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Question #67392

Difficulty: 8/10 | 🟢 Free

The Question

Let $\vec{a}, \vec{b}, \vec{c}$ be three non-coplanar vectors and $\vec{d} = \vec{a} + \vec{b} + \vec{c}$ . Define $\vec{r}_1 = \vec{a} \times \vec{b}$ , $\vec{r}_2 = \vec{b} \times \vec{c}$ , and $\vec{r}_3 = \vec{c} \times \vec{a}$ . Let $M$ be a $3 \times 3$ matrix whose elements are given by $M_{ij} = [\vec{r}_i \ \vec{r}_j \ \vec{d}]$ for $1 \le i, j \le 3$ , and let $\Delta = \det(M)$ . Consider the matrix $A = \begin{pmatrix} 1 & 2 & 3 \\ \Delta & 1 & 4 \\ 0 & \Delta & 1 \end{pmatrix}$ . If $S$ is the sum of all elements of the matrix $A^{10}$ , which of the following is NOT a factor of $S + 27$ ?

Options

A.
12
B.
16
C.
32
D.
64

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

Row/Column Exchange Parity|Vector Product Anti-Commutativity|Symmetry vs. Antisymmetry

Why This Question Is Tricky

Examiner Levers

Negative Thinking

Uses "NOT", "EXCEPT", or "INCORRECT" to flip the logic gate.

Difficulty Boost : +2

Recursive Logic

The answer to part A is required to solve part B (Step-wise failure test).

Difficulty Boost : +5

Common Mistakes

This question tests the student's ability to recognize skew-symmetric structures hidden within vector operations.

The trap lies in attempting to expand the scalar triple products $M_{ij}$ algebraically, which leads to a massive, time-consuming calculation (the 'Lethal Collision').

By identifying that the anti-commutativity of the cross product ( $\vec{r}_i \times \vec{r}_j = -\vec{r}_j \times \vec{r}_i$ ) forces $M_{ji} = -M_{ij}$ , students can instantly deduce that $M$ is a $3 \times 3$ skew-symmetric matrix, making its determinant $\Delta = 0$ .

The problem then recursively uses this result in a matrix exponentiation task, employing negative thinking by asking for what is NOT a factor of the final sum.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Determinants - Class 12
Concept	Row/Column Exchange Parity
Difficulty	8
Cognitive Load	8
Target Solve Time	180 sec
Access	Free