Before You Begin

Question #41295

Difficulty: 8/10 | 🟢 Free

The Question

Let $\vec{a} = \hat{i} + x\hat{j} + x^2\hat{k}$ , $\vec{b} = \hat{i} + |x|\hat{j} + |x|^2\hat{k}$ , and $\vec{c} = \hat{i} + [x]\hat{j} + [x]^2\hat{k}$ , where $[x]$ denotes the greatest integer less than or equal to $x$ , and $\{x\}$ denotes the fractional part of $x$ . Let $f(x) = [\vec{a} \times \vec{b} \ \ \vec{b} \times \vec{c} \ \ \vec{c} \times \vec{a}]$ . Find the number of non-integer solutions to the equation $f(x) = 4x^2 (x + 2)^2 \{x\}^2$ .

Options

A.
0
B.
1
C.
2
D.
3

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

Factor Theorem in Determinants|Vector Triple Product Expansion|Modulus Function Symmetry|Integral Part and Fractional Part

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to handle piecewise functions within determinant properties.

The primary trap is blindly expanding the Vandermonde determinant as a polynomial and canceling terms without considering the piecewise nature of $|x|$ .

For $x \ge 0$ , the determinant is identically zero, but for $x < 0$ , $|x| = -x$ fundamentally changes the factors, breaking the standard cyclic symmetry.

Furthermore, students often fail to restrict the fractional part $\{x\} \in (0, 1)$ , which leads to extraneous roots when solving the resulting linear Diophantine-like equations for $[x]$ and $\{x\}$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Determinants - Class 12
Concept	Factor Theorem in Determinants
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free