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Quadratic Equation and Inequalities question

2021 · 27 Jul · Shift 1 · Q36
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  5. /2021 · 27 Jul · Shift 1 · Q36

Quadratic Equation and Inequalities question

2021 · 27 Jul · Shift 1 · Q36

JEE MainMathematicsQuadratic Equation and InequalitiesMCQ+4 / −1
Let α\alphaα, β\betaβ be two roots of the equation x2 + (20)1/4x + (5)1/2 = 0. Then α\alphaα 8 + β\betaβ 8 is equal to
  1. A
    10
  2. B
    100
  3. C
    50
  4. D
    160
View written solutionFree

Correct answer: C

  1. Given quadratic equation

The equation is x2+204 x+5=0.x^2 + \sqrt[4]{20}\,x + \sqrt{5}=0.x2+420​x+5​=0. Let its roots be α,β\alpha,\betaα,β.

So, by Vieta's formulas, α+β=−204,\alpha+\beta=-\sqrt[4]{20},α+β=−420​, αβ=5.\alpha\beta=\sqrt{5}.αβ=5​.

  1. Simplify the coefficients

Note that 204=(20)1/4=(4⋅5)1/4=41/451/4=2 51/4.\sqrt[4]{20}=(20)^{1/4}=(4\cdot 5)^{1/4}=4^{1/4}5^{1/4}=\sqrt{2}\,5^{1/4}.420​=(20)1/4=(4⋅5)1/4=41/451/4=2​51/4. Also, 5=51/2.\sqrt{5}=5^{1/2}.5​=51/2.

Now observe: (α+β)2=20=25=2αβ.(\alpha+\beta)^2=\sqrt{20}=2\sqrt{5}=2\alpha\beta.(α+β)2=20​=25​=2αβ.

Thus, α2+2αβ+β2=2αβ\alpha^2+2\alpha\beta+\beta^2=2\alpha\betaα2+2αβ+β2=2αβ which gives α2+β2=0.\alpha^2+\beta^2=0.α2+β2=0.

  1. Use recurrence / power identities

We need α8+β8.\alpha^8+\beta^8.α8+β8. Let Sn=αn+βn.S_n=\alpha^n+\beta^n.Sn​=αn+βn. We already have S2=α2+β2=0.S_2=\alpha^2+\beta^2=0.S2​=α2+β2=0.

Now, S4=α4+β4=(α2+β2)2−2α2β2.S_4=\alpha^4+\beta^4=(\alpha^2+\beta^2)^2-2\alpha^2\beta^2.S4​=α4+β4=(α2+β2)2−2α2β2. Since α2+β2=0\alpha^2+\beta^2=0α2+β2=0 and αβ=5\alpha\beta=\sqrt{5}αβ=5​, S4=0−2(αβ)2=−2(5)2=−2⋅5=−10.S_4=0-2(\alpha\beta)^2=-2(\sqrt{5})^2=-2\cdot 5=-10.S4​=0−2(αβ)2=−2(5​)2=−2⋅5=−10.

Then, S8=α8+β8=(α4+β4)2−2α4β4.S_8=\alpha^8+\beta^8=(\alpha^4+\beta^4)^2-2\alpha^4\beta^4.S8​=α8+β8=(α4+β4)2−2α4β4. Now, α4β4=(αβ)4=(5)4=25.\alpha^4\beta^4=(\alpha\beta)^4=(\sqrt{5})^4=25.α4β4=(αβ)4=(5​)4=25. Hence, S8=(−10)2−2(25)=100−50=50.S_8=(-10)^2-2(25)=100-50=50.S8​=(−10)2−2(25)=100−50=50.

  1. Check options

Thus, α8+β8=50.\alpha^8+\beta^8=50.α8+β8=50. So the correct option is:

C: 50

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