By signing in you confirm you are 18 or older, or have your parent’s or guardian’s consent.

NDA 1 2024 Maths, question 45

Topic: Trigonometry

Q45

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] ?

  1. (a) 1−cos⁡β<01-\cos \beta<0
  2. (b) 1−cos⁡β≤01-\cos \beta\le 0
  3. (c) 1−cos⁡β>01-\cos \beta>0
  4. (d) 1−cos⁡β≥01-\cos \beta\ge 0Correct answer

Solution.

Answer: (d) 1−cos⁡β≥01-\cos \beta\ge 0

  1. For 0 ≤ β ≤ π, sin β ≥ 0 and 1 − cos β ≥ 0.
  2. The quadratic discriminant is cos²β + 4(1 − cos β)sin β, which is nonnegative throughout this interval.
  3. At the degenerate endpoint β = 0, the surviving linear equation also has a real solution.
  4. Thus the full interval in option (d) works.