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Question #41488
The Question
The curves and intersect at a real point . A third curve is defined implicitly by the equation Find the slope of the tangent to the curve at the point .
Options
- A.
-31
- B.
1
- C.
31
- D.
-1
Check Your Answer
Correct Answer
Step-by-Step Solution
Explanation
How the Examiner Thinks
Specialist
Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.
Concepts Behind This Question
Intersection Point Accuracy|Implicit Differentiation Rate|Logarithmic Differentiation Domain
Why This Question Is Tricky
Common Mistakes
This question tests the student's vigilance regarding the domain of logarithmic functions during algebraic manipulation.
When solving the system of equations, combining the logarithms yields a quadratic equation with two roots, producing points (4,3) and (0,-1).
The trap lies in blindly accepting both points or arbitrarily choosing one without verifying against the initial domain constraints (x>2, y>1).
To make the trap mathematically punishing, the third curve C_3 is specifically constructed to pass through both the valid and the extraneous intersection points.
A student who uses the extraneous point (0,-1) will successfully find a tangent slope of 1, which is present in the options, thus filtering out procedural solvers from conceptual thinkers.
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 |