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Question #40741

Difficulty: 8/10 | 🟢 Free

The Question

Let $A = \begin{pmatrix} 2 & 1 & 0 \\ 1 & 3 & 0 \\ 0 & 0 & 1 \end{pmatrix}$ and $B = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}$ . A point $P(x, y)$ in the Cartesian plane is represented by the column vector $P = \begin{pmatrix} x \\ y \\ 1 \end{pmatrix}$ . A line $L$ with equation $ax + by + c = 0$ is represented by the column vector $V = \begin{pmatrix} a \\ b \\ c \end{pmatrix}$ such that $V^T P = 0$ . Under the transformation $P' = M P$ , where $M = (AB)^T$ , the line $L: 3x - 4y + 5 = 0$ is transformed to a new line $L': a'x + b'y + c' = 0$ . If $\gcd(|a'|, |b'|, |c'|) = 1$ and $a' > 0$ , find the value of $a' + b' + c'$ .

Options

A.
$-27$
B.
$-4$
C.
$2$
D.
$-1$

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

General Equation Transformations|Minors and Cofactors Sign Convention|Transpose of Product

Why This Question Is Tricky

Common Mistakes

This question tests the student's understanding of general equation transformations using matrices and the properties of the transpose and inverse of matrix products.

The primary trap lies in the transformation of the line vector: students often incorrectly assume $V' = M V$ or $V' = M^{-1} V$ , failing to apply the covariant transformation rule $V' = (M^{-1})^T V$ .

Furthermore, the expression $(M^{-1})^T$ simplifies elegantly to $(AB)^{-1}$ due to the property $((AB)^T)^{-1} = ((AB)^{-1})^T$ .

A secondary trap involves the incorrect expansion of $(AB)^{-1}$ as $A^{-1} B^{-1}$ instead of $B^{-1} A^{-1}$ , which leads to the incorrect sum $-4$ .

The inclusion of the reflection matrix $B$ also tests the correct application of cofactor sign conventions when computing the inverse.

Quick Facts About This Question

Question Information

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