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Question #72209

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \int_0^x ([t] + 25) dt$ , where $[ \cdot ]$ denotes the greatest integer function. Let $L$ be the left-hand derivative of $f(x)$ at $x = 2$ . A continuous function $y(x)$ defined on $[1, 4]$ satisfies the differential equation $f(x) \frac{dy}{dx} + ([x] + 25) y = 161 x^2$ for all non-integer $x \in (1, 4)$ . If $y(1) = L - 26$ , then the value of $y(3)$ is:

Options

A.
$\frac{161}{9}$
B.
$\frac{4261}{234}$
C.
$\frac{4111}{234}$
D.
$\frac{483}{26}$

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

Exact Differential Recognition|Newton-Leibniz Formula with variable limits|Greatest Integer Function (GIF)|L'Hopital's Pitfall

Why This Question Is Tricky

Examiner Levers

Recursive Logic

The answer to part A is required to solve part B (Step-wise failure test).

Difficulty Boost : +5

Common Mistakes

The problem tests the evaluation of the derivative of an integral with a discontinuous integrand (GIF) and exact differential equations.

The primary trap lies in finding the left-hand derivative $L$ .

Students often blindly apply the Fundamental Theorem of Calculus, assuming $f'(2) = [2] + 25 = 27$ , failing to realize that for the left-hand derivative, the integrand must be evaluated as $x \to 2^-$ , yielding $1 + 25 = 26$ .

This incorrect $L$ cascades into the initial condition.

Furthermore, students must recognize the exact differential $d(y f(x))$ and integrate across integer discontinuities, which is valid since $y(x)f(x)$ is continuous.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Differential Equations - Class 12
Concept	Exact Differential Recognition
Difficulty	8
Cognitive Load	7
Target Solve Time	170 sec
Access	Free