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NDA 2023
Exam
NDA
Year
2023
Subject
Mathematics
Chapter
Quadratic Equation

NDA 2023 · Mathematics · Quadratic Equation

α and β are distinct real roots of the quadratic equation...

α\alpha and β\beta are distinct real roots of the quadratic equation x2+ax+b=0\mathrm{x}^2+\mathrm{ax}+\mathrm{b}=0. Which of the following statements is/are sufficient to find α\alpha ? 1. α+β=0,α2+β2=22.αβ2=−1,a=0\alpha+\beta=0, \alpha^2+\beta^2=2 2. \alpha \beta^2=-1, a=0 Select the correct answer using the code given below:

Options

  1. A1 only
  2. B2 only
  3. CBoth 1 and 2
  4. DNeither 1 nor 2

Correct answer

C. Both 1 and 2

Step-by-step solution

x2+ax+b=0x^2+a x+b=0 (i) α+β=0⇒β=−αα2+β2=2(α+β)2−2αβ=20−2αβ=2αβ=−1⇒α⋅(−α)=−1⇒α2=1\alpha+\beta=0 \Rightarrow \beta=-\alpha\alpha^2+\beta^2=2(\alpha+\beta)^2-2 \alpha \beta=20-2 \alpha \beta=2\alpha \beta=-1 \Rightarrow \alpha \cdot(-\alpha)=-1 \Rightarrow \alpha^2=1 (ii) αβ2=−1a=0⇒α+β=0⇒β=−αα(−α)2=−1⇒α3=−1⇒α=−1∴\alpha \beta^2=-1a=0 \Rightarrow \alpha+\beta=0 \Rightarrow \beta=-\alpha\alpha(-\alpha)^2=-1 \Rightarrow \alpha^3=-1 \Rightarrow \alpha=-1\therefore \quad Both 1 and 2 are sufficient.
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