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NDA 2022
Exam
NDA
Year
2022
Subject
Mathematics
Chapter
Quadratic Equation

NDA 2022 · Mathematics · Quadratic Equation

Let α and β be the roots of the equation x^2+p x+q=0 . If α^3 and...

Let α\alpha and β\beta be the roots of the equation x2+px+q=0x^2+p x+q=0. If α3\alpha^3 and β3\beta^3 are the roots of the equation x2+mx+n=0x^2+m x+n=0, then what is the value of m+nm+n ?

Options

  1. Ap3+q3+pqp^3+q^3+p q
  2. Bp3+q3−pqp^3+q^3-p q
  3. Cp3+q3+3pqp^3+q^3+3 p q
  4. Dp3+q3−3pqp^3+q^3-3 p q

Correct answer

D. p3+q3−3pqp^3+q^3-3 p q

Step-by-step solution

Given that α,β\alpha, \beta are roots of equation x2+px+q=0∴α+β=−px^2+p x+q=0\therefore \alpha+\beta=-p \quad and α.β=q\quad \alpha. \beta=q and α3\alpha^3 and β3\beta^3 are roots of equation x2+mn+n=0∴α3+β3=(α+β)3−3αβ(α+β)=−m⇒−p3+3pq=−m⇒m=p3−3pqα3⋅β3=(αβ)3=q3=n∴m+n=p3+q3−3pq[x^2+m n+n=0\therefore \quad \alpha^3+\beta^3=(\alpha+\beta)^3-3 \alpha \beta(\alpha+\beta)=-m\Rightarrow \quad-p^3+3 p q=-m \Rightarrow m=p^3-3 p q\alpha^3 \cdot \beta^3=(\alpha \beta)^3=q^3=n\therefore \quad m+n=p^3+q^3-3 \mathrm{pq}[ from (i)(i) and (ii)](i i)]
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