Quadratic Equations – Assertion and Reason Questions

Q 1 :

Assertion (A): 9x² - 3x - 20 = 0 ⇒ (3x - 5)(3x + 4) = 0.The roots are calculated by splitting the middle term.

Reason (R): To factorise ax² + bx + c = 0, we write in the form $a x ² + b ₁ x + b ₂ x + c = 0$ such that $b ₁ + b ₂ = b a n d b ₁ b ₂ = a c$ .

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

1

2

3

4

(1)

We have, 9x² - 3x - 20 = 0
⇒ 9x² - 15x + 12x - 20 = 0 ⇒ 3x(3x - 5) + 4(3x - 5) = 0
⇒ (3x - 5)(3x + 4) = 0
Hence, both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

Q 2 :

Assertion (A): The roots of equation x² – 7x + 12 = 0 are x = 3 and x = 4.

Reason (R): The splitting of middle term of the equation x² – 7x + 12 = 0 is
$(x ² - 7 / 2 x - 7 / 2 x + 12 = 0) .$

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

1

2

3

4

(3)

We have, x² – 7x + 12 = 0 ⇒ x² – 4x – 3x + 12 = 0 ⇒ x(x – 4) – 3(x – 4) = 0
⇒ (x – 4)(x – 3) = 0 ⇒ x = 3, 4. ∴ Assertion (A) is true.

In Reason (R): Given, x² – 7x + 12 = x² – (7/2)x – (7/2)x + 12 = 0
This is not middle term splitting because
(–7/2)x × (–7/2)x ≠ 12x²

Q 3 :

Assertion (A): If $5 + \sqrt 7$ is a root of a quadratic equation with rational coefficients, then its other root is $5 - \sqrt 7$

Reason (R): Irrational roots of a quadratic equation with rational coefficients occur in conjugate pairs.

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

1

2

3

4

(1)

Since irrational roots of a quadratic equation always occur in conjugate pairs, both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A).

Q 4 :

Assertion (A): $2 x ² - 4 x + 3 = 0$ is a quadratic equation.

Reason (R): All polynomials of degree n, when n is a whole number, can be treated as quadratic equations.

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

1

2

3

4

(3)

Yes, $2 x ² - 4 x + 3 = 0$ is a quadratic equation because the degree of the equation is 2. However, not all polynomials of degree n can be treated as quadratic equations.

If n = 1, the equation is called a linear equation, and if n = 3, it is called a cubic equation.

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

Q 5 :

Assertion (A): If the difference of roots of the equation $x ² - 2 p x + q = 0$ is same as the difference of the roots of the equation $x ² - 2 r x + s = 0$ , then $s - q = r ² - p ² .$

Reason (R): The roots of the quadratic equation $a x ² + b x + c = 0$ are given by $x = \frac{- b \pm \sqrt D}{2 a}$ , where D is the discriminant.

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

1

2

3

4

(2)

Clearly, Reason (R) is true. Let α, β be the roots of $x ² - 2 p x + q = 0 a n d γ, δ b e t h e r o o t s o f x ² - 2 r x + s = 0 .$

Then, $α + β = 2 p, α β = q, a n d γ + δ = 2 r, γ δ = s$ . It is given that $α - β = γ - δ .$

$\Rightarrow (α - β) ² = (γ - δ) ²$

$\Rightarrow (α + β) ² - 4 α β = (γ + δ) ² - 4 γ δ$

$\Rightarrow (2 p) ² - 4 q = (2 r) ² - 4 s$

$\Rightarrow 4 p ² - 4 q = 4 r ² - 4 s \Rightarrow s - q = r ² - p ² .$

So, Assertion (A) is true, but Reason (R) is not the correct explanation of Assertion (A).

Q 6 :

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.

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

1

2

3

4

(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).

Q 7 :

Assertion (A): The quadratic equation $4 x ² - 12 x + 9 = 0$ has repeated roots.

Reason (R): The quadratic equation $a x ² + b x + c = 0, a \neq 0$ , has equal roots if D > 0.

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

1

2

3

4

(3)

Given, $4 x ² - 12 x + 9 = 0 . D i s c r i m i n a n t, D = (- 12) ² - 4 \times 4 \times 9 = 144 - 144 = 0 .$

Hence, the equation has equal (repeated) roots. Assertion (A) is true but Reason (R) is false.

Q 8 :

Assertion (A): The equation $a x ² + b x + c = 0$ has two real roots always.

Reason (R): The sum of roots of $a x ² + b x + c = 0 i s - b / a$ and the product of roots is c/a.

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

1

2

3

4

(4)

For a quadratic equation $a x ² + b x + c = 0,$ we know:
(i) $I f b ² - 4 a c > 0,$ then the equation has two real and distinct roots.
(ii) $I f b ² - 4 a c = 0$ , then the equation has two real and equal roots.
(iii) $I f b ² - 4 a c < 0$ , then the equation has no real roots.

And, sum of roots = −b/a and product of roots = c/a.
Hence, Assertion (A) is false but Reason (R) is true.

Topic Question Set

Q 1 :

Assertion (A): 9x² - 3x - 20 = 0 ⇒ (3x - 5)(3x + 4) = 0.The roots are calculated by splitting the middle term.

Reason (R): To factorise ax² + bx + c = 0, we write in the form $a x ² + b ₁ x + b ₂ x + c = 0$ such that $b ₁ + b ₂ = b a n d b ₁ b ₂ = a c$ .

Q 2 :

Assertion (A): The roots of equation x² – 7x + 12 = 0 are x = 3 and x = 4.

Reason (R): The splitting of middle term of the equation x² – 7x + 12 = 0 is
$(x ² - 7 / 2 x - 7 / 2 x + 12 = 0) .$

Q 3 :

Assertion (A): If $5 + \sqrt 7$ is a root of a quadratic equation with rational coefficients, then its other root is $5 - \sqrt 7$

Reason (R): Irrational roots of a quadratic equation with rational coefficients occur in conjugate pairs.

Q 4 :

Assertion (A): $2 x ² - 4 x + 3 = 0$ is a quadratic equation.

Reason (R): All polynomials of degree n, when n is a whole number, can be treated as quadratic equations.

Q 6 :

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.

Q 7 :

Assertion (A): The quadratic equation $4 x ² - 12 x + 9 = 0$ has repeated roots.

Reason (R): The quadratic equation $a x ² + b x + c = 0, a \neq 0$ , has equal roots if D > 0.

Q 8 :

Assertion (A): The equation $a x ² + b x + c = 0$ has two real roots always.

Reason (R): The sum of roots of $a x ² + b x + c = 0 i s - b / a$ and the product of roots is c/a.

Want Us to Email you About Special Offers & Updates?