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Question #40330

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all differentiable functions $f: (1, \infty) \to (1, \infty)$ . A relation $R$ on $S$ is defined as $(f, g) \in R$ if for every homogeneous function $H(x, y)$ of degree $k > 0$ (where $H(x, y) > 1$ ), the composite function $U(x, y) = f(g(H(x, y)))$ satisfies the partial differential equation: $x \frac{\partial U}{\partial x} + y \frac{\partial U}{\partial y} = 2k U \ln U$ It is given that $(f, g) \in R$ and $(g, h) \in R$ . If $g(x) = e^{x^2}$ , $h(x) = 2x$ , and $f(e) = e^3$ , what is the value of $f(h(2))$ ?

Options

A.
64
B.
128
C.
256
D.
512

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Relation as Subset|Homogeneous Function Homogeneity|Implicit Differentiation Rate

Why This Question Is Tricky

Common Mistakes

This question tests the Implicit Differentiation Rate and Chain Rule under a homogeneity constraint.

The primary trap lies in the application of Euler's Theorem.

Students often incorrectly assume $U(x,y)$ itself is homogeneous because of the PDE's structure, or they misapply the chain rule as $f'(x)g'(x)$ instead of $f'(g(H))g'(H)H_x$ .

This leads to an incorrect differential equation, trapping them in an algebraic loop where they cannot deduce the equivalence class form $f(g(x)) = e^{Ax^2}$ .

The inclusion of $(g,h) \in R$ serves as a conceptual distractor to test if they truly understand the relation's condition rather than just pattern matching.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 11
Concept	Relation as Subset
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free