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Question #40324

Difficulty: 8/10 | 🟢 Free

The Question

Let $t_1, t_2, t_3$ be the three distinct real roots of the equation $t^3 - 10t^2 + 29t - 22 = 0$ . For $i = 1, 2, 3$ , let $P_i(x_i, y_i)$ be points in the Cartesian plane such that the ordered pairs $(x_i, y_i)$ satisfy the equality: $(x_i + 2y_i, 2x_i - y_i) = (7t_i^3 - 7t_i^2 + 7t_i - 4, -t_i^3 + t_i^2 - t_i + 2)$ Let $\Delta$ be the area of the triangle formed by the points $P_1, P_2,$ and $P_3$ . The value of $\Delta$ is:

Options

A.
0
B.
10
C.
22
D.
29

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

Ordered Pair Equality|Area of Triangle via Determinants|Condition of Collinearity

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to verify the condition of collinearity (degeneracy) before blindly applying the determinant formula for the area of a triangle.

The ordered pair equality is deliberately designed to yield coordinates that are cubic polynomials of the roots of an irreducible cubic equation.

A student falling into the trap will attempt to use Vieta's formulas (symmetric sums of roots) to evaluate the $3 \times 3$ area determinant, leading to a massive, error-prone calculation.

The logical bypass is to identify the linear dependence between the $x$ and $y$ coordinates ( $y_i = 3x_i - 2$ ), which immediately proves the points are collinear and the area is zero.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 11
Concept	Ordered Pair Equality
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free