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Question #40290

Difficulty: 8/10 | 🟢 Free

The Question

Let $\vec{u} = (k^3 + k^2 - 4k - 4)\hat{i} + (k^3 - 2k^2 - 4k + 8)\hat{j} + (k^3 - 3k^2 - 4k + 12)\hat{k}$ and $\vec{v} = 2\hat{i} - \hat{j} - 2\hat{k}$ . Let $S$ be the set of all real values of $k$ for which the vector $\vec{w} = \frac{\vec{u}}{|\vec{u}|} + \frac{\vec{v}}{|\vec{v}|}$ does NOT represent a valid direction for the internal angle bisector of $\vec{u}$ and $\vec{v}$ . Find the sum of the squares of the elements of $S$ .

Options

A.
5
B.
9
C.
10
D.
14

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

Well-definedness|Linear Independence vs Dependence|Angle Bisector Direction

Why This Question Is Tricky

Common Mistakes

This question tests the well-definedness of the standard angle bisector formula $\vec{w} = \hat{u} + \hat{v}$ .

Students often blindly apply this formula without considering the boundary constraints.

The formula fails to provide a valid direction in two degenerate cases: (1) when either vector is the null vector (causing division by zero), and (2) when the vectors are anti-parallel, meaning $\hat{u} = -\hat{v}$ , which results in $\vec{w} = \vec{0}$ (a vector with no defined direction).

By forcing students to check for linear dependence and zero-magnitude conditions, this problem acts as a trap for those who ignore the geometric and algebraic constraints of vector directions.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sets - Class 11
Concept	Well-definedness
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free