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Standard XII

Mathematics

Properties of Conjugate of a Complex Number

Assertion :If...

Question

Assertion :If $a^{2} + b^{2} + c^{2} < 0$ , then if roots of the equation $a x^{2} + b x + c = 0$ are imaginary, then they are not complex conjugates. Reason: Equation $a x^{2} + b x + c = 0$ has complex conjugate roots when $a, b, c$ are real.

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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Both Assertion and Reason are incorrect

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Solution

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
If $a^{2} + b^{2} + c^{2} < 0,$ then all $a, b, c$ are not real or at least one of $a, b, c$ is imaginary number.
Hence roots of equation $a x^{2} + b x +$ $c = 0$ has no complex conjugate roots, even though the roots are complex.

Hence statement 1 is true. Statement 2 is obviously true (see the theory).
Also, statement 2 is correct explanation of statement 1.

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Similar questions

Q. Assertion (A)
The equation

2 x^{2} - 4 x + 3 = 0

has no real roots.

Reason (R)
The equation

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

has no real 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.

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 .

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.

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.

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.

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Assertion :If a2+b2+c2<0, then if roots of the equation ax2+bx+c=0 are imaginary, then they are not complex conjugates. Reason: Equation ax2+bx+c=0 has complex conjugate roots when a,b,c are real.

Assertion :If $a^{2} + b^{2} + c^{2} < 0$ , then if roots of the equation $a x^{2} + b x + c = 0$ are imaginary, then they are not complex conjugates. Reason: Equation $a x^{2} + b x + c = 0$ has complex conjugate roots when $a, b, c$ are real.