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Question #40613

Difficulty: 8/10 | 🟢 Free

The Question

Let $U_n$ be the set of all $n$ -th roots of unity in the Argand plane, where $n \ge 4$ is an even integer. Let $T$ and $Q$ denote the number of distinct triangles and quadrilaterals, respectively, whose vertices belong to $U_n$ . Let $d$ be the number of diagonals of the regular $n$ -gon formed by $U_n$ that pass through the origin. If $S$ is the set of all valid $n$ satisfying the inequality $\frac{Q - 5T}{d^2 - 13d + 40} \le 0$ find the sum of all elements in $S$ .

Options

A.
78
B.
104
C.
159
D.
26

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Geometry-based Combinations|Roots of Unity Properties|Wavy Curve (Method of Intervals)

Why This Question Is Tricky

Common Mistakes

This question tests the integration of geometric properties of roots of unity with combinatorics and rational inequalities.

The primary trap lies in the Wavy Curve method (Method of Intervals).

Students often forget to exclude the roots of the denominator ( $n=10, 16$ ), leading to an incorrect sum of 104.

Additionally, they may overlook the parity constraint ( $n$ must be even) derived from the existence of diagonals passing through the origin, erroneously including odd integers in the solution set.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Permutations and Combinations - Class 11
Concept	Geometry-based Combinations
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free