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Question #41172

Difficulty: 8/10 | 🟢 Free

The Question

Let y=y(x)y = y(x) be a differentiable function on an interval containing x=−1x = -1, satisfying the implicit equation y2−x2=3exp⁡(3πcos⁡−1(x∣y∣)−2)y^2 - x^2 = 3 \exp\left( \frac{3}{\pi} \cos^{-1}\left(\frac{x}{|y|}\right) - 2 \right) If y(−1)=−2y(-1) = -2, then the value of dydx\frac{dy}{dx} at x=−1x = -1 is

Options

  • A.

    2(33+2π33+8π)2 \left( \frac{3\sqrt{3} + 2\pi}{3\sqrt{3} + 8\pi} \right)

  • B.

    2(33−2π33−8π)2 \left( \frac{3\sqrt{3} - 2\pi}{3\sqrt{3} - 8\pi} \right)

  • C.

    2(33+2π33−8π)2 \left( \frac{3\sqrt{3} + 2\pi}{3\sqrt{3} - 8\pi} \right)

  • D.

    2(33−2π33+8π)2 \left( \frac{3\sqrt{3} - 2\pi}{3\sqrt{3} + 8\pi} \right)

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Correct Answer

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Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Derivative of Implicit Inverse Forms|Implicit Differentiation Rate|The Modulus Trap

Why This Question Is Tricky

Common Mistakes

This problem tests implicit differentiation combined with the 'Modulus Trap' inside an inverse trigonometric function.

Students often blindly differentiate ∣y∣|y| as yy or ignore the modulus entirely, leading to the incorrect chain rule application for cos⁡−1(x/y)\cos^{-1}(x/y).

Because y(−1)=−2<0y(-1) = -2 < 0, the correct local branch is ∣y∣=−y|y| = -y, which flips the sign of the argument to cos⁡−1(−x/y)\cos^{-1}(-x/y).

This sign flip fundamentally alters the derivative's value at the point, catching students who fail to resolve the absolute value locally before differentiating.

Quick Facts About This Question

Question Information

SubjectMathematics
ChapterInverse Trigonometric Functions - Class 12
ConceptDerivative of Implicit Inverse Forms
Difficulty8
Cognitive Load7
Target Solve Time160 sec
AccessFree