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Question #40666

Difficulty: 8/10 | 🟢 Free

The Question

Let $\vec{a}, \vec{b}, \vec{c}$ be three vectors in $\mathbb{R}^3$ such that $|\vec{a}| + \sqrt{2}|\vec{b}| + \sqrt{3}|\vec{c}| = 9$ . If $V$ represents the volume of the parallelepiped formed by the vectors $\vec{a}, \vec{b}, \vec{c}$ as coterminous edges, find the maximum possible value of $[V]$ , where $[x]$ denotes the greatest integer less than or equal to $x$ .

Options

A.
10
B.
11
C.
12
D.
13

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

Arithmetic Mean Inequality|Scalar Triple Product Volume Logic|Greatest Integer Function Traps

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of vector algebra (scalar triple product) and algebraic inequalities (AM-GM) while embedding a precision trap via the Greatest Integer Function.

Students must recognize that volume is maximized when vectors are orthogonal, reducing the problem to maximizing a product given a sum.

The critical trap lies in evaluating the irrational maximum volume $9\sqrt{6}/2$ .

A premature or rough approximation (e.g., taking $\sqrt{6} \approx 2.4$ ) yields $10.8$ , leading to an incorrect GIF of 10.

Squaring the irrational expression to exactly $121.5$ rigorously proves the value lies strictly between 11 and 12, punishing students who lack numerical discipline and rely on loose decimal estimations.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sequences and Series - Class 11
Concept	Arithmetic Mean Inequality
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free