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Question #40294

Difficulty: 8/10 | 🟢 Free

The Question

Let $\vec{a}, \vec{b}, \vec{c}$ be three mutually orthogonal unit vectors in $\mathbb{R}^3$ . Let $S$ be the set of all vectors $\vec{r}$ satisfying the following conditions: (i) $[\vec{r}-\vec{a} \quad \vec{a}-\vec{b} \quad \vec{b}-\vec{c}] = 0$ (ii) $|\vec{r} - \vec{a}|^2 + |\vec{r} - \vec{b}|^2 + |\vec{r} - \vec{c}|^2 = \frac{9}{5}$ Find the number of elements in the set $S$ .

Options

A.
0
B.
1
C.
2
D.
$\infty$

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

The Empty Set Trap|Scalar Triple Product Volume Logic|Geometrical Interpretation of Modulus|Locus via Distance Constraints

Why This Question Is Tricky

Common Mistakes

This question deploys the 'Empty Set Trap' via an 'Algebraic-Geometric Duality'.

Students must first decode a scalar triple product to find that the locus of $\vec{r}$ is a plane ( $x+y+z=1$ ).

The second condition represents a sphere.

The trap triggers when students blindly solve the algebraic equations without checking the geometric consistency (the minimum distance from the origin to the plane).

By setting the sphere's radius such that it falls short of the plane's perpendicular distance from the origin ( $4/15 < 1/3$ ), the intersection is strictly empty.

Students who assume a valid intersection will likely guess 1, 2, or infinite, failing to recognize the null locus.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sets - Class 11
Concept	The Empty Set Trap
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free