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Question #74554

Difficulty: 5/10 | 🟢 Free

The Question

Let $n = 2023$ and $\omega = e^{i 2\pi/n}$ . For $k = 0, 1, 2, \dots, n$ , define the points $P_k = (\text{Re}(\omega^k), \text{Im}(\omega^k), k)$ in $\mathbb{R}^3$ . Let $L_1$ be the line passing through the points $A = \sum_{k=1}^{1011} P_k$ and $B = \sum_{k=1012}^{2022} P_k$ . Let $L_2$ be the line passing through $P_0$ and $P_n$ . Let $S$ be the set of all possible values of $MQ + NQ$ , where $M$ is any point on $L_1$ , $N$ is any point on $L_2$ , and $Q$ is any point in $\mathbb{R}^3$ . Which of the following values is NOT an element of $S$ ?

Options

A.
$\frac{5}{4}$
B.
$\frac{7}{4}$
C.
$\frac{9}{4}$
D.
$\frac{11}{4}$

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

Shortest distance between skew lines|Triangle Inequality Mind Games|Roots of Unity Properties

Why This Question Is Tricky

Examiner Levers

Negative Thinking

Uses "NOT", "EXCEPT", or "INCORRECT" to flip the logic gate.

Difficulty Boost : +2

Common Mistakes

This problem baits students into brute-forcing the shortest distance formula using complex geometric progression sums, leading to an inevitable algebraic collapse.

The trap is bypassed by recognizing the cyclic symmetry of the roots of unity (sum of roots = 0), which elegantly reveals that L1 and L2 lie in parallel planes x = -1/2 and x = 1.

Furthermore, it tests the Triangle Inequality (MQ + NQ >= MN), forcing students to establish a theoretical lower bound rather than just calculating a distance.

The 'Negative Thinking' lever is pulled by asking which value is NOT possible, requiring absolute confidence in the established lower bound of 3/2.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Three-Dimensional Geometry - Class 12
Concept	Shortest distance between skew lines
Difficulty	5
Cognitive Load	6
Target Solve Time	140 sec
Access	Free