To solve the question, we need to analyze both statements provided and determine their validity.
### Step 1: Analyze Statement-I
**Statement-I**: If \( a, b, c \) are in A.P., then \( bc, ca, ab \) are also in A.P.
1. **Definition of A.P.**: Three numbers \( a, b, c \) are in A.P. if \( 2b = a + c \).
2. **Assume values**: Let's take \( a = 2, b = 3, c = 4 \).
3. **Check A.P. condition**:
- Here, \( 2b = 2 \times 3 = 6 \) and \( a + c = 2 + 4 = 6 \).
- Thus, \( a, b, c \) are indeed in A.P.
4. **Calculate \( bc, ca, ab \)**:
- \( bc = 3 \times 4 = 12 \)
- \( ca = 4 \times 2 = 8 \)
- \( ab = 2 \times 3 = 6 \)
5. **Check if \( bc, ca, ab \) are in A.P.**:
- We need to check if \( 2ca = bc + ab \).
- Calculate \( 2ca = 2 \times 8 = 16 \) and \( bc + ab = 12 + 6 = 18 \).
- Since \( 16 \neq 18 \), \( bc, ca, ab \) are not in A.P.
**Conclusion for Statement-I**: False.
### Step 2: Analyze Statement-II
**Statement-II**: If a constant number is subtracted from each term of an A.P., then the resulting pattern of numbers also forms an A.P.
1. **Assume an A.P.**: Let’s take an A.P. with terms \( 4, 6, 8 \).
2. **Calculate the common difference**:
- \( 6 - 4 = 2 \)
- \( 8 - 6 = 2 \)
3. **Subtract a constant**: Let’s subtract \( 2 \) from each term.
- New terms: \( 4 - 2 = 2, 6 - 2 = 4, 8 - 2 = 6 \).
4. **Check if new terms are in A.P.**:
- Calculate the common difference for new terms:
- \( 4 - 2 = 2 \)
- \( 6 - 4 = 2 \)
5. **Conclusion**: The new terms \( 2, 4, 6 \) also have a common difference of \( 2 \), thus they form an A.P.
**Conclusion for Statement-II**: True.
### Final Conclusion:
- **Statement-I**: False
- **Statement-II**: True
### Correct Option:
The correct option is that Statement-I is false and Statement-II is true.
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