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Question #41257

Difficulty: 8/10 | 🟢 Free

The Question

Let $\Delta(x) = \begin{vmatrix} \cos x & x^2 & 1 \\ x^3 & x & x \\ [x] & [-x] & [x]+[-x] \end{vmatrix}$ , where $[t]$ denotes the greatest integer less than or equal to $t$ . Evaluate the integral $I = \int_{-\pi/2}^{\pi/2} \Delta(x) \, dx$

Options

A.
$\frac{3\pi}{2} - \frac{3\pi^2}{8} - 2(\sin 1 + \cos 1)$
B.
$\frac{\pi}{2} - \frac{\pi^2}{8} - 1$
C.
$\frac{3\pi}{2} - \frac{\pi^2}{8} - 2\sin 1$
D.
$\frac{\pi}{2} - \frac{3\pi^2}{8} - 2\cos 1$

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

Row/Column Exchange Parity|Even and Odd Function Symmetry|Greatest Integer Function Traps

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of determinant parity, definite integral symmetry, and Greatest Integer Function (GIF) discontinuities.

The primary trap lies in assuming that $[x] - [-x]$ simplifies to a constant $\text{sgn}(x)$ or $1$ over the entire positive domain.

Students often blindly apply the property $[-x] = -[x]-1$ and assume the resulting factor is $1$ for all $x > 0$ , failing to recognize that $[x]$ steps up at integer boundaries (here, $x=1$ ).

This forces the integral to be split, punishing those who integrate over $[0, \pi/2]$ without accounting for the GIF jump.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Determinants - Class 12
Concept	Row/Column Exchange Parity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free