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Question #40944

Difficulty: 8/10 | 🟢 Free

The Question

Let $y = g(x)$ be a real-valued differentiable function defined implicitly by the equation $y^5 + 2y^3 + 3y = x$ . For each $x \in \mathbb{R}$ , a line $L(x)$ is given by the symmetric equations: $\frac{X - 1}{g(x)} = \frac{Y - 2}{g'(x)} = \frac{Z - 3}{\lambda(x)}$ If the line $L(x)$ lies entirely in the plane $7X + 49Y + 2Z = 111$ for all $x \in \mathbb{R}$ , then the value of $\lambda'(6)$ is:

Options

A.
$\frac{1}{28}$
B.
$\frac{15}{4}$
C.
$-\frac{1}{28}$
D.
$-\frac{65}{64}$

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

Collinearity and Coplanarity|Line lying in a plane|Derivative of Implicit Inverse Forms

Why This Question Is Tricky

Common Mistakes

This problem fuses 3D geometry (coplanarity and lines in a plane) with differential calculus (implicit inverse functions).

The geometric condition $7g(x) + 49g'(x) + 2\lambda(x) = 0$ forces students to find the second derivative of an inverse function.

The primary trap lies in calculating $g''(x)$ : students frequently forget to apply the chain rule (multiplying by $dy/dx$ ) when differentiating $(f'(y))^{-1}$ with respect to $x$ , leading to the incorrect value $-f''(y)/(f'(y))^2$ instead of $-f''(y)/(f'(y))^3$ .

This specific chain rule failure results in the trap option 15/4.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Introduction to Three-dimensional Geometry - Class 11
Concept	Collinearity and Coplanarity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free