Assertion -If - - - are the zeroes of x3-2x2-qx-r and -0- then 2q-r Reason- If - - - are the zeroes of ax3-bx2-cx-d- then -b-a- -c-a- - -d-a

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Standard X

Mathematics

Relationship between Zeroes and Coefficients of a Polynomial

Assertion :If...

Question

Assertion :If $α, β, γ$ are the zeroes of $x^{3} - 2 x^{2} + q x - r$ and $α + β = 0$ , then $2 q = r$ Reason: If $α, β, γ$ are the zeroes of $a x^{3} + b x^{2} + c x + d$ , then $α + β + γ = - \frac{b}{a}, α β + β γ + γ α = \frac{c}{a}, α β, γ = - \frac{d}{a}$

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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Assertion is incorrect but Reason is correct

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Solution

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
Clearly, Reason is true. [Standard Result]
$α + β + γ = - (- 2) = 2$
$\Rightarrow 0 + γ = 2, ∴ γ = 2$
$\Rightarrow α β γ = - (- r) = r$
$\Rightarrow α β (2) = r$
$\Rightarrow α β = \frac{r}{2}$
$\Rightarrow α β + β γ + γ α = q$
$\Rightarrow \frac{r}{2} + γ (α + β) = q$
$\Rightarrow \frac{r}{2} + γ (0) = q$
$\Rightarrow r = 2 q ∴$ Assertion is true
Since, Reason gives Assertion

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Similar questions

Q. Assertion (A)
A monic quadratic polynomial having 4 and −2 as zeroes is x² − 2x − 8.

Reason (R)
The monic quadratic polynomial having α and β as zeroes is given by p(x) = x² − (α + β) x + αβ.

(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. If

α, β, γ

are non zero roots of

x^{3} + p x^{2} + q x + r = 0

, then the equation whose roots are

α (β + γ), β (γ + α), γ (α + β)

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.

Q. Assertion (A)
If one zero of the polynomial p(x) = (k² + 4) x² + 9x + 4k is the reciprocal of the other zero, then k = 2.

Reason (R)
If (x − α) is a factor of the polynomial p(x), then α is a zero of p(x).

(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.

If $α, β, γ$ be the zeroes of the polynomial $a x^{3} + b x^{2} + c x + d$ , then the value of $α β + β γ + γ α$ is-

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Assertion :If α,β,γ are the zeroes of x3−2x2+qx−r and α+β=0, then 2q=r Reason: If α,β,γ are the zeroes of ax3+bx2+cx+d, then α+β+γ=−ba,αβ+βγ+γα=ca,αβ,γ=−da

Assertion :If $α, β, γ$ are the zeroes of $x^{3} - 2 x^{2} + q x - r$ and $α + β = 0$ , then $2 q = r$ Reason: If $α, β, γ$ are the zeroes of $a x^{3} + b x^{2} + c x + d$ , then $α + β + γ = - \frac{b}{a}, α β + β γ + γ α = \frac{c}{a}, α β, γ = - \frac{d}{a}$