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Question #40365

Difficulty: 8/10 | 🟢 Free

The Question

Let $C$ be the curve $y = x^3 - 9x^2 + 24x - 18$ and $L$ be the tangent to $C$ at its point of inflection. Let $R$ be the closed region bounded by $C$ , $L$ , and the vertical lines $x = 0$ and $x = 7$ . If $S$ is the set of all points $(x, y)$ with integer coordinates lying strictly inside the region $R$ (i.e., not on any of its boundaries), find the number of elements in $S$ .

Options

A.
40
B.
33
C.
7
D.
51

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

Subset vs. Proper Subset|Bounded vs Unbounded Regions|Concavity Analysis

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to correctly identify bounded regions when a curve changes concavity.

The trap lies in assuming the bounding inequality remains constant (e.g., $f(x) \le y \le L(x)$ ) across the entire interval.

Because the tangent is drawn at the inflection point, it crosses the curve, flipping the upper and lower bounds.

Students who ignore this concavity change will only calculate the points in one lobe (yielding 7 or 33) or incorrectly include boundary points (yielding 51), missing the proper subset definition of the interior.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sets - Class 11
Concept	Subset vs. Proper Subset
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free