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NDA 2019 Maths · Mathematics · Quadratic Equations · medium

If α,β are the roots of x^2-3x+1=0, then (α^2/(β)+(β^2/(α) equals

Single correct option

If α,β\alpha,\beta are the roots of x2−3x+1=0x^2-3x+1=0, then α2β+β2α\dfrac{\alpha^2}{\beta}+\dfrac{\beta^2}{\alpha} equals
  1. A. 99
  2. B. 1818
  3. C. 77
  4. D. 2121
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Answer: (B) 1818

Combine the fractions: α3+β3αβ\dfrac{\alpha^3+\beta^3}{\alpha\beta}. Here α+β=3\alpha+\beta=3 and αβ=1\alpha\beta=1, so α3+β3=(α+β)3−3αβ(α+β)=27−9=18\alpha^3+\beta^3=(\alpha+\beta)^3-3\alpha\beta(\alpha+\beta)=27-9=18. Dividing by αβ=1\alpha\beta=1 gives 1818. The value 77 is α2+β2\alpha^2+\beta^2, and 99 is (α+β)2(\alpha+\beta)^2, both tempting but not asked.
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