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Question #40848

Difficulty: 8/10 | 🟢 Free

The Question

Let $A(x_1, y_1, z_1)$ , $B(x_2, y_2, z_2)$ , and $C(x_3, y_3, z_3)$ be three non-collinear points in $\mathbb{R}^3$ such that the area of $\triangle ABC$ is $3\sqrt{5}$ . Let $M = \begin{pmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{pmatrix}$ . It is given that $\text{adj}(\text{adj}(M))$ is a singular matrix. Let $\Delta_x$ and $\Delta_y$ be the areas of the orthogonal projections of $\triangle ABC$ onto the $yz$ -plane and $zx$ -plane, respectively. If $\Delta_x = 2\Delta_y$ , what is the value of $\Delta_x$ ?

Options

A.
6
B.
$2\sqrt{5}$
C.
$\sqrt{30}$
D.
$\sqrt{15}$

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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

Projection vs. Coordinate|Area of Triangle via Determinants|Singular vs Non-Singular Matrices

Why This Question Is Tricky

Common Mistakes

This question tests the geometric interpretation of determinants and the transition from 3D coordinates to 2D projections.

The 'Lethal Collision' occurs when students fail to connect the singularity of the adjoint matrix to the vanishing area of the 2D projection.

By stating that adj(adj(M)) is singular, it implicitly forces det(M) = 0, which geometrically means the projection of the triangle onto the xy-plane collapses into a line (collinear vertices), yielding an area of zero.

Students who ignore this non-singular constraint will find the system underdetermined or incorrectly assume a symmetric non-zero area for the xy-projection, missing the collinearity trap entirely.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Introduction to Three-dimensional Geometry - Class 11
Concept	Projection vs. Coordinate
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free