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NDA 2 2025 Maths, question 78

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}}.

Q78

What is the slope of the tangent to the curve y=f(x)y=f(x) at x=0.5x=0.5?

  1. (a) 4π3/274\pi\sqrt{3}/27Correct answer
  2. (b) 8π3/278\pi\sqrt{3}/27
  3. (c) 4π4\pi
  4. (d) 8π8\pi

Solution.

Answer: (a) 4π3/274\pi\sqrt{3}/27

  1. Differentiating, the two x/(1−x²) terms cancel and y′=sin⁻¹x/(1−x²)^(3/2).
  2. At x=1/2, this is (π/6)/(3/4)^(3/2)=4π√3/27.