Factorisation — Assertion-Reason Type Questions

Assertion(A): One of the factors of (5x + 1)² + (25x² - 1) is 2x.

Reason(R): (a + b)² = (a + b)(a + b) and a² - b² = (a + b)(a - b)

1. A is true, R is false

2. A is false, R is true

3. Both A and R are true

4. Both A and R are false

Answer

Given,

⇒ (5x + 1)² + (25x² - 1)

⇒ (5x + 1)² + (5x)² - (1)²

⇒ (5x + 1)² + (5x + 1)(5x - 1)

⇒ (5x + 1)[(5x + 1) + (5x - 1)]

⇒ (5x + 1)(10x)

⇒ 2x(5)(5x + 1).

Assertion (A) is true.

⇒ (a + b)² can be written as (a + b)(a + b)

By identity,

a² - b² = (a + b)(a - b)

Reason (R) is true.

Thus, both A and R are true.

Hence, option 3 is correct option.

Assertion(A): $a^3 - 2\sqrt{2}b^3$ can be factorized as $(a - 2\sqrt{2})(a^2 + \sqrt{2}ab + 2b^2)$

Reason(R): x³ - y³ = (x - y)(x² + xy + y²).

1. A is true, R is false

2. A is false, R is true

3. Both A and R are true

4. Both A and R are false

Answer

Given,

$a^3 - 2\sqrt{2}b^3$

By using identity,

a³ - b³ = (a - b)(a² + ab + b²)

$\Rightarrow (a)^3 - (\sqrt{2}b)^3 \\[1em] \Rightarrow (a - \sqrt{2}b)(a^2 + a(\sqrt{2}b) + (\sqrt{2}b)^2) \\[1em] \Rightarrow (a - \sqrt{2}b)(a^2 + \sqrt{2}ab + 2b^2) \\[1em]$

Assertion (A) is false.

Given,

x³ - y³ = (x - y)(x² + xy + y²)

This is a identity.

Reason (R) is true.

A is false, R is true.

Hence, option 2 is correct option.