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Question #40909

Difficulty: 8/10 | 🟢 Free

The Question

Let $k = \lim_{x \to 0^+} \frac{1}{x^2} \int_{\sqrt{3}x}^{x^{3/2}} \frac{2 \ln(1+u^2)}{u} du$ . A point $P$ divides the line segment joining the points $A(1, 2, 3)$ and $B(3, 0, -1)$ in the ratio $k:1$ . Find the square of the distance of point $P$ from the origin.

Options

A.
26
B.
$\frac{13}{2}$
C.
6
D.
146

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

Ratio of Division (k:1)|Substitution (Change of Variable)|Newton-Leibniz Formula with variable limits

Why This Question Is Tricky

Common Mistakes

The question tests the application of the Newton-Leibniz formula under a limit, combined with a variable substitution that masks the true limits of integration.

The primary trap lies in the evaluation of the limit: the upper limit ( $x^3$ ) contributes $0$ to the final value, while the lower limit ( $3x^2$ ) dictates the result.

Because it is the lower bound, the Leibniz rule introduces a negative sign, yielding $k = -3$ .

Students who blindly subtract or ignore the lower limit's derivative will obtain a positive ratio (e.g., $k = 3$ ), incorrectly assuming internal division and arriving at the trap option $13/2$ .

This effectively filters candidates who lack rigorous sign tracking in calculus and 3D geometry vector division.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Introduction to Three-dimensional Geometry - Class 11
Concept	Ratio of Division (k:1)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free