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Question #68630

Difficulty: 8/10 | 🟢 Free

The Question

A curve $y = f(x)$ passes through the point $(1, 0)$ . For all $x \ge 1$ , the slope of the normal to the curve at any point $(x, y)$ is given by $-\left( \int_0^x [t] dt \right)^{-1}$ , where $[ \cdot ]$ denotes the greatest integer function. Let $L$ be the tangent to the curve at $x = \frac{5}{2}$ . If $L$ intersects the x-axis at $A$ and the y-axis at $B$ , find the length of the line segment $AB$ .

Options

A.
$\frac{15\sqrt{5}}{8}$
B.
$\frac{25\sqrt{5}}{8}$
C.
$\frac{23\sqrt{5}}{8}$
D.
$\frac{17\sqrt{5}}{8}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Integration Constant Trap|Slope Sensitivity|Tangent/Normal Slope Interplay|Greatest Integer Function (GIF)

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

This question tests the student's ability to handle differential equations involving piecewise continuous functions, specifically the Greatest Integer Function (GIF).

The core trap lies in the integration constant.

Students often incorrectly assume that a single global integration constant applies to the entire domain, or they mistakenly use the initial condition $f(1)=0$ directly in the interval $[2, 3)$ to find $C_2=0$ .

To succeed, the student must recognize that integrating a piecewise derivative requires patching the solutions together using the continuity of the curve $f(x_n^+) = f(x_n^-)$ at each jump discontinuity of the GIF.

The final geometric calculation of the tangent's intercepts acts as a step-wise failure test, heavily penalizing any error in the local integration constant.

Quick Facts About This Question

Question Information

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