For r-1 roots are - - for r-2 roots are - - for r-3 roots are - - then which of the following is true-

The correct options are A α= 1 /P_1+P_2 P_3 B β= 1 /P_1 P_2+P_3 C 1/α+1/β+1/γ=P_1+P_2+P_3 D γ= 1 / P_1+P_2+P_3 From the given data, when ...

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Byju's Answer

Standard XII

Mathematics

Relations between Roots and Coefficients : Higher Order Equations

For r=1 roo...

Question

For $r = 1$ roots are $α, β$ & for $r = 2$ roots are $α, γ$ & for $r = 3$ roots are $β, γ$ , then which of the following is true?

α = \frac{1}{P_{1} + P_{2} - P_{3}}

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β = \frac{1}{P_{1} - P_{2} + P_{3}}

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γ = \frac{1}{- P_{1} + P_{2} + P_{3}}

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\frac{1}{α} + \frac{1}{β} + \frac{1}{γ} = P_{1} + P_{2} + P_{3}

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Solution

The correct options are
A $α = \frac{1}{P_{1} + P_{2} - P_{3}}$
B $β = \frac{1}{P_{1} - P_{2} + P_{3}}$
C $\frac{1}{α} + \frac{1}{β} + \frac{1}{γ} = P_{1} + P_{2} + P_{3}$
D $γ = \frac{1}{- P_{1} + P_{2} + P_{3}}$
From the given data, when $r = 1, 2, 3$ , we get
$x^{2} - 2 P_{1} x + 1 = 0 (r o o t s, α, β) . . . . . . . . . . . . . . . . . . (A)$
$x^{2} - 2 P_{2} x + 2 = 0 (r o o t s, α, γ) . . . . . . . . . . . . . . . . . . . (B)$
$x^{3} - 2 P_{3} x + 3 = 0 (r o o t s β, γ) . . . . . . . . . . . . . . . . . . . (C)$

${\begin{matrix} α + β = 2 P_{1}, α β = 1, \end{matrix}$

${\begin{matrix} α + γ = 4 P_{2}, α γ = 2, \end{matrix}$

${\begin{matrix} β + γ = 6 P_{3}, β γ = 3, \end{matrix}$
For $r = 1, α, β$ are roots of $x^{2} - 2 P_{1} x + 1 = 0$
$α + β = 2 P_{1}$ and $α β = 1$
$\frac{α + β}{α β} = 2 P_{1}$
$\Rightarrow \frac{1}{α} + \frac{1}{β} = 2 P_{1},$
Similarly for $r = 2,$ we have $\frac{1}{α} + \frac{1}{γ} = 2 P_{2},$
for $r = 3,$ we have $\frac{1}{β} + \frac{1}{γ} = 2 P_{3} .$
Add and get $2 (\frac{1}{α} + \frac{1}{β} + \frac{1}{γ}) = 2 (P_{1} + P_{2} + P_{3})$
$\Rightarrow \frac{1}{α} + \frac{1}{β} + \frac{1}{γ} = P_{1} + P_{2} + P_{3}$

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Similar questions

Q. If

α, β

are roots of

p_{1} x^{2} + q_{1} x + r_{1} = 0, β, γ

are roots of

p_{2} x^{2} + q_{2} x + r_{2} = 0

and

α, β

are the roots of

p_{3} x^{2} + q_{3} x + r_{3} = 0

then

Q. In

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P_{1} : y = 0

and

P_{2} : x + z = 1

. Let

P_{3}

be a plane, different from

P_{1}

and

P_{2}

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P_{1}

and

P_{2}

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(0, 1, 0)

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P_{3}

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and the distance of a point

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Q. If

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x^{2} + q x + 1 = 0

show that

q^{2} - p^{2} - (α - γ) (β - γ) (α + δ) (β + δ) = 0

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Q. If

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For r=1 roots are α,β & for r=2 roots are α,γ & for r=3 roots are β,γ, then which of the following is true?

For $r = 1$ roots are $α, β$ & for $r = 2$ roots are $α, γ$ & for $r = 3$ roots are $β, γ$ , then which of the following is true?