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NDA 2 2024 Maths, question 21

Topic: Trigonometry

Q21

If 4sin⁡−1x+cos⁡−1x=π4\sin ^{-1}x+\cos ^{-1}x=\pi, then what is sin⁡−1x+4cos⁡−1x\sin ^{-1}x+4\cos ^{-1}x equal to?

  1. (a) π/2\pi/2
  2. (b) π\pi
  3. (c) 3π/23\pi/2Correct answer
  4. (d) 2π2\pi

Solution.

Answer: (c) 3π/23\pi/2

  1. Put A=sin⁻¹x and B=cos⁻¹x.
  2. Principal values satisfy A+B=π/2.
  3. Combining with 4A+B=π gives A=π/6 and B=π/3.
  4. Thus A+4B=3π/2; x=1/2 checks the original equation.