`{:("List - I","List - II"),("(A) x-t graph of simple harmonic oscillator","(E) Straight line with +ve slope"),("(B) a-x graph of SHO","(F) Straight line with -ve slope"),("(C) l-T graph of a simple pendulum","(G) Sine curve"),("(D) l-"T^(2) "graph of a simple pendulum","(H) Parabola"):}`

Options

A. a - e, b - f, c - g, d - h
B. a - g, b - f, c - h, d - e
C. a - g, b - e, c - h, d - f
D. a - g, b - f, c - e, d - h

Answer

To solve the problem of matching the items in List I with those in List II, we will analyze each item in List I and determine the corresponding item in List II based on the characteristics of the graphs involved.

### Step-by-Step Solution:

1. **Identify the x-t graph of a simple harmonic oscillator (SHO)**:
- The displacement \( x \) of a simple harmonic oscillator varies sinusoidally with time \( t \). The general equation is given by:
\[ x(t) = A \sin(\omega t + \phi) \]
- This graph is a sine curve, which oscillates between \( +A \) and \( -A \).
- **Match**: (A) corresponds to (G) Sine curve.

2. **Identify the a-x graph of SHO**:
- The acceleration \( a \) of a simple harmonic oscillator is proportional to the negative of the displacement \( x \):
\[ a = -\omega^2 x \]
- This relationship indicates that the graph of acceleration versus displacement is a straight line with a negative slope.
- **Match**: (B) corresponds to (F) Straight line with -ve slope.

3. **Identify the l-T graph of a simple pendulum**:
- The time period \( T \) of a simple pendulum is given by:
\[ T = 2\pi \sqrt{\frac{L}{g}} \]
- When plotting \( L \) against \( T \), it resembles a square root function, which is a part of a parabola.
- **Match**: (C) corresponds to (H) Parabola.

4. **Identify the l-T² graph of a simple pendulum**:
- If we square the time period equation, we get:
\[ T^2 = \frac{4\pi^2}{g} L \]
- This is a linear relationship between \( T^2 \) and \( L \), which can be expressed in the form \( y = mx \) where \( m \) is positive.
- **Match**: (D) corresponds to (E) Straight line with +ve slope.

### Final Matches:
- (A) → (G) Sine curve
- (B) → (F) Straight line with -ve slope
- (C) → (H) Parabola
- (D) → (E) Straight line with +ve slope

### Summary of Matches:
- A - G
- B - F
- C - H
- D - E