2=32$ (including origin).
Similarly, points of intersection in the $3^{rd}$ quadrant is also $32$ (including origin).
So, the total number of points of intersection is $32\times 2-1=63$ (since origin has to be counted once).