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Question #40460

Difficulty: 8/10 | 🟢 Free

The Question

Let $z = \frac{1+i\sqrt{3}}{2}$ . A real-valued differentiable function $f(x)$ is defined on $(0, 2) \cup (2, \infty)$ and satisfies the differential equation $x f'(x) + f(x) = \text{Re} \left( \frac{4x^3}{x^4 + 16 z^3} \right)$ for all $x \in (0, 2) \cup (2, \infty)$ . It is given that $\lim_{x \to \infty} (x f(x) - 4\ln x) = -\ln 2$ and $f(1) = \ln\left(\frac{15}{2}\right)$ . Find the sum of the $4^{\text{th}}$ powers of all distinct real roots of the equation $f(x) = 0$ .

Options

A.
32
B.
18
C.
64
D.
36

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

Algebraic Manipulation of i|Integrating f(x) + xf'(x) forms|Logarithmic Integration Modulus|Chain Rule Inner Function Differentiability

Why This Question Is Tricky

Common Mistakes

This question tests the 'Logarithmic Integration Modulus' trap combined with piecewise constant evaluation.

Students often integrate $g'(x)/g(x)$ as $\ln(g(x))$ without the absolute value.

If they omit the modulus here, they will incorrectly assume $x f(x) = \ln(x^4 - 16) - \ln 2$ for the entire domain.

When solving $f(x) = 0$ , this flawed function only yields the root $x^4 = 18$ , leading them to the trap answer of 18.

The correct approach requires acknowledging the modulus $\ln|x^4 - 16|$ and independently verifying the constants of integration for the disconnected intervals $(0, 2)$ and $(2, \infty)$ , which reveals a second valid root $x^4 = 14$ hidden in the negative branch of the argument.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Complex Numbers and Quadratic Equations - Class 11
Concept	Algebraic Manipulation of i
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free