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Question #41560

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x, y)$ be a polynomial function satisfying $f(tx, ty) = t^5 f(x, y)$ for all $t, x, y \in \mathbb{R}$ . Given $f(1, y) = 1 - 5y^2 + 4y^4$ , evaluate the integral $I = \int_{-3}^{3} \left( |f(x, 1)| + f(x, 1) \right) dx$

Options

A.
$\frac{171}{2}$
B.
$\frac{171}{4}$
C.
$\frac{153}{2}$
D.
$\frac{153}{4}$

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

The Modulus Trap|Homogeneous Function Homogeneity|Even and Odd Function Symmetry|Wavy Curve (Method of Intervals)

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of homogeneous functions, even/odd symmetry, and the modulus wavy curve trap.

Students must first decode the homogeneous property to extract the single-variable polynomial.

The primary trap lies in the modulus integration over a symmetric interval.

A common mistake is failing to apply the Wavy Curve method to partition the integral at the roots $x=1$ and $x=2$ .

If they blindly integrate without toggling the modulus sign based on the intervals $[0,1]$ , $[1,2]$ , and $[2,3]$ , they will incorrectly compute the area (yielding the trap answer 153/2).

Furthermore, forgetting that the modulus of an odd function is even leads to missing the multiplier of 2, yielding another trap answer.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Integrals - Class 12
Concept	The Modulus Trap
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free