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Question #40491

Difficulty: 8/10 | 🟢 Free

The Question

Let $z_1(x)$ and $z_2(x)$ be the roots of the quadratic equation in $z$ : $z^2 - 2xz + x^2 + x^4 (-1)^{[x]} = 0$ where $[x]$ denotes the greatest integer less than or equal to $x$ . Let $f(x) = \frac{1}{2} \left( (\text{Im}(z_1(x)))^2 + (\text{Im}(z_2(x)))^2 \right)$ , where $\text{Im}(z)$ denotes the imaginary part of $z$ . Let $y(x)$ be the continuous solution to the differential equation: $x^2 y \frac{dy}{dx} - x y^2 = x^3 - f(x) \quad \text{for } x \ge 1$ If $y(1) = \sqrt{2}$ , then the value of $y(e)^2$ is:

Options

A.
$2e^2(4 - e)$
B.
$2e^2$
C.
$2e^2(3 - e)$
D.
$4e^2$

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

Quadratic Equations with Complex Coefficients|Exact Differential Recognition|Greatest Integer Function Traps

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to link the nature of roots of a complex quadratic to an exact differential equation, while navigating a Greatest Integer Function (GIF) trap.

The common mistake is assuming the discriminant magnitude directly gives the imaginary parts, leading students to integrate $x^4$ over the entire domain $[1, e]$ .

By defining $f(x)$ strictly in terms of the imaginary parts, the ODE's forcing function bifurcates: it is zero when the discriminant is positive (real roots on $[1, 2)$ ) and non-zero when negative (complex roots on $[2, e]$ ).

Failing to perform this case analysis across integer boundaries results in incorrect integration limits and selects one of the trap options.

Quick Facts About This Question

Question Information

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