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Question #41173

Difficulty: 8/10 | 🟢 Free

The Question

A curve $C$ in $\mathbb{R}^3$ is defined by the intersection of the surfaces $x^3 + 3x = 4y$ and $y^3 + y = 2z$ . Let $L$ be the tangent line to the curve $C$ at the point $P(1, 1, 1)$ . If $\theta$ is the acute angle between the line $L$ and the plane $6x - 2y + 3z = 10$ , find the value of $49 \sin \theta$ .

Options

A.
15
B.
18
C.
24
D.
36

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Derivative of Implicit Inverse Forms|Angle between line and plane|Parametric form pitfalls

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to correctly extract the direction vector of a 3D curve defined by implicit inverse forms.

The primary trap lies in the 'Parametric form pitfall.' Students often incorrectly assume the direction vector is simply $\langle \frac{dx}{dy}, \frac{dy}{dz}, 1 \rangle$ or $\langle 1, \frac{dy}{dx}, \frac{dz}{dy} \rangle$ without applying the chain rule properly across the parameters, leading to erroneous vectors like $\langle 4, 3, 6 \rangle$ (which yields the trap answer 36) or $\langle 2, 3, 4 \rangle$ (yielding 18).

Furthermore, students might confuse the gradients of the implicit surfaces with the direction vector of the curve itself.

Proper parametrization or the cross-product of gradients is required to preserve the orientation parameter.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Inverse Trigonometric Functions - Class 12
Concept	Derivative of Implicit Inverse Forms
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free