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NDA 1 2025 Maths, question 92

Topic: Differential calculus

Passage · Q91 to Q92

Let y=sin⁡−1(x−4x327)y = \sin^{- 1}\left( x - \frac{4x^{3}}{27} \right).

Q92

What is dy/dxdy/dx equal to?

  1. (a) 19−x2\frac{1}{\sqrt{9 - x^{2}}}
  2. (b) 13−x2\frac{1}{\sqrt{3 - x^{2}}}
  3. (c) 39−x2\frac{3}{\sqrt{9 - x^{2}}}Correct answer
  4. (d) 99−x2\frac{9}{\sqrt{9 - x^{2}}}

Solution.

Answer: (c) 39−x2\frac{3}{\sqrt{9 - x^{2}}}

  1. From question 91, y=3sin⁡−1(x/3)y = 3\sin^{-1}(x/3).
  2. dydx=3×11−x2/9×13=39−x2\frac{dy}{dx} = 3 \times \frac{1}{\sqrt{1 - x^{2}/9}} \times \frac{1}{3} = \frac{3}{\sqrt{9 - x^{2}}}.

Trap: Like question 91, this is valid for ∣x∣<3/2|x| < 3/2; for 3/2<∣x∣<33/2 < |x| < 3 the derivative changes sign.