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Question #41048

Difficulty: 8/10 | 🟢 Free

The Question

Three independent trials are performed. In each trial, a fair die is rolled. If the die shows a multiple of $3$ , a success is recorded. Let $x, y, z \in \{1, -1\}$ such that the variable takes the value $1$ if the corresponding trial (1st, 2nd, 3rd) is a success, and $-1$ if it is a failure. Consider the vectors $\vec{u} = \hat{i} + \hat{j}$ , $\vec{v} = x\hat{i} + y\hat{j} + z\hat{k}$ , and $\vec{w} = \hat{i} + \hat{k}$ . Let $A$ be the event that the volume of the parallelepiped formed by $\vec{u}, \vec{v}, \vec{w}$ is exactly $1$ . Given that event $A$ has occurred, what is the probability that the acute angle between the planes containing the pairs of vectors $(\vec{u}, \vec{v})$ and $(\vec{v}, \vec{w})$ is $\frac{\pi}{6}$ ?

Options

A.
$\frac{4}{7}$
B.
$\frac{8}{9}$
C.
$\frac{1}{3}$
D.
$\frac{4}{9}$

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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 Construction|Bernoulli Trials Logic|Scalar Triple Product Volume Logic|Angle between two planes

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of 3D geometry and probability, specifically targeting the 'sign-ambiguity' trap of the Scalar Triple Product (STP).

Students frequently forget that the volume of a parallelepiped is the absolute value of the STP ( $V = |[\vec{u} \vec{v} \vec{w}]|$ ).

If a student incorrectly assumes $[\vec{u} \vec{v} \vec{w}] = 1$ , they will only consider half of the valid sample space.

Because the Bernoulli trials have asymmetric probabilities ( $1/3$ vs $2/3$ ), this mistake does not cancel out in the conditional probability fraction, leading directly to the trap answer of $8/9$ .

Furthermore, the angle between two planes requires the absolute value of the dot product of their normal vectors; missing this will cause students to incorrectly classify acute/obtuse angles, further filtering out valid outcomes.

Quick Facts About This Question

Question Information

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