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Question #72164

Difficulty: 8/10 | 🟢 Free

The Question

Let a line $L$ have direction ratios $(l, m, n)$ given by the following limits: $l = \lim_{x \to 0} \frac{6(ax - \sin(ax))}{x^2 \sin x}$ $m = \lim_{x \to 0} \frac{3(\tan(bx) - bx)}{2x(1 - \cos x)}$ $n = \lim_{x \to 0} \frac{6(\arcsin(cx) - cx)}{x^2 \tan x}$ where $a, b, c$ are non-zero real parameters. If $\theta$ is the acute angle between the line $L$ and the plane $\Pi: \frac{x}{a^3} + \frac{y}{b^3} + \frac{z}{c^3} = 1$ , which of the following represents $\theta$ ?

Options

A.
$\arcsin \left( \frac{3}{\sqrt{a^6 + b^6 + c^6} \sqrt{a^{-6} + b^{-6} + c^{-6}}} \right)$
B.
$\arccos \left( \frac{3}{\sqrt{a^6 + b^6 + c^6} \sqrt{a^{-6} + b^{-6} + c^{-6}}} \right)$
C.
$\arcsin \left( \frac{3}{\sqrt{a^2 + b^2 + c^2} \sqrt{a^{-2} + b^{-2} + c^{-2}}} \right)$
D.
$\arcsin \left( \frac{1}{\sqrt{a^6 + b^6 + c^6} \sqrt{a^{-6} + b^{-6} + c^{-6}}} \right)$

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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 line and plane|Indeterminate Forms|Trigonometric Standard Limits

Why This Question Is Tricky

Examiner Levers

Variable Manipulation

Algebraic solutions only (No numbers); tests pure symbolic logic.

Difficulty Boost : +4

Common Mistakes

This question tests the strategic choice of Taylor series expansions over L'Hôpital's rule for evaluating complex $0/0$ trigonometric indeterminate forms.

Students who reflexively apply L'Hôpital's rule will face massive algebraic expansion when differentiating denominators like $x^2 \sin x$ or $2x(1-\cos x)$ multiple times, leading to a high probability of arithmetic errors.

Furthermore, it tests the geometric distinction between the angle between two vectors (which uses cosine) and the angle between a line and a plane (which uses sine), a classic conceptual trap represented in the options.

Quick Facts About This Question

Question Information

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