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NDA 2 2024 Maths, question 24

Topic: Trigonometry

Q24

The roots of the equation 7x2−6x+1=07x^{2}-6x+1=0 are tan⁡α\tan \alpha and tan⁡β\tan \beta, where 2α2\alpha and 2β2\beta are the angles of a triangle. Which one of the following is correct?

  1. (a) The triangle is equilateral
  2. (b) The triangle is isosceles but not right-angled
  3. (c) The triangle is right-angledCorrect answer
  4. (d) The triangle is right-angled isosceles

Solution.

Answer: (c) The triangle is right-angled

  1. Vieta gives tan α+tan β=6/7 and tan α tan β=1/7.
  2. Hence tan(α+β)=(6/7)/(1−1/7)=1.
  3. Since α,β are positive half-angles with α+β<π/2, α+β=π/4.
  4. Thus the third triangle angle is π−2(α+β)=π/2.
  5. The two roots differ, so it is not isosceles.