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Question #41756

Difficulty: 8/10 | 🟢 Free

The Question

Let $a, b, c$ be three distinct positive real numbers which are the roots of the equation $t^3 - pt^2 + qt - r = 0$ . A variable plane $\Pi$ in 3D space is defined by the determinant equation: $\begin{vmatrix} x & y & z & 1 \\ a & a^2 & a^3 & 1 \\ b & b^2 & b^3 & 1 \\ c & c^2 & c^3 & 1 \end{vmatrix} = 0$ If the plane $\Pi$ passes through the point $(1, 1, 1)$ and the sum of the roots of the cubic equation is $6$ , then the value of $q - r$ is:

Options

A.
4
B.
5
C.
6
D.
7

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

Variable plane intercepts|Adjoint-Determinant Relationship|Factor Theorem in Determinants

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to recognize and apply the Factor Theorem to large determinants (Trap 241) instead of falling into the trap of brute-force expansion.

The 'Lethal Collision' occurs when the variable plane's determinant equation is evaluated.

Students who attempt to expand the $4 \times 4$ determinant will face severe algebraic exhaustion and likely make sign errors.

The advanced paradigm requires treating the $3 \times 3$ sub-determinants as alternating symmetric polynomials (Vandermonde determinants) of the roots $a, b, c$ .

By factoring out the common difference product $\Delta = (a-b)(b-c)(c-a)$ , the plane equation elegantly collapses to $qx - py + z = r$ .

The condition of passing through a fixed point then directly yields the required relation without ever needing the individual root values.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Three-Dimensional Geometry - Class 12
Concept	Variable plane intercepts
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free