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Question #41029

Difficulty: 8/10 | 🟢 Free

The Question

Let $\Omega$ be the region in the first octant bounded by the coordinate planes and the tangent plane to the surface $xyz = 8$ at the point on the surface closest to the origin. A point $(x,y,z)$ is chosen uniformly at random from $\Omega$ . Let $f(t) = |t-x| + |t-y| + |t-z|$ . If $E$ is the event that the absolute minimum value of $f(t)$ is at most $4$ , find the probability of $E$ .

Options

A.
$\frac{25}{27}$
B.
$\frac{2}{27}$
C.
$\frac{31}{36}$
D.
$\frac{13}{18}$

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

Complementary Counting|Exhaustive Events Concept|Geometry Constraint Modeling|Critical Point vs Stationary Point

Why This Question Is Tricky

Common Mistakes

This question tests the interaction between geometric probability, complementary counting, and optimization.

A classic trap is assuming the minimum of a sum of absolute values can be found via the first derivative test (stationary points).

However, $f'(t)$ is never zero; the minimum strictly occurs at a non-differentiable critical point (the median).

Students who fail to visualize this will incorrectly evaluate the minimum.

Furthermore, calculating the volume for $M-m \le 4$ directly involves complex piecewise boundaries, forcing the student to use complementary counting on the exhaustive sample space to find the volume of $M-m > 4$ instead.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 11
Concept	Complementary Counting
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free