Before You Begin

Question #41031

Difficulty: 8/10 | 🟢 Free

The Question

Let $y = f(x)$ be the continuous solution to the differential equation $\frac{dy}{dx} + |x-2| y = |x-2|$ satisfying $f(2) = 2$ . Let $k$ be the sum of the $x$ -coordinates of all points of inflection of the curve $y = f(x)$ . Three fair six-sided dice are rolled, and their outcomes are used as the coordinates of a point $P(a,b,c)$ in 3D space. Let $\Pi$ be the plane $2x + 2y + z = 3k$ . Find the probability that the perpendicular distance from $P$ to the plane $\Pi$ is strictly greater than 1.

Options

A.
$\frac{121}{216}$
B.
$\frac{147}{216}$
C.
$\frac{179}{216}$
D.
$\frac{95}{216}$

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

Complementary Counting|Distance from point to plane|Integrating Factor (IF) Sign Sensitivity|Point of Inflection Conditions

Why This Question Is Tricky

Common Mistakes

This question integrates differential equations with absolute values, rigorous calculus definitions, and 3D geometric probability.

The primary trap lies in the Integrating Factor sign sensitivity and the definition of a point of inflection.

Students who ignore the absolute value in the DE will incorrectly find POIs at $x=1$ and $x=3$ ( $k=4$ ).

Furthermore, students who correctly solve the DE often miss the POI at $x=2$ because $y''$ is undefined there, forgetting that a POI only requires a change in concavity and a valid tangent line ( $k=3$ ).

Finally, complementary counting is required to efficiently find the probability of the point falling outside the boundary.

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