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Question #40753

Difficulty: 8/10 | 🟢 Free

The Question

Let $A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 0 & -1 \\ 2 & 1 & 0 \\ 0 & -1 & 1 \end{pmatrix}$ . Let $N = BA$ . The rows of $N$ represent the coefficients $(a_i, b_i, c_i)$ of three lines $L_i: a_i x + b_i y + c_i = 0$ for $i \in \{1, 2, 3\}$ . It is given that the lines $L_1, L_2, L_3$ are concurrent at a point $Q$ . Let $S$ be the sum of the eigenvalues of the matrix $M = AB$ . If a line $L^*$ passes through $Q$ and has a slope equal to $S$ , then the equation of $L^*$ is:

Options

A.
$6x - y - 8 = 0$
B.
$6x - y + 2 = 0$
C.
$6x + y - 4 = 0$
D.
$x - 6y - 13 = 0$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Concurrent Lines Condition|Matrix Multiplication Non-Commutativity|Matrix Trace Properties

Why This Question Is Tricky

Common Mistakes

This question tests the non-commutativity of matrix multiplication ( $AB \neq BA$ ) against the invariance of the trace ( $Tr(AB) = Tr(BA)$ ).

The trap is set by defining the slope $S$ using the matrix $M = AB$ , prompting students to compute $AB$ .

Finding $AB = \begin{pmatrix} 5 & -1 & 2 \\ 14 & -1 & 2 \\ 23 & -1 & 2 \end{pmatrix}$ , a student might incorrectly assume that the concurrency point of the lines formed by $AB$ is the same as that of $BA$ (due to shared trace/determinant invariants or assuming commutativity).

This leads them to the false concurrency point $P(0, 2)$ and the trap answer $6x - y + 2 = 0$ .

The correct path requires recognizing that while the trace is cyclic, the null spaces of $AB$ and $BA$ are distinct, necessitating the explicit computation of $BA$ to find the true concurrency point $Q(1, -2)$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Straight Lines - Class 11
Concept	Concurrent Lines Condition
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free