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Question #41935

Difficulty: 8/10 | 🟢 Free

The Question

Let $L_1$ and $L_2$ be two lines in $\mathbb{R}^3$ . The direction vector of $L_1$ is given by $\vec{d_1} = f_1'(1)\hat{i} + f_2'(1)\hat{j} + f_3'(1)\hat{k}$ , where $f_1(x) = \int_{0}^{x} 3 \ln|t-3| dt, \quad f_2(x) = \int_{0}^{x} 2 \ln|t-5| dt, \quad f_3(x) = \int_{0}^{x} 12 \ln|t+1| dt$ The direction vector of $L_2$ is given by $\vec{d_2} = g_1'(2)\hat{i} + g_2'(2)\hat{j} + g_3'(2)\hat{k}$ , where $g_1(x) = \int_{0}^{x} 4 \ln|t| dt, \quad g_2(x) = \int_{0}^{x} 12 \ln|t| dt, \quad g_3(x) = \int_{0}^{x} 3 \ln|t| dt$ If $\theta$ is the acute angle between the lines $L_1$ and $L_2$ , then the value of $\cos \theta$ is:

Options

A.
$\frac{96}{169}$
B.
$\frac{24}{169}$
C.
$\frac{48}{169}$
D.
$\frac{64}{169}$

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

Angle between two lines|Newton-Leibniz Formula with variable limits|Logarithmic Integration Modulus

Why This Question Is Tricky

Common Mistakes

This question tests the application of the Newton-Leibniz formula combined with the properties of the logarithmic function involving a modulus.

The core trap lies in evaluating the derivative of the integral at a point where the argument of the natural logarithm is negative.

Students often incorrectly assume $\ln(-x) = -\ln x$ when trying to resolve a negative argument algebraically, failing to use the proper definition $\ln|x| = \ln(-x)$ for $x < 0$ .

This leads to a sign error in the first two components of the direction vector $\vec{d_1}$ , changing its proportional direction ratios from $(3, 4, 12)$ to $(-3, -4, 12)$ .

Consequently, the dot product calculation yields an incorrect acute angle of $\cos^{-1}(24/169)$ instead of the correct $\cos^{-1}(96/169)$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Three-Dimensional Geometry - Class 12
Concept	Angle between two lines
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free