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Question #41617

Difficulty: 8/10 | 🟢 Free

The Question

Let $A = \begin{pmatrix} 1 & 2 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 0 & 1 \\ 1 & 1 & 0 \\ 0 & 2 & 1 \end{pmatrix}$ . Let $M = \text{adj}(AB)$ . Consider the system of linear equations $M X = \begin{pmatrix} 1 \\ -1 \\ 2 \end{pmatrix}$ , where $X = \begin{pmatrix} x \\ y \\ z \end{pmatrix}$ . Let $P_1$ and $P_2$ be the planes represented by the first and second equations of this system, respectively. If $L$ is the line of intersection of the planes $P_1$ and $P_2$ , and $L$ intersects the plane $2x - 3y + z = 3$ at the point $Q(\alpha, \beta, \gamma)$ , then the value of $9(\alpha + \beta + \gamma)$ is:

Options

A.
78
B.
52
C.
64
D.
86

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

Line-Plane intersection logic|System of Linear Equations (Cramer/Inverse)|Minors and Cofactors Sign Convention|Matrix Multiplication Non-Commutativity

Why This Question Is Tricky

Common Mistakes

This question tests the geometric interpretation of systems of linear equations and properties of the adjugate matrix.

The primary trap targets the non-commutative property of matrix multiplication.

Students often incorrectly assume $\text{adj}(AB) = \text{adj}(A)\text{adj}(B)$ instead of the correct reversal property $\text{adj}(B)\text{adj}(A)$ .

If they use the incorrect order, they effectively solve the system for $BA$ instead of $AB$ , leading to a completely different line of intersection and the trap answer 52.

Additionally, the problem rewards deep conceptual understanding: recognizing that the cross product of the first two rows of $\text{adj}(K)$ is proportional to the third column of $K$ , and that the intersection point can be found via $X = \frac{1}{|K|} K C$ , bypassing tedious cofactor calculations.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Three-Dimensional Geometry - Class 12
Concept	Line-Plane intersection logic
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free