Which of the following statements is/are false.
i. `(a+b)^2 = a^2+b^2+2ab`
ii. `(a+b)^2-(a-b)^2=4ab`
iii. `(a+b)^2+(a-b)^2=2(a^2+b^2)`
iv. `(a-b)^2 = a^2-b^2+2ab`
v. `(a-b)^2 = a^2+b^2`

Options

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

Answer

To determine which of the given statements is/are false, we will evaluate each statement one by one using algebraic identities.

1. **Statement i**: \((a+b)^2 = a^2 + b^2 + 2ab\)

This is a well-known algebraic identity known as the square of a binomial.

**Verification**:
\[ (a+b)^2 = (a+b)(a+b) = a^2 + ab + ab + b^2 = a^2 + b^2 + 2ab \]
**Conclusion**: This statement is **true**.

**Hint**: Remember the formula for the square of a binomial.

2. **Statement ii**: \((a+b)^2 - (a-b)^2 = 4ab\)

We will expand both squares and simplify.

**Verification**:
\[ (a+b)^2 = a^2 + b^2 + 2ab \]
\[ (a-b)^2 = a^2 + b^2 - 2ab \]
Now, substituting these into the equation:
\[ (a+b)^2 - (a-b)^2 = (a^2 + b^2 + 2ab) - (a^2 + b^2 - 2ab) \]
Simplifying:
\[ = a^2 + b^2 + 2ab - a^2 - b^2 + 2ab = 4ab \]
**Conclusion**: This statement is **true**.

**Hint**: Use the difference of squares to simplify.

3. **Statement iii**: \((a+b)^2 + (a-b)^2 = 2(a^2 + b^2)\)

We will again expand both squares and simplify.

**Verification**:
\[ (a+b)^2 = a^2 + b^2 + 2ab \]
\[ (a-b)^2 = a^2 + b^2 - 2ab \]
Now, substituting these into the equation:
\[ (a+b)^2 + (a-b)^2 = (a^2 + b^2 + 2ab) + (a^2 + b^2 - 2ab) \]
Simplifying:
\[ = a^2 + b^2 + 2ab + a^2 + b^2 - 2ab = 2a^2 + 2b^2 = 2(a^2 + b^2) \]
**Conclusion**: This statement is **true**.

**Hint**: Combine like terms carefully.

4. **Statement iv**: \((a-b)^2 = a^2 - b^2 + 2ab\)

We will expand the left side.

**Verification**:
\[ (a-b)^2 = a^2 - 2ab + b^2 \]
The right side is \(a^2 - b^2 + 2ab\). Clearly, these two expressions are not equal.

**Conclusion**: This statement is **false**.

**Hint**: Remember that the square of a difference includes a negative sign with the \(2ab\) term.

5. **Statement v**: \((a-b)^2 = a^2 + b^2\)

We will expand the left side.

**Verification**:
\[ (a-b)^2 = a^2 - 2ab + b^2 \]
The right side is \(a^2 + b^2\). Again, these two expressions are not equal.

**Conclusion**: This statement is **false**.

**Hint**: Check the expansion of the square carefully.

### Final Conclusion:
The false statements are **iv** and **v**.