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Question #40697

Difficulty: 8/10 | 🟢 Free

The Question

Let $a, b, c$ be distinct positive real numbers in Harmonic Progression. Consider the system of linear equations in variables $x, y, z$ : $a x + b y + c z = 0$ $(a^2+ac) x + k y + (c^2+ac) z = 0$ $f(a) x + f(b) y + f(c) z = 0$ If this system has a non-trivial solution for every strictly increasing function $f: \mathbb{R}^+ \to \mathbb{R}$ , then the value of $k$ is:

Options

A.
$2ac$
B.
$ac$
C.
$a^2+c^2$
D.
$\frac{2ac}{a+c}$

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

Harmonic Mean Constraints|Linear Dependency of Rows|Singular vs. Non-Singular Threshold

Why This Question Is Tricky

Common Mistakes

The problem tests the intersection of linear dependency and sequence constraints.

The trap lies in students attempting to evaluate the determinant for a specific function or getting bogged down in algebraic expansion.

The presence of an arbitrary strictly increasing function $f$ forces the realization that the first two rows must be linearly dependent.

Even after deducing $k = b(a+c)$ , students must apply the Harmonic Mean constraint to find the final value.

It punishes brute-force calculation and rewards structural observation of row proportionality.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sequences and Series - Class 11
Concept	Harmonic Mean Constraints
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free