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Question #40418

Difficulty: 8/10 | 🟢 Free

The Question

Let $\vec{u} = \sin(x^\circ)\hat{i} + \cos(x^\circ)\hat{j}$ and $\vec{v} = \sin(x)\hat{i} + \cos(x)\hat{j}$ , where $x > 0$ is a real number. A line $L$ with direction cosines $l, m, n$ is orthogonal to $\vec{u}$ and makes an angle of $\frac{\pi}{4}$ with the positive z-axis. If the acute angle between $L$ and $\vec{v}$ is $\frac{\pi}{3}$ , find the smallest positive value of $x$ .

Options

A.
$\frac{45\pi}{180 - \pi}$
B.
$\frac{30\pi}{180 - \pi}$
C.
$\frac{45\pi}{180 + \pi}$
D.
$\frac{90\pi}{180 - \pi}$

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

Radian-Degree Conversion Identity|Linear Independence vs Dependence|Direction Cosines constraint

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of radian-degree conversions with 3D geometry constraints.

The primary trap lies in the Direction Cosine identity $l^2 + m^2 + n^2 = 1$ .

Students often find the orthogonal components $l \propto -\cos(x^\circ)$ and $m \propto \sin(x^\circ)$ and incorrectly assume $l = -\cos(x^\circ)$ and $m = \sin(x^\circ)$ without normalizing them against the fixed $n = 1/\sqrt{2}$ component.

This 'Identity Failure' leads to an incorrect dot product magnitude, yielding the trap answer $x = \frac{30\pi}{180 - \pi}$ .

Furthermore, the question forces the explicit algebraic conversion $x^\circ = \frac{\pi x}{180}$ to solve the trigonometric equation, penalizing those who treat degrees and radians as interchangeable variables.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Trigonometric Functions - Class 11
Concept	Radian-Degree Conversion Identity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free