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Quadratic Equation and Inequalities question

2022 · 25 Jul · Shift 1 · Q22
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  5. /2022 · 25 Jul · Shift 1 · Q22

Quadratic Equation and Inequalities question

2022 · 25 Jul · Shift 1 · Q22

JEE MainMathematicsQuadratic Equation and InequalitiesMCQ+4 / −1
If α,β,γ,δ\alpha, \beta, \gamma, \deltaα,β,γ,δ are the roots of the equation x4+x3+x2+x+1=0x^{4}+x^{3}+x^{2}+x+1=0x4+x3+x2+x+1=0, then α2021+β2021+γ2021+δ2021\alpha^{2021}+\beta^{2021}+\gamma^{2021}+\delta^{2021}α2021+β2021+γ2021+δ2021 is equal to :
  1. A
    −-− 4
  2. B
    −-− 1
  3. C
    1
  4. D
    4
View written solutionFree

Correct answer: B

  1. Recognize the given polynomial

We have x4+x3+x2+x+1=0.x^4+x^3+x^2+x+1=0.x4+x3+x2+x+1=0.

Using the geometric series identity, x4+x3+x2+x+1=x5−1x−1(x≠1).x^4+x^3+x^2+x+1=\frac{x^5-1}{x-1} \quad (x\ne 1).x4+x3+x2+x+1=x−1x5−1​(x=1).

So the roots of x4+x3+x2+x+1=0x^4+x^3+x^2+x+1=0x4+x3+x2+x+1=0 are the 5th roots of unity other than 111.

Thus, α,β,γ,δ∈{ω,ω2,ω3,ω4},\alpha,\beta,\gamma,\delta \in \{\omega,\omega^2,\omega^3,\omega^4\},α,β,γ,δ∈{ω,ω2,ω3,ω4}, where ω5=1\omega^5=1ω5=1 and ω≠1\omega\ne 1ω=1.

  1. Reduce the exponent modulo 5

We need to find α2021+β2021+γ2021+δ2021.\alpha^{2021}+\beta^{2021}+\gamma^{2021}+\delta^{2021}.α2021+β2021+γ2021+δ2021.

Since each root satisfies r5=1r^5=1r5=1, powers repeat modulo 555.

Now, 2021≡1(mod5)2021 \equiv 1 \pmod{5}2021≡1(mod5) because 2021=5⋅404+1.2021=5\cdot 404+1.2021=5⋅404+1.

Hence for any root rrr, r2021=r5⋅404+1=(r5)404r=1404r=r.r^{2021}=r^{5\cdot 404+1}=(r^5)^{404}r=1^{404}r=r.r2021=r5⋅404+1=(r5)404r=1404r=r.

Therefore, α2021+β2021+γ2021+δ2021=α+β+γ+δ.\alpha^{2021}+\beta^{2021}+\gamma^{2021}+\delta^{2021}=\alpha+\beta+\gamma+\delta.α2021+β2021+γ2021+δ2021=α+β+γ+δ.

  1. Find the sum of the roots

For the polynomial x4+x3+x2+x+1,x^4+x^3+x^2+x+1,x4+x3+x2+x+1, by Vieta's formula, the sum of roots is α+β+γ+δ=−coefficient of x3coefficient of x4=−1.\alpha+\beta+\gamma+\delta=-\frac{\text{coefficient of }x^3}{\text{coefficient of }x^4}=-1.α+β+γ+δ=−coefficient of x4coefficient of x3​=−1.

So, α2021+β2021+γ2021+δ2021=−1.\alpha^{2021}+\beta^{2021}+\gamma^{2021}+\delta^{2021}=-1.α2021+β2021+γ2021+δ2021=−1.

  1. Check options
  • A: −4-4−4
  • B: −1-1−1 ✅
  • C: 111
  • D: 444

So the correct option is B.

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