Before You Begin

Question #41704

Difficulty: 8/10 | 🟢 Free

The Question

Two fair dice are rolled to obtain two numbers $a$ and $b$ respectively, where $a, b \in \{1, 2, 3, 4, 5, 6\}$ . Consider the system of linear equations: $x + 2y + z = 1$ $2x + ay + 3z = b$ $3x + 7y + 4z = 4$ Three urns $U_1, U_2,$ and $U_3$ contain balls as follows: $U_1$ has 3 Red and 2 Black balls, $U_2$ has 2 Red and 3 Black balls, and $U_3$ has 4 Red and 1 Black balls. A ball is drawn at random from $U_1$ if the system has a unique solution, from $U_2$ if the system has infinitely many solutions, and from $U_3$ if the system has no solution. If the drawn ball is Red, what is the probability that the system of equations had no solution?

Options

A.
$\frac{5}{28}$
B.
$\frac{3}{14}$
C.
$\frac{1}{56}$
D.
$\frac{9}{28}$

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

Sample Space Partitioning|System of Linear Equations (Cramer/Inverse)|Singular vs. Non-Singular Threshold

Why This Question Is Tricky

Common Mistakes

This question tests the 'Singular Threshold' trap within a multi-concept framework.

Students often mechanically apply Cramer's rule and assume that a determinant of zero strictly implies no solution, completely missing the geometric consistency check (infinite solutions) when $b=3$ .

This oversight skews the sample space partition.

By linking this algebraic threshold to Bayes' theorem, the problem rigorously evaluates the student's ability to handle conditional probability across dynamically defined, mutually exclusive partitions derived from linear algebra constraints.

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