To solve the problem, we will evaluate each statement one by one and determine whether they are true (T) or false (F).
### Step 1: Evaluate Statement (i)
**Statement:** The sum of a two-digit number and the number formed by reversing its digits is exactly divisible by 11.
Let the two-digit number be represented as \(10A + B\), where \(A\) is the tens digit and \(B\) is the units digit. The number formed by reversing the digits is \(10B + A\).
Now, we add these two numbers:
\[
(10A + B) + (10B + A) = 11A + 11B = 11(A + B)
\]
Since \(11(A + B)\) is clearly divisible by 11, this statement is **True (T)**.
### Step 2: Evaluate Statement (ii)
**Statement:** The generalized form of \(X0YZ\) is \(1000X + 100X0 + 10Y + Z\).
The correct interpretation of \(X0YZ\) is:
- \(X\) is in the thousands place,
- \(0\) is in the hundreds place,
- \(Y\) is in the tens place,
- \(Z\) is in the units place.
Thus, the correct expression should be:
\[
1000X + 0 \cdot 100 + 10Y + Z = 1000X + 10Y + Z
\]
The statement incorrectly includes \(100X0\) which is not valid. Therefore, this statement is **False (F)**.
### Step 3: Evaluate Statement (iii)
**Statement:** \(100A + 10B + C\) is divisible by 3, if \(A + B - C\) is divisible by 3.
For a number to be divisible by 3, the sum of its digits must be divisible by 3. The number \(100A + 10B + C\) has digits \(A\), \(B\), and \(C\). The condition given is \(A + B - C\), which does not directly relate to the divisibility of \(100A + 10B + C\) by 3.
Instead, the correct condition should be \(A + B + C\) is divisible by 3. Therefore, this statement is **False (F)**.
### Step 4: Evaluate Statement (iv)
**Statement:** If \(X9 + 4Y = Y5\), then \(X + Y\) equals 7, where \(X\) and \(Y\) are single digits.
Let’s analyze the equation:
- The left side \(X9 + 4Y\) can be expressed as \(10X + 9 + 4Y\).
- The right side \(Y5\) can be expressed as \(10Y + 5\).
Setting the two expressions equal gives:
\[
10X + 9 + 4Y = 10Y + 5
\]
Rearranging this:
\[
10X + 4Y - 10Y = 5 - 9
\]
\[
10X - 6Y = -4
\]
\[
10X = 6Y - 4
\]
\[
5X = 3Y - 2
\]
Now, we can find integer solutions for \(X\) and \(Y\) that are single digits. Testing values, we find:
- If \(Y = 6\), then \(5X = 3(6) - 2 = 18 - 2 = 16\) → \(X = \frac{16}{5}\) (not an integer)
- If \(Y = 5\), then \(5X = 3(5) - 2 = 15 - 2 = 13\) → \(X = \frac{13}{5}\) (not an integer)
- If \(Y = 4\), then \(5X = 3(4) - 2 = 12 - 2 = 10\) → \(X = 2\) (integer)
- Thus, \(X = 2\) and \(Y = 4\) gives \(X + Y = 2 + 4 = 6\) (not equal to 7).
Continuing this process, we find that \(X + Y = 7\) does not hold for any single-digit values of \(X\) and \(Y\). Therefore, this statement is **False (F)**.
### Final Answers
1. (i) T
2. (ii) F
3. (iii) F
4. (iv) F
