Before You Begin

Question #40910

Difficulty: 8/10 | 🟢 Free

The Question

Let $P_t$ be a variable plane with intercepts $t, 2t,$ and $2t$ on the $x, y,$ and $z$ axes respectively, where $t > 0$ . Let $N(x_t, y_t, z_t)$ be the foot of the perpendicular drawn from the point $A(4, 2, 2)$ to the plane $P_t$ . Evaluate the integral: $I = \int_{1}^{5} \left| \frac{2y_t - 2}{y_t^2 - 2y_t} \cdot \frac{dy_t}{dt} \right| dt$

Options

A.
$\ln\left(\frac{81}{25}\right)$
B.
0
C.
$\ln\left(\frac{9}{5}\right)$
D.
$2\ln\left(\frac{5}{3}\right)$

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

Coordinate Axis Projections|Variable plane intercepts|Logarithmic Integration Modulus

Why This Question Is Tricky

Common Mistakes

This question tests the 'Logarithmic Integration Modulus' trap combined with 3D coordinate geometry.

Students must first determine the locus of the projection of a fixed point onto a variable plane, yielding a parameter-dependent coordinate $y_t$ .

The core trap lies in evaluating the definite integral of the absolute value of the logarithmic derivative, $\left| f'(t)/f(t) \right|$ .

A common mistake is to blindly integrate this as $\ln|f(t)|$ and evaluate at the boundaries, which yields $0$ .

The modulus necessitates splitting the integral at the critical point $t=3$ where the integrand changes sign, forcing students to handle the non-differentiability of the modulus function correctly.

Quick Facts About This Question

Question Information

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