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Byju's Answer

Standard XII

Mathematics

Conjugate of a Complex Number

Assertion :If...

Question

Assertion :If the roots of the equations $x^{2} - b x + c = 0$ and $x^{2} - c x + b = 0$ differ by the same quantity, then $b + c$ is equal to $- 4$ Reason: If $α, β$ are the roots of the equation $A x^{2} + B x + C = 0,$ then $α - β = \frac{\sqrt{B^{2} - 4 A C}}{A}$

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
We know that if $α, β$ are roots of the equation
$A x^{2} + B x + C = 0,$
Then, $α - β = \sqrt{\frac{B^{2} - 4 A C}{A}} .$
Equating the value of $α - β$ from both the given equations,
we get
$\sqrt{b^{2} - 4 c} = \sqrt{c^{2} - 4 b} \Rightarrow b^{2} - 4 c = c^{2} - 4 b$
$\Rightarrow b^{2} - c^{2} = - 4 (b - c) \Rightarrow (b - c) (b + c + 4) = 0$
$\Rightarrow b + c = - 4 (∵ b \neq c)$

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Similar questions

Q. Assertion :If the roots of the equations

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

and

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

differ by the same quantity, then

b + c

is equal to

- 4

Reason: If

α, β

are the roots of the equation

A x^{2} + B x + C = 0

, then

α - β = \frac{\sqrt{B^{2} - 4 A C}}{A}

Q. Assertion (A)
The equation x² + x + 1 = 0 has both real roots.

Reason (R)
The equation ax2 + bx + c = 0, (a ≠ 0) has both real roots, if (b2 − 4ac) > 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 .

(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) Order of the reaction can be zero or fractional.

Reason (R) We cannot determine order from balanced chemical equation.

(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 the roots of the equations x2−bx+c=0 and x2−cx+b=0 differ by the same quantity, then b+c is equal to −4 Reason: If α,β are the roots of the equation Ax2+Bx+C=0, then α−β=√B2−4ACA

Assertion :If the roots of the equations $x^{2} - b x + c = 0$ and $x^{2} - c x + b = 0$ differ by the same quantity, then $b + c$ is equal to $- 4$ Reason: If $α, β$ are the roots of the equation $A x^{2} + B x + C = 0,$ then $α - β = \frac{\sqrt{B^{2} - 4 A C}}{A}$