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Question #41648

Difficulty: 8/10 | 🟢 Free

The Question

Let a curve $L$ be the locus of points $(x, y, z)$ satisfying the equations: $y = 2x - 1 - x \ln x + \int_1^x \ln(t) dt$ $z = x + \frac{11}{4} - \frac{x^2}{2} \ln x + \frac{x^2}{4} + \int_1^x t \ln(t) dt$ Find the length of the segment of $L$ that lies inside the sphere $S: x^2 + y^2 + z^2 - 2x - 4y - 6z = 0$ .

Options

A.
$3\sqrt{3}$
B.
$4\sqrt{3}$
C.
$2\sqrt{3}$
D.
$5\sqrt{3}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

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

Concepts Behind This Question

Line lying in a plane|Exact Differential Recognition|Integration by Parts (LIATE Rule)|Domain Restrictions

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to recognize that simplifying integral equations via Integration by Parts can yield a linear trajectory in 3D space.

The primary trap lies in the domain restriction: the logarithmic terms implicitly restrict the line to a ray ( $x > 0$ ).

Students who blindly simplify the equations and treat the locus as a full infinite line will find two intersection points with the sphere ( $x = 3$ and $x = -1$ ) and incorrectly calculate the full chord length ( $4\sqrt{3}$ ).

By enforcing the domain constraint, the intersection set is truncated, and the true segment length is measured from the ray's origin to the valid boundary point ( $3\sqrt{3}$ ).

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Three-Dimensional Geometry - Class 12
Concept	Line lying in a plane
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free