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Question #69400

Difficulty: 8/10 | 🟢 Free

The Question

Let $A = \begin{bmatrix} t & 1 & 1 \\ 1 & t & 1 \\ 1 & 1 & t \end{bmatrix}$ for $t \in \mathbb{R} \setminus \{0, 1, -2\}$ . Let $C$ be the cofactor matrix of $A$ . Consider the region $\mathcal{R}$ in the $xy$ -plane bounded by the three lines represented by the rows of the matrix equation $C \begin{bmatrix} x \\ y \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ . Let $S$ be the set of all real values of $t$ for which the area of the region $\mathcal{R}$ is exactly $2$ . Find the sum of all elements in the set $S$ .

Options

A.
-1
B.
0
C.
2
D.
4

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

Bounded vs Unbounded Regions|Area of Triangle via Determinants|Minors and Cofactors Sign Convention

Why This Question Is Tricky

Examiner Levers

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 geometric interpretation of cofactor matrices and the absolute value constraint in determinant-based area formulas.

Students must recognize that the intersection of lines defined by the rows of a cofactor matrix corresponds to the projective coordinates given by the rows of the original matrix.

The primary trap lies in solving the area equation $\Delta = 2$ : students who forget the absolute value will only solve $(t-1)^2(t+2) = 4t$ , yielding a sum of roots equal to 0.

By enforcing the absolute value $|t+2|/|t|$ , a second case emerges for $t \in (-2, 0)$ , revealing the hidden root $t=-1$ and shifting the correct sum to -1.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Integrals - Class 12
Concept	Bounded vs Unbounded Regions
Difficulty	8
Cognitive Load	7
Target Solve Time	170 sec
Access	Free