Which of the following statements is/are true?
i.`28^2=27^2+27+28`
ii. `71^2=72^2-71-72`
iii.`61^2=60^2+61+62`
iv.`(x+1)^2=x^2+x+1`
v.`24^2=25^2-24-23`

Options

A. i only
B. i and ii only
C. i and iv only
D. iii and v only

Answer

To determine which of the given statements are true, we will evaluate each statement one by one.

### Step-by-Step Solution:

1. **Evaluate Statement i: `28^2 = 27^2 + 27 + 28`**
- Calculate `28^2`:
\[ 28^2 = 784 \]
- Calculate `27^2 + 27 + 28`:
\[ 27^2 = 729 \]
\[ 27^2 + 27 + 28 = 729 + 27 + 28 = 729 + 55 = 784 \]
- Conclusion:
\[ 28^2 = 27^2 + 27 + 28 \quad \text{(True)} \]

2. **Evaluate Statement ii: `71^2 = 72^2 - 71 - 72`**
- Calculate `71^2`:
\[ 71^2 = 5041 \]
- Calculate `72^2 - 71 - 72`:
\[ 72^2 = 5184 \]
\[ 72^2 - 71 - 72 = 5184 - 71 - 72 = 5184 - 143 = 5041 \]
- Conclusion:
\[ 71^2 = 72^2 - 71 - 72 \quad \text{(True)} \]

3. **Evaluate Statement iii: `61^2 = 60^2 + 61 + 62`**
- Calculate `61^2`:
\[ 61^2 = 3721 \]
- Calculate `60^2 + 61 + 62`:
\[ 60^2 = 3600 \]
\[ 60^2 + 61 + 62 = 3600 + 61 + 62 = 3600 + 123 = 3723 \]
- Conclusion:
\[ 61^2 = 60^2 + 61 + 62 \quad \text{(False)} \]

4. **Evaluate Statement iv: `(x + 1)^2 = x^2 + x + 1`**
- Expand `(x + 1)^2`:
\[ (x + 1)^2 = x^2 + 2x + 1 \]
- Compare with `x^2 + x + 1`:
\[ x^2 + 2x + 1 \neq x^2 + x + 1 \]
- Conclusion:
\[ (x + 1)^2 = x^2 + x + 1 \quad \text{(False)} \]

5. **Evaluate Statement v: `24^2 = 25^2 - 24 - 23`**
- Calculate `24^2`:
\[ 24^2 = 576 \]
- Calculate `25^2 - 24 - 23`:
\[ 25^2 = 625 \]
\[ 25^2 - 24 - 23 = 625 - 24 - 23 = 625 - 47 = 578 \]
- Conclusion:
\[ 24^2 = 25^2 - 24 - 23 \quad \text{(False)} \]

### Summary of Results:
- Statement i: True
- Statement ii: True
- Statement iii: False
- Statement iv: False
- Statement v: False

### Final Answer:
The true statements are **i and ii**.