Assertion (A): The quadratic equation

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Q.
Assertion (A): The quadratic equation $9 x ² + 3 k x + 4 = 0 h a s e q u a l r o o t s f o r k = \pm 4 .$

Reason (R): If the discriminant of a quadratic equation is equal to zero, then the roots of the equation are real and equal.

1 Both A and R are true, and R is the correct explanation of A.

2 Both A and R are true, but R is not the correct explanation of A

3 A is true, but R is false

4 A is false, but R is true

Ans.
(1)

We have the quadratic equation $9 x ² + 3 k x + 4 = 0 .$
For roots to be equal, D = 0.

⇒ $(3 k) ² - 4 \times 9 \times 4 = 0 \Rightarrow 9 k ² - 144 = 0$

$\Rightarrow 9 k ² = 144 \Rightarrow k ² = 16$

$\Rightarrow k = \pm 4$ .

Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A).

Solution

Q.
Assertion (A): The quadratic equation $9 x ² + 3 k x + 4 = 0 h a s e q u a l r o o t s f o r k = \pm 4 .$

Reason (R): If the discriminant of a quadratic equation is equal to zero, then the roots of the equation are real and equal.

1 Both A and R are true, and R is the correct explanation of A.

2 Both A and R are true, but R is not the correct explanation of A

3 A is true, but R is false

4 A is false, but R is true

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Solution

Q. Assertion (A): The quadratic equation 9x² + 3kx + 4 = 0 has equal roots for k = ±4. Reason (R): If the discriminant of a quadratic equation is equal to zero, then the roots of the equation are real and equal.

1 Both A and R are true, and R is the correct explanation of A.

2 Both A and R are true, but R is not the correct explanation of A

3 A is true, but R is false

4 A is false, but R is true

Ans. (1) We have the quadratic equation 9x² + 3kx + 4 = 0. For roots to be equal, D = 0. ⇒ (3k)² − 4×9×4 = 0 ⇒ 9k² − 144 = 0 ⇒ 9k² = 144 ⇒ k² = 16 ⇒ k = ±4. Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A).

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Q.
Assertion (A): The quadratic equation $9 x ² + 3 k x + 4 = 0 h a s e q u a l r o o t s f o r k = \pm 4 .$

Reason (R): If the discriminant of a quadratic equation is equal to zero, then the roots of the equation are real and equal.