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Question #41467

Difficulty: 8/10 | 🟢 Free

The Question

Let $A(x) = \begin{pmatrix} 2x & \sin(\pi x) \\ \cos(\pi x) & 3x^2 [x] \end{pmatrix}$ , where $[x]$ denotes the greatest integer less than or equal to $x$ . Let $f: (0, \infty) \to \mathbb{R}$ be a continuous function, differentiable for all non-integer $x > 0$ , satisfying $x f'(x) + f(x) = \text{Tr}(A(x))$ for all non-integer $x > 0$ . If $f(1) = 1$ , then the value of $10 f\left(\frac{5}{2}\right)$ is:

Options

A.
114
B.
146
C.
110
D.
142

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

Integrating f(x) + xf'(x) forms|Matrix Trace Properties|Greatest Integer Function Traps

Why This Question Is Tricky

Common Mistakes

The problem tests the recognition of the reverse product rule $\frac{d}{dx}(xf(x)) = xf'(x) + f(x)$ disguised within a matrix trace.

The 'Lethal Collision' is the integration of a term involving the Greatest Integer Function, $3x^2[x]$ .

Students who blindly apply the Fundamental Theorem of Calculus might incorrectly assume $\int 3x^2[x] dx = x^3[x]$ and evaluate it directly from 1 to 5/2, ignoring the jump discontinuity at $x=2$ .

This traps them into missing the constant of integration shift that occurs when partitioning the integral over intervals where $[x]$ is constant, leading to the incorrect option 146.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Integrals - Class 12
Concept	Integrating f(x) + xf'(x) forms
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free