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Question #40603

Difficulty: 8/10 | 🟢 Free

The Question

Two students independently select a subset of items from a common pool of $n$ distinct items. Let $k$ be the number of ways they can make their selections such that their chosen subsets have at least one item in common.

A point $P$ divides the line segment joining $A(3, -10, -2)$ and $B\left(\frac{1}{3}, \frac{2}{3}, \frac{2}{3}\right)$ internally in the ratio $k:1$ . If the coordinates of $P$ represent the direction cosines of a line, find the value of $n$ .

Options

A.
2
B.
3
C.
4
D.
5

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

Restricted Selection (At least one common)|Ratio of Division (k:1)|Direction Cosines constraint

Why This Question Is Tricky

Common Mistakes

The question fuses combinatorics with 3D geometry, specifically targeting the conceptual understanding of direction cosines.

The primary trap lies in the geometric constraint: students often attempt to algebraically expand the sum of squares of the coordinates of P in terms of k, leading to a massive, error-prone quadratic equation.

The logical bypass is recognizing that 'direction cosines' simply means the point lies on the unit sphere (|P|=1).

By parameterizing the line segment AB with a parameter t = k/(k+1) and using vector dot products, the intersection with the unit sphere is found elegantly.

Furthermore, the combinatorial setup subtly tests if students realize that subsets sharing an element are inherently non-empty, avoiding unnecessary inclusion-exclusion steps.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Permutations and Combinations - Class 11
Concept	Restricted Selection (At least one common)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free