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Question #41132

Difficulty: 8/10 | 🟢 Free

The Question

Let $A = \begin{pmatrix} 2 & 2 \\ 0 & 1 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & -6 \\ 1 & 2 \end{pmatrix}$ . For any $2 \times 2$ matrix $M = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ , define the transformation $T_M(x) = \frac{ax + b}{cx + d}$ . Let $f(x) = T_A(x)$ and $g(x) = T_B(x)$ . Let $S$ be the set of all real numbers $x$ satisfying the equation $f(g(h(x))) = g(f(h(x)))$ , where $h(x) = \sqrt{|[x]|}$ and $[ \cdot ]$ denotes the greatest integer function. Find the value of $\int_{S} (3x^2 + 202x) \, dx$

Options

A.
300
B.
970
C.
122
D.
128

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

Composition Non-Commutativity|Matrix Multiplication Non-Commutativity|Greatest Integer Function Traps|Square Root and Modulus Logic

Why This Question Is Tricky

Common Mistakes

This question tests the non-commutativity of function composition by mapping it to matrix multiplication ( $T_A \circ T_B = T_{AB}$ ).

The primary trap lies in the domain analysis of the Greatest Integer Function combined with a modulus and square root: $h(x) = \sqrt{|[x]|}$ .

Students often incorrectly solve $|[x]| = 4$ as $x \in (-5, -4] \cup [4, 5)$ instead of the correct $[-4, -3) \cup [4, 5)$ , which leads to incorrect integration bounds and the trap answer 122.

Additionally, students who forget the negative branch entirely will arrive at 970.

The integral is carefully constructed with parity-breaking coefficients (202 and 3) to reward exact domain analysis and penalize algebraic shortcuts.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 12
Concept	Composition Non-Commutativity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free