Question

Match the plots in Section I with the corresponding functions in Section II. 
Section I 

Section II 
(1) \( y = \frac{\sin^2x}{x} \) 
(2) \( y = x \sin^2x \) 
(3) \( y = \frac{\sin x}{x} \) 
(4) \( y = x \sin x \) 
 

• A. P – 3, Q – 2, R – 4, S – 1
• B. P – 2, Q – 3, R – 4, S – 1
• C. P – 1, Q – 4, R – 3, S – 2
• D. P – 2, Q – 3, R – 1, S – 4

Hint:

When matching function graphs, start with simple checks: 1. {Symmetry:} Is it even (\(f(-x)=f(x)\), symmetric about y-axis) or odd (\(f(-x)=-f(x)\), symmetric about origin)? 2. {Value at x=0:} Does it pass through the origin or another value? 3. {Amplitude:} Does the amplitude grow, shrink, or stay constant as \(x\) increases? These three checks are usually enough to distinguish between similar-looking functions.

Answer 1

The Correct Option is A

To solve this problem, we need to match each plot in Section I with its corresponding function from Section II. Let's analyze each function and try to understand its behavior.

Analysis of Functions:

1. \( y = \frac{\sin^2x}{x} \): This function will have a singularity at \( x = 0 \) since division by zero is undefined. For \( x \neq 0 \), the behavior is influenced by \(\sin^2x\), which oscillates, and the division by \( x \), which stretches or compresses the amplitude inversely with x.
2. \( y = x \sin^2x \): Here, the function behaves differently than a simple sinusoidal curve due to the \( x \) factor. As \( x \) increases, the amplitude of oscillation increases.
3. \( y = \frac{\sin x}{x} \): Known for its oscillation damped by the linear division by \( x \). This is typically associated with the sinc function and results in decaying oscillations.
4. \( y = x \sin x \): Predicts increased oscillatory behavior with increasing x, unlike a standard sinusoidal function due to the \( x \) multiplication.

Matching:

Plot (Section I)	Function (Section II)
P	\( y = \frac{\sin x}{x} \)
Q	\( y = x \sin^2x \)
R	\( y = x \sin x \)
S	\( y = \frac{\sin^2x}{x} \)

Based on the analysis, we can match the options as follows:

• P – 3
• Q – 2
• R – 4
• S – 1

This matches the option P – 3, Q – 2, R – 4, S – 1 which is the correct answer.