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Question #40835

Difficulty: 8/10 | 🟢 Free

The Question

Find the sum of all integers $k$ such that the conics $C_1: y = x^2 - kx$ and $C_2: x = y^2 - ky$ intersect in exactly four distinct real points with $x$ -coordinates $x_1, x_2, x_3, x_4$ , and satisfy the inequality $\sum_{i=1}^4 [x_i]^2 \le 53$ , where $[ \cdot ]$ denotes the greatest integer function.

Options

A.
5
B.
14
C.
0
D.
9

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Intersection of Conics|Intersection Point Accuracy|Greatest Integer Function Traps|Wavy Curve (Method of Intervals)

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of conics, reducing the system to a quartic equation that factors symmetrically.

The primary trap lies in the Wavy Curve analysis of the discriminant to ensure four distinct real roots, which restricts the integer parameter $k$ to $k \ge 2$ or $k \le -4$ .

Students often forget the negative domain.

Furthermore, evaluating the Greatest Integer Function (GIF) of the irrational roots requires tight algebraic bounding (e.g., $k < \sqrt{k^2+2k-3} < k+1$ ).

The GIF behaves asymmetrically for positive and negative values of $k$ , leading to two different quadratic inequalities.

Failing to account for the negative branch or miscalculating the GIF of negative irrational numbers will lead to the trap answer 14.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Conic Sections - Class 11
Concept	Intersection of Conics
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free