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Byju's Answer

Standard XII

Mathematics

Roots of a Quadratic Equation

Assertion :If...

Question

Assertion :If $z_{1}, z_{2}$ are the roots of the quadratic equation $a z^{2} + b z + c = 0$ such that at least one of a, b, c is imaginary then $z_{1}$ and $z_{2}$ are conjugate of each other Reason: If quadratic equation having real coefficients has complex roots, then roots are always conjugate to each other

Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

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Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

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Assertion is correct but Reason is incorrect

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Assertion is incorrect and Reason is correct

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Solution

The correct option is D Assertion is incorrect and Reason is correct
Consider the quadratic equation
$a x^{2} + b x + c = 0$
Therefore the roots will be
$x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$
Now
If $a, b, c$ are all real and $b^{2} - 4 a c < 0$
Then
$x = \frac{- b \pm i \sqrt{| b^{2} - 4 a c |}}{2 a}$

Hence we get two conjugate roots
$x = \frac{- b + i \sqrt{| b^{2} - 4 a c |}}{2 a}$ and $x = \frac{- b - i \sqrt{| b^{2} - 4 a c |}}{2 a}$
Hence reason is correct, but assertion is wrong.

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Similar questions

Q. Assertion :If the equation

a x^{2} + b x + c = 0

0 < a < b < c,

has non real complex roots

z_{1}

and

z_{2}

, then

| z_{1} | > 1

| z_{2} | > 1

. Reason: Complex roots always occur in conjugate pairs.

Q. Assertion (A)
If

p x^{2} - 2 x + 2 = 0

has real roots, then

p \leq \frac{1}{2} .

Reason (R)
The equation

(a^{2} +^{b}) x^{2} + 2 (a c + b d) x + (c^{2} + d^{2}) = 0

has no real root, if ad ≠ bc.

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).
(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).
(c) Assertion (A) is true and Reason (R) is false.
(d) Assertion (A) is false and Reason (R) is true.

Q. Assertion (A)
If −5 is a root of

2 x^{2} + 2 p x - 15 = 0 and p (x^{2} + x) + k = 0

has equal roots, then

k = \frac{7}{8} .

Reason (R)
The equation

a x^{2} + b x + c = 0, a \neq 0

has equal roots, if

(b^{2} - 4 a c) = 0 .

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).
(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).
(c) Assertion (A) is true and Reason (R) is false.
(d) Assertion (A) is false and Reason (R) is true.

Assertion: All the points (1, 0), (-1, 0), (2, 0) and (5, 0) lie on the x – axis.

Reason: Equation of the x – axis is y = 0
Now, choose the correct option.

Assertion (A) Rate constant determined from Arrhenius equation are fairly accurate for simple as well as complex molecules.

Reason (R) Reactant molecules undergo chemical change irrespective of their orientation during collision.

(a) Both assertion and reason are correct and the reason is correct explanation of assertion
(b) Both assertion and reason are correct, but reason does not explain assertion.
(c) Assertion is correct, but reason is incorrect
(d) Both assertion and reason are incorrect
(e) Assertion is incorrect but reason is correct.

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Assertion :If z1,z2 are the roots of the quadratic equation az2+bz+c=0 such that at least one of a, b, c is imaginary then z1 and z2 are conjugate of each other Reason: If quadratic equation having real coefficients has complex roots, then roots are always conjugate to each other