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Question #40779

Difficulty: 8/10 | 🟢 Free

The Question

Four distinct collinear points $P_1, P_2, P_3, P_4$ lie on the curve $C: y = (x^2 - 6x + 5)^{2/3} + 2x$ and the line $L: y = 2x + 1$ . Let their $x$ -coordinates be $x_1 < x_2 < x_3 < x_4$ . Let $N$ be the number of points on $C$ with $x$ -coordinate in the interval $(x_1, x_4)$ at which the tangent to $C$ is parallel to $L$ . Find the value of $N$ .

Options

A.
1
B.
2
C.
3
D.
4

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

Condition of Collinearity|Critical Point vs Stationary Point|Rolle's Theorem Anchors

Why This Question Is Tricky

Common Mistakes

This question tests the precise conditions required for Rolle's Theorem and the Mean Value Theorem.

Students often blindly apply the theorem, assuming that a line intersecting a curve at four distinct collinear points guarantees at least three points where the tangent is parallel to the secant line (roots of the derivative).

The trap lies in the non-differentiability of the function at $x=1$ and $x=5$ , which fall exactly inside the outer intervals $(x_1, x_2)$ and $(x_3, x_4)$ .

Because differentiability fails at these critical points, the derivative does not vanish in those intervals, reducing the actual number of parallel tangents from the erroneously expected 3 down to just 1.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Straight Lines - Class 11
Concept	Condition of Collinearity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free