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Question #41102

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all points that divide the line segment joining $A(2, -3, 4)$ and $B(-1, 1, -1)$ strictly internally. A relation $R_n$ on $S$ is defined as $(P, Q) \in R_n$ if and only if both $P$ and $Q$ lie strictly inside the $n^{\text{th}}$ octant. Let $E$ be the set of all octant numbers $n \in \{1, 2, \dots, 8\}$ for which $R_n$ is an empty relation. If $P_\alpha$ and $P_\beta$ are the boundary points of the segment of $AB$ that lies in the $3^{\text{rd}}$ octant, find the value of $\sum_{n \in E} n + 9 \times (\text{sum of all coordinates of } P_\alpha \text{ and } P_\beta)$ .

Options

A.
24
B.
21
C.
27
D.
18

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

Empty and Universal Relation|Ratio of Division (k:1)|Octant Sign Convention

Why This Question Is Tricky

Common Mistakes

This question fuses 3D coordinate geometry with set theory (empty relations).

The core trap lies in the 'Empty Relation' condition, which forces students to systematically analyze the sign of the section formula coordinates as the ratio $k$ varies continuously from $0$ to $\infty$ .

Students often mistakenly assume a line segment between two points only passes through the octants containing the endpoints, ignoring the intermediate octants crossed when coordinates change signs at different rates.

Furthermore, identifying the boundary points requires recognizing that the roots of the coordinate numerators ( $k=2, 3, 4$ ) act as the exact transition thresholds between octants.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 12
Concept	Empty and Universal Relation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free