Q173 / 270 · NDA 2023 (Phase 1) · Mathematics

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

  • A
    1 only
  • B
    2 only
  • Correct answer
    C
    Both 1 and 2
  • D
    Neither 1 nor 2

Solution

$x^2+a x+b=0$ (i) $\alpha+\beta=0 \Rightarrow \beta=-\alpha$ $\alpha^2+\beta^2=2$ $(\alpha+\beta)^2-2 \alpha \beta=2$ $0-2 \alpha \beta=2$ $\alpha \beta=-1 \Rightarrow \alpha \cdot(-\alpha)=-1 \Rightarrow \alpha^2=1$ (ii) $\alpha \beta^2=-1$ $a=0 \Rightarrow \alpha+\beta=0 \Rightarrow \beta=-\alpha$ $\alpha(-\alpha)^2=-1 \Rightarrow \alpha^3=-1 \Rightarrow \alpha=-1$ $\therefore \quad$ Both 1 and 2 are sufficient.

Hence, the correct answer is option: 'C'.

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