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Question #41562

Difficulty: 8/10 | 🟢 Free

The Question

Let $M$ be a $3 \times 3$ matrix with real entries such that $\det(M) = 2$ , and let $A = \text{adj}(M)$ . Let $O$ be the origin $(0,0)$ . For any real number $x$ , let $P(x)$ be the point $(x, \det(xA))$ and $Q$ be the point $(2, \det(2A))$ in the $xy$ -plane. If $S(x)$ denotes the area of the triangle $OP(x)Q$ , find the value of the integral $\int_{-1}^{2} S(x) \, dx$

Options

A.
23
B.
9
C.
$\frac{32}{3}$
D.
$\frac{23}{2}$

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

The Modulus Trap|Area of Triangle via Determinants|Determinant Scaling Property

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of linear algebra and integral calculus, specifically targeting the 'Determinant Scaling Trap' and the 'Modulus Trap'.

Students frequently make two critical errors: first, they incorrectly scale the determinant by assuming $\det(xA) = x^2\det(A)$ (associating the 2D triangle context with a $2 \times 2$ matrix) or $\det(xA) = x\det(A)$ , forgetting that $A$ is a $3 \times 3$ matrix requiring $k^n$ scaling.

Second, when integrating the area function $S(x)$ , they drop the absolute value bars, failing to recognize that the determinant's sign changes over the interval $[-1, 2]$ .

Splitting the integral at $x=0$ is mandatory to respect the geometric definition of area.

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