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Question #41612

Difficulty: 8/10 | 🟢 Free

The Question

Let the quadratic equations $x^2 - ax + 6 = 0$ and $x^2 - bx + 12 = 0$ have a common root $\alpha$ . Let their other roots be $\beta$ and $\gamma$ respectively. It is given that $\beta$ and $\gamma$ are positive integers such that $\beta + \gamma = 6$ .

Let $P$ be a point whose position vector with respect to the origin $O(0,0,0)$ is $\vec{p} = \alpha \hat{i} + \beta \hat{j} + \gamma \hat{k}$ . The origin is then shifted to the point $Q(2, 0, 2)$ . Let $\vec{v} = v_1 \hat{i} + v_2 \hat{j} + v_3 \hat{k}$ be the position vector of $P$ with respect to the new origin.

If a matrix $M$ is defined as $M = \begin{pmatrix} 2 & v_1 & v_2 \\ -v_1 & 2 & v_3 \\ -v_2 & -v_3 & 2 \end{pmatrix}$ , then the value of $\det(\text{adj}(M))$ is:

Options

A.
26
B.
676
C.
4356
D.
5184

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

Position Vector Origin Dependence|Adjoint-Determinant Relation|Skew-Symmetric Determinant|Condition for Common Roots

Why This Question Is Tricky

Common Mistakes

This question integrates the theory of equations, vector origin dependence, and determinant properties of skew-symmetric matrices.

The primary trap lies in the 'Origin Dependence' of position vectors.

Students often calculate the initial position vector $\vec{p}$ using the common root conditions and immediately substitute its components into the matrix $M$ , ignoring the coordinate transformation (origin shift to $Q$ ).

This leads to the highly plausible distractor 4356.

Furthermore, the matrix $M$ is structured as $\lambda I + S$ (where $S$ is skew-symmetric), testing the student's ability to bypass tedious Sarrus rule expansions by recognizing that $\det(M) = \lambda^3 + \lambda(v_1^2 + v_2^2 + v_3^2)$ .

Finally, the adjoint-determinant relation $|adj(M)| = |M|^2$ acts as the final computational lock.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Vector Algebra - Class 12
Concept	Position Vector Origin Dependence
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free