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Question #40530

Difficulty: 8/10 | 🟢 Free

The Question

Let $S = \{-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5\}$ . A subset $\{a, b, c\}$ of three distinct elements is chosen from $S$ . Find the number of such subsets for which the lines $L_a, L_b,$ and $L_c$ are concurrent, where the family of lines is given by $L_m : x + \frac{2}{m}y + m^2 - m = 0$

Options

A.
8
B.
10
C.
12
D.
14

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

Denominator Restriction|Concurrent Lines Condition|Collinearity vs Coplanarity|Polar and Pole Duality

Why This Question Is Tricky

Common Mistakes

This question tests the 'Denominator Restriction' trap within the context of concurrent lines.

Students typically set up the determinant for concurrency and correctly derive the condition $a+b+c=1$ .

However, they often mechanically count all subsets summing to 1, including those containing 0 (which gives 12 subsets).

They fail to recognize that the original line equation $L_m$ contains $m$ in the denominator, making $m=0$ geometrically undefined.

Enforcing $m \neq 0$ eliminates 4 extraneous subsets, leaving the correct count of 8.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Inequalities - Class 11
Concept	Denominator Restriction
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free