Before You Begin

Question #41344

Difficulty: 8/10 | 🟢 Free

The Question

A family of curves $C_1$ satisfies the differential equation $x dy - \lambda y dx = 0$ , where $\lambda = \frac{(\alpha-2)(\alpha-5)^2 (\alpha-8)^4}{\alpha+3}$ and $\alpha \in \mathbb{R} \setminus \{-3\}$ . Let $C_2$ be the family of orthogonal trajectories to $C_1$ . If every real, non-degenerate curve in $C_2$ is unbounded, find the sum of all positive integers $\alpha$ .

Options

A.
16
B.
8
C.
3
D.
14

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

Rate of Change Dependency|Orthogonal Trajectories|Wavy Curve (Method of Intervals)

Why This Question Is Tricky

Common Mistakes

This question integrates the formation of orthogonal trajectories with a complex inequality constraint.

The primary trap lies in the 'Wavy Curve' (Method of Intervals) applied to the parameter $\lambda$ .

Students often correctly deduce that $\lambda \le 0$ for the curves to be unbounded (hyperbolas or lines instead of bounded ellipses).

However, when solving the inequality $\frac{(\alpha-2)(\alpha-5)^2 (\alpha-8)^4}{\alpha+3} \le 0$ , they frequently miss the isolated points $\alpha = 5$ and $\alpha = 8$ where the expression equals zero, or they mishandle the closed bracket at $\alpha = 2$ , leading to an incomplete solution set and an incorrect sum.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Derivatives - Class 12
Concept	Rate of Change Dependency
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free