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Question #40836

Difficulty: 8/10 | 🟢 Free

The Question

A circle $C: x^2 + y^2 - 10x - 2y = 0$ intersects the parabola $y^2 = 4x$ at the origin $O$ and three other distinct points $P, Q, R$ . Let $t_1, t_2, t_3$ be the parameters of $P, Q, R$ respectively, where the coordinates of any point on the parabola are given by $(t^2, 2t)$ . Let $M = \begin{pmatrix} t_1 & t_2 & t_3 \\ t_2 & t_3 & t_1 \\ t_3 & t_1 & t_2 \end{pmatrix}$ . If $B = \text{adj}(M - 2I)$ , where $I$ is the $3 \times 3$ identity matrix, find the value of $|B|$ .

Options

A.
784
B.
576
C.
400
D.
1024

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

Intersection of Conics|Normal to Conic|Properties of Transpose Determinant|Adjoint-Determinant Relation

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of coordinate geometry (intersection of conics) with advanced matrix determinant properties.

The primary trap is the brute-force calculation: students might attempt to explicitly solve the cubic $t^3 - 6t - 4 = 0$ (which has irrational roots) and substitute them into the $3 \times 3$ matrix, leading to a massive time-sink and high probability of algebraic error.

The elegant, intended pathway requires recognizing that the matrix $M$ is a symmetric circulant-like matrix whose characteristic polynomial can be expressed entirely in terms of the elementary symmetric polynomials of its entries.

By leveraging Vieta's formulas and the adjoint-determinant relation $|adj(A)| = |A|^{n-1}$ , the problem bypasses raw coordinate computation entirely.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Conic Sections - Class 11
Concept	Intersection of Conics
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free