Question

(i) (x + y + 1)(x – y – 1) = x² + (y + 1)²

(ii) (x + 9y)(x – 9y) = x² – 9y²

Options

• Only (i)

• Only (ii)

• Both (i) and (ii)

• Neither (i) nor (ii)

Answer 1

Neither (i) nor (ii)

Explanation:

Let’s analyze both expressions given:

(i) (x + y + 1)(x – y – 1) = x² + (y + 1)²

Expanding the left side:

(x + y + 1)(x – y – 1) = x(x – y – 1) + y(x – y – 1) + 1(x – y – 1) 

(x + y + 1)(x – y – 1) = x² – xy – x + xy – y² – y + x – y – 1 

Simplifying:

x² – y² – 2y – 1

Compare with the right side:

x² + (y + 1)² = x² + y² + 2y + 1

Clearly, x² – y² – 2y – 1 ≠ x² + y² + 2y + 1

So, the first expression (i) is incorrect.

(ii) (x + 9y)(x – 9y) = x² – 9y² 

This is the standard difference of squares:

(x + a)(x – a) = x² – a²

Here a = 9y, so the right side should be: 

x² – (9y)² = x² – 81y²

The given right side is x² – 9y², which is incorrect. 

The correct expression would be:

(x + 9y)(x – 9y) = x² – 81y²

Hence, (ii) is also incorrect.