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Question #41658

Difficulty: 8/10 | 🟢 Free

The Question

Let $\omega = e^{i \frac{2\pi}{5}}$ . Let $\text{Arg}(z)$ denote the principal argument of a complex number $z$ , such that $-\pi < \text{Arg}(z) \le \pi$ . Consider two planes $P_1$ and $P_2$ in 3D space defined by the equations: $P_1: x \text{Arg}(\omega^2) + y \text{Arg}(\omega^3) + z \text{Arg}(\omega) = 1$ $P_2: x \text{Arg}(\omega^2) + y \text{Arg}(\omega^4) + z \text{Arg}(\omega^3) = 2$ If $\theta$ is the acute angle between the planes $P_1$ and $P_2$ , then the value of $\cos \theta$ is:

Options

A.
$\frac{4}{9}$
B.
$\frac{8}{9}$
C.
$\frac{19}{\sqrt{406}}$
D.
$\frac{5}{9}$

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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 two planes|Roots of Unity Properties|Argument and Principal Value

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of 3D coordinate geometry with the properties of the n-th roots of unity, specifically targeting the principal argument branch constraint.

The primary trap lies in the definition of $\text{Arg}(z) \in (-\pi, \pi]$ .

Students frequently ignore this and use standard polar angles like $\frac{6\pi}{5}$ and $\frac{8\pi}{5}$ for $\omega^3$ and $\omega^4$ , which leads to incorrect normal vectors and the distractor $19/\sqrt{406}$ .

Another common error is taking the absolute values of the arguments, which yields the distractor $8/9$ .

The problem strictly enforces branch cut awareness to correctly evaluate the dot product and the resulting acute angle.

Quick Facts About This Question

Question Information

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