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Question #41288

Difficulty: 8/10 | 🟢 Free

The Question

Let $P$ be the set of all polynomials $p(x) = ax^2 + bx + c$ of degree exactly 2 ( $a \neq 0$ ) with integer coefficients. For each $p \in P$ , define a $3 \times 3$ matrix $A_p$ derived from function composition as: $A_p = \begin{pmatrix} p(1) & p(p(1)) & p(p(p(1))) \\ 0 & p(2) & p(p(2)) \\ 0 & 0 & p(3) \end{pmatrix}$ Let $T_p : \mathbb{Z}^3 \to \mathbb{Z}^3$ be the linear transformation defined by $T_p(X) = A_p X$ . If $S$ is the subset of $P$ containing all polynomials for which $T_p$ is a bijective mapping, find the value of $\sum_{p \in S} (a^2 + b^2 + c^2)$ .

Options

A.
358
B.
179
C.
716
D.
234

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

Triangular/Diagonal Matrix Shortcut|Singular vs. Non-Singular Threshold|Bijectivity and Inverse|Composite Injectivity/Surjectivity

Why This Question Is Tricky

Common Mistakes

This question tests the threshold between injectivity and surjectivity for linear transformations over restricted domains (like $\mathbb{Z}^3$ ).

The matrix is upper-triangular, allowing an immediate determinant shortcut ( $\det(A) = p(1)p(2)p(3)$ ).

The core trap: Students mechanically equate 'Non-Singular' ( $\det \neq 0$ ) with 'Bijective', forgetting that over the integers, a transformation is only surjective if its inverse mapping space is also defined over integers.

This requires the determinant to be exactly $\pm 1$ .

Failing to recognize this leads to an infinite number of solutions instead of the constrained finite set.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Determinants - Class 12
Concept	Triangular/Diagonal Matrix Shortcut
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free