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Question #40752

Difficulty: 8/10 | 🟢 Free

The Question

Consider three lines in the $X-Y$ plane parameterized by a real number $x$ : $L_1: X - Y = x - \{x\}(1-\{x\})$ $L_2: X + Y = x + \{x\}(1-\{x\})$ $L_3: X - 2Y = [x]$ where $\{x\}$ and $[x]$ denote the fractional part and greatest integer function, respectively. Let $C$ be the locus of the intersection point of $L_1$ and $L_2$ as $x$ varies over $\mathbb{R}$ . Let $S$ be the set of all $x \in [0, 20]$ such that $L_1, L_2$ , and $L_3$ are concurrent at a point $P$ , and the curve $C$ (expressed as $Y = f(X)$ ) is differentiable at $P$ . Find the sum of all elements in $S$ .

Options

A.
200
B.
410
C.
210
D.
400

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

Concurrent Lines Condition|Integral Part and Fractional Part|Differentiability vs Continuity

Why This Question Is Tricky

Common Mistakes

The question tests the intersection of concurrency conditions with the calculus of non-differentiable functions.

Students typically solve the algebraic concurrency condition to find {x} = 0 and {x} = 1/2, yielding both integer and half-integer solutions.

The trap lies in ignoring the geometric constraint: the locus of the intersection point Y = {X}(1-{X}) forms a sequence of parabolic arcs that are continuous but have sharp corners (non-differentiable) at integer values of X.

By failing to reject the integer solutions where the derivative is undefined, students will incorrectly sum both sets of roots, falling for the classic over-counting trap (yielding 410 instead of 200).

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Straight Lines - Class 11
Concept	Concurrent Lines Condition
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free