Q.10 Which of the following curves represents the function
y = ln(|e|sin(|x|)||) for |x| < 2π?
Here, x represents the abscissa and y represents the ordinate.
Introduction
In this article, we analyze the function:
y = ln(e|sin(|x|)|) for |x| < 2π
and determine which given graph represents it. We simplify the function and compare it to the provided curves A, B, C, and D.
Step-by-Step Function Simplification
Starting with:
y = ln(e|sin(|x|)|)
Using the property:
ln(ea) = a
Therefore:
y = |sin(|x|)|
Since sin(|x|) is symmetric:
y = |sin(x)|
Final Simplified Function
y = |sin(x)|
Properties of y = |sin(x)|
| Property | Value |
|---|---|
| Range | 0 to 1 |
| Period | π |
| Symmetry | Even function |
| Minima | At multiples of π |
| Maxima | At π/2, 3π/2, etc. |
Graph Comparison of Options
Option A
Goes negative. Impossible since |sin(x)| ≥ 0.
❌ Eliminated
Option B
Still crosses negative values.
❌ Wrong
Option C
Always positive but amplitude not exactly 1.
⚠️ Close but incorrect
Option D
Always above 0, peaks repeat consistently, touches 0 at multiples of π.
✔️ Perfect match for |sin(x)|
Conclusion
The simplified function y = |sin(x)| is always non-negative,
periodic and symmetric. Among all given curves, only:
✅ Option D correctly represents the graph.
