Q195 / 270 · NDA 2024 (Phase 1) · Mathematics

Under which one of the following conditions does the equation $(\cos \beta-1) x^2+(\cos \beta) x+\sin \beta=0$ in $x$ have a real root for $\beta \in[0, \pi]$ ?

  • A
    $1-\cos \beta \lt 0$
  • Correct answer
    B
    $1-\cos \beta \leq 0$
  • C
    $1-\cos \beta \gt 0$
  • D
    $1-\cos \beta \geq 0$

Solution

For real roots $(\cos \beta)^2-4 \sin \beta(\cos \beta-1) \geq 0$ Since, $-1 \leq \cos \beta \leq 1$ and $\sin \beta \geq 0$ for $\beta \in[0, \pi]$ So, it is only possible when, $\cos \beta-1 \geq 0$.

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

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