Quadratic Equations — Assertion-Reason Type Questions

Assertion (A): The discriminant of the quadratic equation $x^2 + 2\sqrt{2}x + 1 = 0$ is greater than zero.

Reason (R): If the discriminant of a quadratic equation is greater than zero, the quadratic equation has real and distinct roots.

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.

Answer

Given,

$x^2 + 2\sqrt{2}x + 1 = 0$

Comparing $x^2 + 2\sqrt{2}x + 1 = 0$ with ax² + bx + c = 0 we get,

a = 1, b = $2\sqrt{2}$ and c = 1.

We know that,

Discriminant (D) = b² - 4ac = $(2\sqrt{2})^2$ - 4 × (1) × (1)

= 8 - 4 = 4; which is positive.

Since, D = 4 > 0, the discriminant is greater than zero.

So, Assertion (A) is true.

D > 0 Real and distinct roots

D = 0 Real and equal roots

D < 0 Imaginary roots

If Discriminant is grater than zero,

Real and distinct roots

So, Reason (R) is true.

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

Hence, option 2 is the correct option.

Assertion (A): The quadratic equation 3kx² - 4kx + 4 = 0 has equal roots, if k = 3.

Reason (R): For equal roots of a quadratic equation, we must have D = 0.

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.

Answer

Given,

⇒ 3kx² - 4kx + 4 = 0

when k = 3

⇒ 3 × (3) × x² - 4 × (3) × x + 4 = 0

⇒ 9x² - 12x + 4 = 0

Comparing 9x² - 12x + 4 = 0 with ax² + bx + c = 0 we get,

a = 9, b = -12 and c = 4.

We know that,

Discriminant (D) = b² - 4ac

= (-12)² - 4 × (9) × (4)

= 144 - 144 = 0.

Therefore, the equation has rational and equal roots.

So, Assertion (A) is true.

The Discriminant is given by b² - 4ac, if the discriminant of any quadratic equation is zero. Then it is said have equal and real roots.

So, Reason (R) is true.

Thus, both A and R are true and R is the correct explanation of A.

Hence, option 1 is the correct option.

Assertion (A): The roots of the quadratic equation 3x² + 7x + 8 = 0 are imaginary.

Reason (R): The discriminant of a quadratic equation is always positive.

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.

Answer

Given,

⇒ 3x² + 7x + 8 = 0

Comparing 3x² + 7x + 8 = 0 with ax² + bx + c = 0 we get,

a = 3, b = 7 and c = 8.

We know that,

Discriminant (D) = b² - 4ac

= (7)² - 4 × (3) × (8)

= 49 - 96 = -47; which is negative.

Therefore, the equation has imaginary and unequal roots.

So, Assertion (A) is true.

The Discriminant of quadratic equation can be positive, negative or equal to zero.

So, Reason (R) is false.

A is true, R is false.

Hence, option 3 is the correct option.

Assertion (A): The roots of the quadratic equation 8x² + 2x - 3 = 0 are -$\dfrac{1}{2}$ and $\dfrac{3}{4}$.

Reason (R): The roots of the quadratic equation ax² + bx + c = 0 are given by $x = \dfrac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$.

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.

Answer

Given,

⇒ 8x² + 2x - 3 = 0

Comparing equation 8x² + 2x - 3 = 0 with ax² + bx + c = 0, we get :

a = 8, b = 2 and c = -3.

By formula,

x = $\dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Substituting values we get :

$\Rightarrow x = \dfrac{-(2) \pm \sqrt{(2)^2 - 4 \times (8) \times (-3)}}{2 \times (8)} \\[1em] = \dfrac{-2 \pm \sqrt{4 + 96}}{16} \\[1em] = \dfrac{-2 \pm \sqrt{100}}{16} \\[1em] = \dfrac{-2 \pm 10}{16} \\[1em] = \dfrac{-2 + 10}{16} \text{ or } \dfrac{-2 - 10}{16} \\[1em] = \dfrac{8}{16} \text{ or } \dfrac{-12}{16} \\[1em] = \dfrac{1}{2} \text{ or } \dfrac{-3}{4}.$

Thus, A is false, R is true.

Hence, option 4 is the correct option.