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Question #41065

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of 6 points in 3D space whose position vectors are formed by all permutations of $(1, 2, 4)$ . Let $P$ be the point with position vector $\hat{i} + \hat{j} + \hat{k}$ . Let $V$ be the maximum possible volume of a tetrahedron formed by $P$ and three distinct points chosen from $S$ . If $N$ is the number of unordered triplets of points from $S$ that form a tetrahedron with $P$ of volume $V$ , find the value of $3V + N$ .

Options

A.
16
B.
18
C.
26
D.
14

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

Combinatorial Counting (P&C Integration)|Scalar Triple Product Volume Logic|Symmetric vs Non-symmetric form|Position Vector Origin Dependence

Why This Question Is Tricky

Common Mistakes

This question tests the 'Position Vector Origin Dependence' trap combined with combinatorial geometry.

Students often incorrectly assume that the maximum volume of a tetrahedron formed by points from a fixed set is invariant under shifting the reference point $P$ .

However, the set of available difference vectors changes.

By shifting from the origin to $P(1,1,1)$ , the maximum determinant drops from 49 to 28.

Failing to recalculate the scalar triple product with the shifted vectors leads to incorrect volume maximization, while the combinatorial aspect forces students to systematically classify the determinants of permutations.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 11
Concept	Combinatorial Counting (P&C Integration)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free