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Question #40976

Difficulty: 8/10 | 🟢 Free

The Question

Let a line $L$ be parallel to the vector $\vec{v} = (\cos \alpha - \sin \alpha)\hat{i} + (\cos \alpha + \sin \alpha)\hat{j} + \sqrt{6}\hat{k}$ , where $\alpha \in [0, 2\pi)$ . Let $\theta$ be the angle between the line $L$ and the plane $P: x + y + \sqrt{6}z = 2024$ . If $R$ and $C_R$ denote the range and the coefficient of range of $\theta$ respectively, find the ordered pair $(R, C_R)$ .

Options

A.
$\left( \frac{\pi}{3}, \frac{1}{2} \right)$
B.
$\left( \frac{\pi}{3}, 1 \right)$
C.
$\left( \frac{\pi}{2}, \frac{1}{2} \right)$
D.
$\left( \frac{\pi}{6}, \frac{1}{3} \right)$

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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

Range and Coefficient of Range|Angle between line and plane|Parametric form pitfalls

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to correctly identify the angle between a line and a plane using parametric direction ratios.

The primary trap lies in the formula for the angle: students frequently confuse the angle between the line and the plane ( $\sin \theta$ ) with the angle between the line and the normal vector ( $\cos \theta$ ).

If a student incorrectly uses $\cos \theta = \frac{\vec{v} \cdot \vec{n}}{|\vec{v}||\vec{n}|}$ , they will obtain $\cos \theta \in [1/2, 1]$ , which implies $\theta \in [0, \pi/3]$ .

While this yields the exact same Range ( $R = \pi/3$ ), it results in an incorrect Coefficient of Range ( $C_R = 1$ ), leading them directly to the trap option.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Statistics - Class 11
Concept	Range and Coefficient of Range
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free