Statement-1 If A = {x |g(x) = 0} and B = {x| f(x) = 0}, then `A nn B` be a root of `{f(x)}^(2) + {g(x)}^(2)=0`
Statement-2 `x inAnnBimpliesx inAorx inB`.

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

Explanation

To solve the problem, we need to analyze the two statements provided and determine their validity.

### Step-by-Step Solution:

**Step 1: Understand the Sets A and B.**
- We define set A as \( A = \{ x \mid g(x) = 0 \} \). This means that A contains all values of \( x \) for which the function \( g(x) \) equals zero.
- We define set B as \( B = \{ x \mid f(x) = 0 \} \). Similarly, B contains all values of \( x \) for which the function \( f(x) \) equals zero.

**Hint for Step 1:** Identify what the sets A and B represent in terms of their respective functions.

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**Step 2: Analyze Statement 1.**
- Statement 1 claims that if \( A \cap B \) (the intersection of A and B) is a root of the equation \( f(x)^2 + g(x)^2 = 0 \), then we need to check if this is true.
- For \( f(x)^2 + g(x)^2 = 0 \) to hold, both \( f(x) \) and \( g(x) \) must be zero because both terms are squares and cannot be negative.

**Hint for Step 2:** Consider the implications of the equation \( f(x)^2 + g(x)^2 = 0 \) and what it means for \( f(x) \) and \( g(x) \).

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**Step 3: Connect the Roots to the Sets.**
- If \( x \) is in \( A \cap B \), then \( x \) must satisfy both \( g(x) = 0 \) and \( f(x) = 0 \).
- Therefore, if \( x \in A \cap B \), it implies that \( f(x) = 0 \) and \( g(x) = 0 \), which means \( f(x)^2 + g(x)^2 = 0 \) is satisfied.

**Hint for Step 3:** Verify how being in the intersection of sets A and B relates to the equation provided.

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**Step 4: Analyze Statement 2.**
- Statement 2 claims that \( x \in A \cap B \) implies \( x \in A \) or \( x \in B \).
- However, this is incorrect because \( x \in A \cap B \) means \( x \) must be in both A and B, not just one of them. This is an "and" condition, not an "or" condition.

**Hint for Step 4:** Recall the definition of intersection and how it differs from union.

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**Step 5: Conclusion.**
- From our analysis, we conclude that:
- Statement 1 is true because if \( x \) is in \( A \cap B \), then it satisfies \( f(x)^2 + g(x)^2 = 0 \).
- Statement 2 is false because it misrepresents the relationship of intersection.

Thus, the correct option is that Statement 1 is true and Statement 2 is false.

**Final Answer:** Statement 1 is true, Statement 2 is false.

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