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NDA 2 2025 Maths, question 79

Topic: Differential calculus

Passage · Q78 to Q80

Let y=f(x)=xsin⁡−1x1−x2+ln⁡1−x2y=f(x)=\frac{x\sin ^{-1}x}{\sqrt{1-x^{2}}}+\ln \sqrt{1-x^{2}}.

Q79

What is d2ydx2\frac{d^{2}y}{dx^{2}} at x=0x=0 equal to?

  1. (a) 00
  2. (b) 0.50.5
  3. (c) 11Correct answer
  4. (d) 1.51.5

Solution.

Answer: (c) 11

  1. From y′=sin⁻¹x/(1−x²)^(3/2), differentiate to obtain y″=(1−x²)⁻²+3x sin⁻¹x(1−x²)^(−5/2).
  2. At x=0 the first term is 1 and the second is 0, so y″(0)=1.