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Question #41765

Difficulty: 8/10 | 🟢 Free

The Question

Let $T(t)$ be a triangle in the $xy$ -plane with vertices $A(t, 0)$ , $B(0, t-1)$ , and $C(|t|, |t-1|)$ . If $S(t)$ denotes the area of the triangle $T(t)$ , find the value of $12 \int_{-1}^{2} S(t) dt$

Options

A.
21
B.
9
C.
11
D.
15

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

Feasible Region Constancy|Area of Triangle via Determinants|The Modulus Trap

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to rigorously apply the determinant formula for the area of a triangle when the vertices are defined by parametric equations involving absolute values.

The primary trap (The Modulus Trap) occurs on two levels.

First, students may incorrectly assume $|t|=t$ and $|t-1|=t-1$ over the entire domain, leading to an incorrect area function and an answer of 11.

Second, even if they correctly partition the domain to resolve the absolute values in the coordinates, they often forget that the area formula requires the absolute value of the determinant ( $S = \frac{1}{2}|\Delta|$ ).

Integrating the signed determinant directly yields 9.

This forces a careful piecewise evaluation of both the coordinates and the resulting determinant's sign.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Programming - Class 12
Concept	Feasible Region Constancy
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free