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Question #40692

Difficulty: 8/10 | 🟢 Free

The Question

Let $T_n$ be a sequence of distinct real numbers in a Geometric Progression. Let $T_1 = 1$ and $r$ be its common ratio. Define the determinant $\Delta(r) = \begin{vmatrix} 1 & 1 & 1 \\ T_1+T_2 & T_2+T_3 & T_3+T_4 \\ T_1^2+T_2^2 & T_2^2+T_3^2 & T_3^2+T_4^2 \end{vmatrix}$ If $\Delta(r) \ge 0$ , which of the following represents the complete set of valid values for $r$ ?

Options

A.
$(-\infty, -1) \cup (-1, 0) \cup (1, \infty)$
B.
$(-1, 0) \cup (1, \infty)$
C.
$(-\infty, 0) \cup (1, \infty)$
D.
$(-\infty, 0] \cup [1, \infty)$

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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

Selection of GP Terms|Factor Theorem in Determinants|Wavy Curve (Method of Intervals)|Polynomial Function Parity

Why This Question Is Tricky

Common Mistakes

This question tests the application of the Factor Theorem in determinants and the Wavy Curve method, specifically targeting two lethal traps.

First, the determinant simplifies to a polynomial with an even power factor $(r+1)^2$ .

Students frequently make sign-inversion errors (Secondary Trap 21) by alternating signs across all roots, incorrectly assuming odd parity for the root $r=-1$ , which leads to the erroneous exclusion of the interval $(-\infty, -1)$ .

Second, the problem requires the geometric progression to have distinct terms, acting as a hidden domain restriction that forces the exclusion of $r \in \{0, 1, -1\}$ .

Forgetting this degeneracy constraint leads to incorrect inclusion of these boundary points.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sequences and Series - Class 11
Concept	Selection of GP Terms
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free