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Question #41108

Difficulty: 8/10 | 🟢 Free

The Question

Let $M(x) = \begin{pmatrix} [x] & [x+\frac{1}{3}] & [x+\frac{2}{3}] \\ [x+\frac{1}{3}] & [x+\frac{2}{3}] & [x+1] \\ [x+\frac{2}{3}] & [x+1] & [x+\frac{4}{3}] \end{pmatrix}$ , where $[t]$ denotes the greatest integer less than or equal to $t$ . Let $S = \{ M(x) \mid x \in [0, 10] \}$ . A relation $R$ is defined on $S$ such that $M(x) R M(y) \iff \det(M(x)) = \det(M(y))$ . Let $N$ be the number of distinct equivalence classes of $R$ , and $I = \int_{0}^{10} |\det(M(x))| \, dx$ . Find the value of $N + 3I$ .

Options

A.
166
B.
165
C.
155
D.
176

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

Transitivity of Identity|Linear Dependency of Rows|Greatest Integer Function Traps

Why This Question Is Tricky

Common Mistakes

This problem tests the intersection of equivalence relations, matrix determinants, and step functions.

The primary trap lies in the apparent linear dependency of the matrix rows.

Students often attempt to apply continuous row operations (like $R_3 - 2R_2 + R_1$ ) and incorrectly conclude the determinant is zero.

However, the Greatest Integer Function introduces jump discontinuities at fractional boundaries ( $x \equiv 1/3, 2/3 \pmod{1}$ ), which breaks the linear dependency and forces the determinant to collapse into a discrete step function $\det(M(x)) = -[x+2/3]$ .

The equivalence classes are then determined by the invariant intervals of this step function, requiring careful integration over the discontinuous boundaries.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 12
Concept	Transitivity of Identity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free