Question

The number of solutions of $\sin x = \frac{x}{10}$ is

• A. 10
• B. 3
• C. 5
• D. 7

Hint:

For $\sin x = ax$, count intersections on one side and double them, then add 1 for the origin.

Answer 1

The Correct Option is D

Step 1: Concept
The number of solutions can be found by finding the intersections of the graphs $y = \sin x$ and $y = x/10$.

Step 2: Meaning
The line $y = x/10$ reaches the maximum value of the sine curve ($y=1$) at $x=10$. Note that $10 \approx 3.18\pi$.

Step 3: Analysis
On the positive x-axis, intersections occur in $(0, \pi)$, $(\pi, 2\pi)$, and $(2\pi, 3\pi)$. This gives 3 positive solutions. By symmetry, there are 3 negative solutions. $x=0$ is also a solution.

Step 4: Conclusion
Total solutions = 3 (positive) + 3 (negative) + 1 (zero) = 7. Final Answer: (D)