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Question #41414

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(t) = \pi \sin(\pi t)$ , $g(t) = e^t$ , and $h(t) = 12t^2$ . Define the function $\Delta(x)$ as the determinant: $\Delta(x) = \begin{vmatrix} \int_0^x f(t) dt & \int_0^{x^2} g(t) dt & \int_0^{x^3} h(t) dt \\ x & x^2 & x^3 \\ 1 & 2x & 3x^2 \end{vmatrix}$ Evaluate the integral $I = \int_0^1 x \Delta''(x) dx$ .

Options

A.
$50 - 8e$
B.
$26 - 6e$
C.
$58 - 10e$
D.
$34 - 8e$

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

Integration by Parts (LIATE Rule)|Newton-Leibniz Formula with variable limits|Differentiation of Determinants

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of Integration by Parts (LIATE) and the Newton-Leibniz formula applied to a determinant.

The primary trap is the 'Lethal Collision' where students either fail to apply the chain rule when differentiating the variable upper limits of the integrals (e.g., differentiating $\int_0^{x^2} g(t) dt$ as $g(x^2)$ instead of $2x g(x^2)$ ), or they incorrectly execute the row-by-row differentiation of the determinant.

Forgetting the chain rule leads directly to the distractor $26 - 6e$ .

Additionally, the problem requires recognizing that the second determinant in the row-by-row expansion vanishes identically, rewarding structural observation over brute-force expansion.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Integrals - Class 12
Concept	Integration by Parts (LIATE Rule)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free