Quadratic Equations — Assertion-Reason Type Questions

Assertion (A): Every quadratic equation ax² + bx + c = 0, a ≠ 0, a, b and c are all real numbers has two real roots.

Reason (R): Every quadratic equation ax² + bx + c = 0, a ≠ 0, a, b and c are all real numbers has two real roots if b² - 4ac ≥ 0.

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

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

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

4. Both Assertion (A) and Reason (R) are correct, and Reason (R) is incorrect reason for Assertion (A).

Answer

The quadratic equation: ax² + bx + c = 0.

The expression b² - 4ac is called the discriminant (D).

When,

1. D > 0; two distinct real roots

2. D = 0; real and equal roots

3. D < 0; then roots are imaginary

thus, assertion (A) is false but reason(R) is true.

Hence, option 2 is the correct option.

Assertion (A): The quadratic equation 4x² + 12x + 15 = 0, has no real roots.

Reason (R): The quadratic equation ax² + bx + c = 0, has real roots iff its 'discriminant' = b² - 4ac ≥ 0.

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

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

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

4. Both Assertion (A) and Reason (R) are correct, and Reason (R) is incorrect reason for Assertion (A).

Answer

For, the quadratic equation: ax² + bx + c = 0. The equation has real roots if

b² - 4ac ≥ 0

So, reason (R) is true.

Comparing equation 4x² + 12x + 15 = 0, with ax² + bx + c = 0, we get :

a = 4, b = 12, c = 15

D = b² - 4ac

= 12² - 4 x 4 x 15

= 144 - 240 = -96.

Since, D < 0, so the equation has no real roots.

So, assertion (A) is true and reason (R) correctly explains assertion (A).

Thus, both Assertion (A) and Reason (R) are correct, and Reason (R) is the correct reason for Assertion (A).

Hence, option 3 is the correct option.

Assertion (A): The equation 9x² + 6x - k = 0 has real roots if k ≥ -1.

Reason (R): The quadratic equation ax² + bx + c = 0 has real roots if 'discriminant' = b² - 4ac > 0.

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

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

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

4. Both Assertion (A) and Reason (R) are correct, and Reason (R) is incorrect reason for Assertion (A).

Answer

We know that,

The quadratic equation ax² + bx + c = 0 has real roots if 'discriminant' = b² - 4ac ≥ 0.

So, reason (R) is false.

Given, 9x² + 6x - k = 0

Comparing above equation with ax² + bx + c = 0, we get :

a = 9, b = 6 and c = -k

If the equation has real roots, then D ≥ 0

⇒ b² - 4ac ≥ 0

⇒ 6² - 4 x 9 x (-k) ≥ 0

⇒ 36 + 36k ≥ 0

⇒ 36k ≥ -36

⇒ k ≥ -$\dfrac{36}{36}$

⇒ k ≥ -1

So, assertion (A) is true.

Thus, Assertion (A) is true, but Reason (R) is false.

Hence, option 1 is the correct option.

Consider the polynomial 2x² - 3x + 5

Assertion (A): Factorisation of the above polynomial is not possible.

Reason (R): Discriminant 'b² - 4ac' is negative.

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

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

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

4. Both Assertion (A) and Reason (R) are correct, and Reason (R) is incorrect reason for Assertion (A).

Answer

Given,

Polynomial : 2x² - 3x + 5

Discriminant (D) = b² - 4ac

= (-3)² - 4 x 2 x 5

= 9 - 40

= -31.

So, reason (R) is true.

Since the discriminant is negative, this quadratic has no real roots and cannot be factorized into linear factors with real coefficients.

So, assertion (A) is true and reason (R) correctly explains assertion (A).

Thus, both Assertion (A) and Reason (R) are correct, and Reason (R) is the correct reason for Assertion (A).

Hence, option 3 is the correct option.

Consider the following equation k²x² - 2kx + 1 = 0

Assertion (A): This equation has real roots for all non-zero values of k.

Reason (R): The discriminant of this equation is zero.

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

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

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

4. Both Assertion (A) and Reason (R) are correct, and Reason (R) is incorrect reason for Assertion (A).

Answer

Given,

Equation : k²x² - 2kx + 1 = 0

Comparing above equation with ax² + bx + c = 0, we get :

a = k², b = -2k and c = 1

Discriminant (D) = b² - 4ac

= (-2k)² - 4 x k² x 1

= 4k² - 4k²

= 0.

So, reason (R) is true.

Since, D = 0 this means the equation has one repeated real root.

So, assertion (A) is true and reason (R) correctly explains assertion (A).

Thus, both Assertion (A) and Reason (R) are correct, and Reason (R) is the correct reason for Assertion (A).

Hence, option 3 is the correct option.