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Question #41673

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all vectors $\vec{v} = l\hat{i} + m\hat{j} + n\hat{k}$ where $l, m, n$ are the direction cosines of a ray originating from the origin $O$ . The ray intersects the plane $P_1: 3x + 4y + 12z = 68$ at point $A$ and the plane $P_2: x + 2y + 2z = 6$ at point $B$ . If $OA = 5$ and $OB = 3$ , then the number of elements in $S$ is:

Options

A.
0
B.
1
C.
2
D.
4

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

Feasible Region Constancy|Line-Plane intersection logic|Direction Cosines constraint

Why This Question Is Tricky

Common Mistakes

The question tests the hidden constraint of direction cosines ( $l^2+m^2+n^2=1$ ) in 3D geometry.

Students often mechanically solve the system of linear equations formed by the plane intersections without verifying if the resulting vector is valid.

By setting the required distance such that the plane equation demands a value exceeding the Cauchy-Schwarz maximum for the direction cosines, the feasible region collapses to an empty set.

This traps candidates who bypass the fundamental existence check of the vector and attempt to find roots of a non-existent intersection.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Programming - Class 12
Concept	Feasible Region Constancy
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free