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Question #41454

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \det \begin{pmatrix} x^2+1 & x & x^2 \\ x^3+2x & x^2+1 & x^3+x \\ x & 1 & e^{x^2} \end{pmatrix}$ . If $I = \int_0^1 f(x) dx$ , then which of the following inequalities is true?

Options

A.
$\frac{5}{6} \le I \le \frac{2e+1}{6}$
B.
$\frac{3e-5}{6} \le I \le \frac{e}{2}$
C.
$\frac{4}{3} \le I \le \frac{e+2}{3}$
D.
$\frac{2}{3} \le I \le \frac{2e-1}{6}$

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

Inequalities in Definite Integrals|Determinant as Geometric Area/Volume|Elementary Row Operations

Why This Question Is Tricky

Common Mistakes

The core trap lies in the misuse of elementary row operations.

Students often perform operations like $R_1 \to x R_1 - R_2$ to create zeros, forgetting that multiplying a row by a variable $x$ scales the entire determinant by $x$ .

This transforms the integrand from $f(x)$ to $x f(x)$ , fundamentally altering the integral's value and leading to incorrect bounds (Option B).

Furthermore, the problem tests the rigorous application of convexity-based integral inequalities ( $e^{x^2} \le 1 + (e-1)x^2$ ) to bound non-integrable functions, a hallmark of JEE Advanced calculus.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Integrals - Class 12
Concept	Inequalities in Definite Integrals
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free