STATEMENT-1: The straight lines `2x+3y+5=0and 3x-2y+1=0` are perpendicular to each other.
STATEMENT-2: Two lines `y=m_(1)x+c_(1)and y=m_(2)x+c_(2)` where `m_(1),m_(2)in` R are perpendicular if `m_(1),m_(2)=-1.`
STATEMENT-2: Two lines `y=m_(1)xx+x_(1)and y=m_(2)x+c_(2)where m_(1),m_(2)inR` are peprendicular if `m_(1)m_(2)=-1.`

Options

A. Statement-1 is True, Statemetn-2 is True, Statemetn-2 is a correct explanation for Statement-1
B. Statement-1 is True, Statemetn-2 is True, Statement-2 is NOT a correct explanation for Statement-1
C. Statement-1 is True, Statement-2 is False
D. Statement-1 is False, Statement-2 is True

Answer

To determine if the statements regarding the straight lines are true and if one is a correct explanation for the other, we will analyze both statements step by step.

### Step 1: Identify the slopes of the given lines

The equations of the lines are:
1. \( 2x + 3y + 5 = 0 \)
2. \( 3x - 2y + 1 = 0 \)

To find the slopes, we need to rewrite these equations in the slope-intercept form \( y = mx + c \), where \( m \) is the slope.

**For the first line:**
\[ 3y = -2x - 5 \\ y = -\frac{2}{3}x - \frac{5}{3} \]
Thus, the slope \( m_1 = -\frac{2}{3} \).

**For the second line:**
\[ -2y = -3x - 1 \\ y = \frac{3}{2}x + \frac{1}{2} \]
Thus, the slope \( m_2 = \frac{3}{2} \).

### Step 2: Check if the lines are perpendicular

Two lines are perpendicular if the product of their slopes is equal to -1:
\[ m_1 \cdot m_2 = -1 \]
Substituting the values we found:
\[ -\frac{2}{3} \cdot \frac{3}{2} = -1 \]
Since the product equals -1, **Statement 1 is true**.

### Step 3: Analyze Statement 2

Statement 2 claims that two lines \( y = m_1x + c_1 \) and \( y = m_2x + c_2 \) are perpendicular if \( m_1 \cdot m_2 = -1 \).

From our analysis in Step 2, we see that this condition holds true. Therefore, **Statement 2 is also true**.

### Step 4: Determine if Statement 2 is a correct explanation for Statement 1

Since Statement 2 correctly describes the condition for two lines to be perpendicular and this condition was satisfied in Statement 1, we conclude that **Statement 2 is a correct explanation for Statement 1**.

### Final Conclusion

Both statements are true, and Statement 2 provides the correct explanation for Statement 1. Therefore, the answer is:
- Statement 1 is true.
- Statement 2 is true.
- Statement 2 is a correct explanation for Statement 1.