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Question #40874

Difficulty: 8/10 | 🟢 Free

The Question

Let $A = (1, -1, 0)$ , $B = (0, 1, -1)$ , and $C = (-1, 0, 1)$ be three points in $\mathbb{R}^3$ . A fourth point $D$ is given by the coordinates $D = \left( \text{sgn}(x^2 - 2x - 3), \text{sgn}(-2x^2 + x + 1), \frac{x}{\text{sgn}(x)} \right)$ where $\text{sgn}(\cdot)$ denotes the signum function. Let $S$ be the set of all real values of $x$ for which the points $A, B, C$ , and $D$ are coplanar. Find the product of all elements in $S$ .

Options

A.
-2
B.
0
C.
2
D.
-4

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

Collinearity and Coplanarity|Singular vs. Non-Singular Threshold|Signum Function Discontinuity

Why This Question Is Tricky

Common Mistakes

The problem tests the student's ability to handle domain restrictions rigorously when dealing with piecewise and signum functions in coordinate geometry.

The coplanarity condition simplifies to an equation involving $|x|$ .

The trap is deliberately set at $x=0$ , where the limit of the coordinate exists (and satisfies the equation), but the function itself evaluates to the indeterminate form $0/0$ .

Students who blindly simplify $\frac{x}{\text{sgn}(x)}$ to $|x|$ without noting $x \neq 0$ will incorrectly include $0$ in their solution set, leading to a product of $0$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Introduction to Three-dimensional Geometry - Class 11
Concept	Collinearity and Coplanarity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free