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Question #41708

Difficulty: 8/10 | 🟢 Free

The Question

Let $S = \{-2, -1, 0, 1, 2\}$ . A point $Q(a, b, c)$ is chosen uniformly at random such that $a, b, c \in S$ . Let $P$ be the point $(1, 1, 1)$ and $\Pi$ be the plane $x + y + z = 0$ . What is the probability that the line segment $PQ$ intersects the plane $\Pi$ ?

Options

A.
$\frac{72}{125}$
B.
$\frac{21}{25}$
C.
$\frac{23}{25}$
D.
$\frac{53}{125}$

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

Sample Space Partitioning|Line-Plane intersection logic|Parametric form pitfalls

Why This Question Is Tricky

Common Mistakes

This question tests the boundary conditions of parametric line equations in 3D geometry, a common pitfall in sample space partitioning.

Students frequently find the intersection of the line with the plane ( $t \in \mathbb{R}$ ) or the ray ( $t \ge 0$ ), forgetting that for a line segment $PQ$ , the parameter $t$ must strictly satisfy $t \in [0, 1]$ .

Ignoring this upper bound leads to the trap answers 23/25 or 21/25, incorrectly expanding the favorable sample space by including intersections that occur outside the segment.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 12
Concept	Sample Space Partitioning
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free