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GATE 2020 AE – Question 9

General Aptitude · Quantitative Aptitude: Algebra, Geometry and Mensuration · 2 marks · Multiple choice

Real a,b,c define the quadratic $ax^2-bx+c=0$ with equal root β. Which statement follows?

  1. $\beta=b/a$
  2. $\beta^2=ac$
  3. $\beta^3=bc/(2a^2)$
  4. $b^2\ne4ac$

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Show answer and explanation

Correct answer: (C) $\beta^3=bc/(2a^2)$

Explanation

A quadratic has a ≠ 0. Repeated roots give $2\beta=b/a$ and $\beta^2=c/a$. Multiplying these identities yields $\beta^3=[b/(2a)](c/a)=bc/(2a^2)$, C. The discriminant is zero, so D is false.

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