STATEMENT-1 : The dual statement of "xis a perfect square or xis a prime number" is "x is a perfect square and a prime number
and
STATEMENT-2 : If x is a prime number, then it isnot a perfect square.

Options

A. Statement-1 is True, Statement-2 is True, Statement-2 is a correct explanation for Statement-1
B. Statement-1 is True, Statement-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 solve the problem, we need to analyze the two statements and determine their validity and relationship. Let's break it down step by step.

### Step 1: Understanding Statement 1
**Statement 1**: "x is a perfect square or x is a prime number."

This statement is a disjunction, meaning that at least one of the conditions must be true for the entire statement to be true.

### Step 2: Finding the Dual of Statement 1
The dual of a statement involves changing "or" to "and" and vice versa.

- The dual statement of Statement 1 would be: "x is a perfect square and x is a prime number."

### Step 3: Analyzing Statement 2
**Statement 2**: "If x is a prime number, then it is not a perfect square."

To analyze this, we need to understand the definitions of prime numbers and perfect squares:
- A **prime number** has exactly two distinct positive divisors: 1 and itself.
- A **perfect square** is a number that can be expressed as the square of an integer.

### Step 4: Validating Statement 2
To validate Statement 2, we can consider the properties of prime numbers:
- The only perfect square that is also a prime number is 1 (since 1 is not considered a prime number).
- All other prime numbers (2, 3, 5, 7, etc.) are not perfect squares because they cannot be expressed as the square of an integer.

Thus, if x is a prime number, it cannot be a perfect square. Therefore, Statement 2 is true.

### Step 5: Conclusion
- Statement 1 is true because its dual statement is correctly formed.
- Statement 2 is true because a prime number cannot be a perfect square.

### Final Answer
Both statements are true.