Factorisation — Assertion-Reason Type Questions

Assertion (A): For the trinomial 2x² + 8x - 9 cannot be factorised.

Reason (R): For trinomial ax² + bx + c to be factorised, b² - 4ac must be a perfect square.

1. Assertion (A) is true, Reason (R) is false.

2. Assertion (A) is false, Reason (R) is true.

3. Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct reason for Assertion (A).

4. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct reason (or explanation) for Assertion (A).

Answer

For trinomial ax² + bx + c to be factorised, the roots are given by the quadratic equation: x = $\dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

For the trinomial ax² + bx + c to be factorizable into linear factors with rational coefficients, its roots must be rational numbers. This occurs if and only if the expression the discriminant b² - 4ac, is a perfect square.

∴ Reason (R) is true.

For the trinomial 2x² + 8x - 9, a = 2, b = 8 and c = -9.

D = b² - 4ac

= 8² - 4 × 2 × -9

= 64 + 72

= 136.

Since, 136 is not a perfect square. So, 2x² + 8x - 9 cannot be factorised.

∴ Assertion (A) is true.

∴ Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct reason for Assertion (A).

Hence, option 3 is the correct option.

Assertion (A): Factorisation of 4x² + 9y² is (2x - 3y)(2x + 3y).

Reason (R): a² - b² = (a - b)(a + b)

1. Assertion (A) is true, Reason (R) is false.

2. Assertion (A) is false, Reason (R) is true.

3. Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct reason for Assertion (A).

4. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct reason (or explanation) for Assertion (A).

Answer

According to reason: a² - b² = (a - b)(a + b)

Expanding, (a - b)(a + b)

⇒ a(a + b) - b(a + b)

⇒ a² + ab - ab - b²

⇒ a² - b².

Thus, (a - b)(a + b) = a² - b²

∴ Reason (R) is true.

Given,

⇒ (2x - 3y)(2x + 3y)

⇒ 2x(2x + 3y) - 3y(2x + 3y)

⇒ (2x)² + 6xy - 6xy - (3y)²

⇒ 4x² - 9y².

∴ Assertion (A) is false.

∴ Assertion (A) is false, Reason (R) is true.

Hence, option 2 is the correct option.

Assertion (A): Factorisation of x³ - 27 is (x - 3)(x² + 3x + 9).

Reason (R): a³ - b³ = (a - b)(a² + ab + b²).

1. Assertion (A) is true, Reason (R) is false.

2. Assertion (A) is false, Reason (R) is true.

3. Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct reason for Assertion (A).

4. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct reason (or explanation) for Assertion (A).

Answer

We know that,

a³ - b³ = (a - b)(a² + ab + b²)

∴ Reason (R) is true.

Given,

⇒ x³ - 27

⇒ x³ - 3³

⇒ (x - 3)(x² + 3x + 3²)

⇒ (x - 3)(x² + 3x + 9).

∴ Assertion (A) is true.

∴ Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct reason for Assertion (A).

Hence, option 3 is the correct option.