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Question #41544

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all points $(x, y)$ in the Cartesian plane that satisfy the system of equations: $3x^2 + 4y^2 = 16$ $5x^2 - 2y^2 = 18$ Find the number of integer values of $k \in [1, 10]$ such that the inequality $\frac{k|x| - 12}{|y| - 2} \le 3$ holds true for all points $(x, y) \in S$ .

Options

A.
4
B.
5
C.
6
D.
7

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

Intersection Point Accuracy|Intersection of Conics|Modulus Inequality Logic|Cross Multiplication Trap

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to accurately find the intersection of two conics and correctly apply modulus inequalities.

The primary trap lies in the cross-multiplication of the rational inequality.

Students often blindly cross-multiply the term $(|y| - 2)$ to the right side without evaluating its sign.

Since $|y| = 1$ for all intersection points, $(|y| - 2) = -1$ , which is negative.

Cross-multiplying without flipping the inequality sign leads to the erroneous condition $k \le 4.5$ , yielding 4 integer values instead of the correct 6.

This perfectly discriminates students who rely on rote algebraic manipulation from those who maintain rigorous logical checks on domain and signs.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Integrals - Class 12
Concept	Intersection Point Accuracy
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free