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NDA 2024
Exam
NDA
Year
2024
Subject
Mathematics
Chapter
Quadratic Equation

NDA 2024 · Mathematics · Quadratic Equation

Under which one of the following conditions does the equation (...

Under which one of the following conditions does the equation (cos⁡β−1)x2+(cos⁡β)x+sin⁡β=0(\cos \beta-1) x^2+(\cos \beta) x+\sin \beta=0 in xx have a real root for β∈[0,π]\beta \in[0, \pi] ?

Options

  1. A1−cos⁡β<01-\cos \beta < 0
  2. B1−cos⁡β≤01-\cos \beta \leq 0
  3. C1−cos⁡β>01-\cos \beta > 0
  4. D1−cos⁡β≥01-\cos \beta \geq 0

Correct answer

B. 1−cos⁡β≤01-\cos \beta \leq 0

Step-by-step solution

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