If `a^(3)b^(4)c^(7)d^(2)` is negative, where a,b,c,d are integers , all the following statementss are incorrect EXCEPT ?

Options

A. `ab^(2) cd^(2)` is positive
B. `b^(2)d` is positive
C. `bc^(2)d` is positive
D. `a^(2)b^(2)c^(2)` is positive

Answer

To solve the problem, we need to analyze the expression \( a^3 b^4 c^7 d^2 \) and determine the conditions under which it is negative. We will also evaluate the options provided to find the one that is correct.

### Step-by-Step Solution:

1. **Understanding the Expression**:
- The expression \( a^3 b^4 c^7 d^2 \) is given to be negative.
- Here, \( a, b, c, d \) are integers.

2. **Analyzing the Powers**:
- The powers of \( b \) and \( d \) are even (4 and 2 respectively), which means \( b^4 \) and \( d^2 \) are always non-negative (either positive or zero).
- The powers of \( a \) and \( c \) are odd (3 and 7 respectively), which means \( a^3 \) and \( c^7 \) can be either positive or negative depending on the sign of \( a \) and \( c \).

3. **Condition for Negativity**:
- For the entire expression \( a^3 b^4 c^7 d^2 \) to be negative, the product of \( a^3 \) and \( c^7 \) must be negative since \( b^4 \) and \( d^2 \) are positive.
- This implies that \( a \) and \( c \) must have opposite signs:
- If \( a \) is positive, \( c \) must be negative.
- If \( a \) is negative, \( c \) must be positive.

4. **Evaluating the Options**:
- We need to check each option to see which one is consistent with our findings.

**Option A**: \( a b^2 c d^2 \)
- Here, \( b^2 \) and \( d^2 \) are positive.
- Since \( a \) and \( c \) have opposite signs, the product \( a b^2 c d^2 \) is negative.
- Thus, this option is incorrect as it states it is positive.

**Option B**: \( b^2 d \)
- \( b^2 \) is positive, and \( d \) can be either positive or negative.
- Therefore, \( b^2 d \) can be positive or negative, making this option incorrect as it states \( b \) is positive.

**Option C**: \( b c^2 d \)
- \( c^2 \) is positive, and \( b \) can be either positive or negative while \( d \) can also be either positive or negative.
- Thus, \( b c^2 d \) can be positive or negative, making this option incorrect as well.

**Option D**: \( a^2 b^2 c^2 \)
- Since \( a^2, b^2, c^2 \) are all positive (even powers), their product is positive.
- This option claims that \( a^2 b^2 c^2 \) is positive, which is correct.

5. **Conclusion**:
- The only correct statement among the options provided is **Option D**: \( a^2 b^2 c^2 \) is positive.

### Final Answer:
The correct option is **D**: \( a^2 b^2 c^2 \) is positive.