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Question #40852

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all real numbers $x$ such that the perpendicular distance from the point $P([x], [x]+|x|, |x|-[x])$ to the plane $2X - Y + 2Z - 202 = 0$ is exactly $22$ . Find the sum of the absolute values of all elements in $S$ . (Here $[t]$ denotes the greatest integer less than or equal to $t$ , and $|t|$ denotes the absolute value of $t$ )

Options

A.
202
B.
136
C.
268
D.
0

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

Distance Formula in 3D|Distance from point to plane|Modulus Sign Dependency|Greatest Integer Function Traps

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to handle piecewise functions involving both the Greatest Integer Function (GIF) and absolute values within a 3D coordinate geometry context.

The primary trap lies in the modulus sign dependency: students frequently and incorrectly assume that $|x| - [x] = \{x\}$ for all $x \in \mathbb{R}$ .

If they make this assumption, they will conclude that the expression is bounded in $[0, 1)$ , leading to zero solutions.

The problem forces a rigorous piecewise analysis, revealing that for $x < 0$ , the expression $|x| - [x]$ can take large integer values, specifically $2I$ when $x$ is a negative integer.

The constants 66 and 202 are carefully chosen to obfuscate the integer solutions and reward students who correctly evaluate the boundary conditions of the GIF.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Introduction to Three-dimensional Geometry - Class 11
Concept	Distance Formula in 3D
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free