This question asks us to find a set of numbers among the given options that shares the same relationship between its elements as seen in the two provided sets. We need to identify the logical rule or pattern connecting the numbers in the example sets and then apply it to the options.

Analyzing the Given Sets to Find the Relationship

The given sets are:

• (12, 313, 13)
• (11, 185, 8)

Let's look at the numbers in the first set: 12, 313, and 13. Let's call the numbers A, B, and C respectively. So, A=12, B=313, and C=13.

We need to find a relationship between 12, 313, and 13. The middle number, 313, is significantly larger than 12 and 13. This suggests operations like multiplication, squaring, or cubing might be involved.

Let's try common operations involving the first and third numbers (A and C) to see if they produce the middle number (B).

• Sum: $12 + 13 = 25$ (Not 313)
• Product: $12 \times 13 = 156$ (Not 313)
• Difference of squares: $|13^2 - 12^2| = |169 - 144| = 25$ (Not 313)
• Sum of squares: $12^2 + 13^2 = 144 + 169 = 313$ (This matches the middle number!)

So, the potential relationship is: The middle number is the sum of the squares of the first and third numbers.

Let's test this relationship with the second given set: (11, 185, 8). Here, A=11, B=185, and C=8.

According to our potential rule, $B = A^2 + C^2$.

Let's calculate $11^2 + 8^2$:

$11^2 = 121$

$8^2 = 64$}

$11^2 + 8^2 = 121 + 64 = 185$

This matches the middle number (185) in the second set. The relationship $B = A^2 + C^2$ seems to be the correct pattern.

Checking Options Based on the Identified Relationship

Now we will apply the relationship $B = A^2 + C^2$ to each of the given options to find the set that follows the same rule.

Option 1: (12, 158, 12)

A=12, B=158, C=12

Check if $B = A^2 + C^2$:

$12^2 + 12^2 = 144 + 144 = 288$

Is $288 = 158$? No. This set does not follow the relationship.

Option 2: (4, 125, 11)

A=4, B=125, C=11

Check if $B = A^2 + C^2$:

$4^2 + 11^2 = 16 + 121 = 137$

Is $137 = 125$? No. This set does not follow the relationship.

Option 3: (9, 90, 3)

A=9, B=90, C=3

Check if $B = A^2 + C^2$:

$9^2 + 3^2 = 81 + 9 = 90$

Is $90 = 90$? Yes. This set follows the relationship $B = A^2 + C^2$.

Option 4: (15, 150, 10)

A=15, B=150, C=10

Check if $B = A^2 + C^2$:

$15^2 + 10^2 = 225 + 100 = 325$

Is $325 = 150$? No. This set does not follow the relationship.

Conclusion: Identifying the Correct Set

Based on our analysis, only Option 3 follows the same numerical relationship ($B = A^2 + C^2$) as the two sets provided in the question. Therefore, the set (9, 90, 3) is the correct answer.

Relationship Analysis and Revision Table

Here is a summary of how the relationship $B = A^2 + C^2$ applies to the given sets and the options:

Additional Information: Number Analogies and Patterns

Number analogy or number pattern questions are common in logical reasoning and quantitative aptitude tests. They require you to identify the relationship between numbers in a given set and apply that same relationship to find a missing number or a matching set from options.

Strategies to solve such problems often involve considering:

• Basic arithmetic operations: addition, subtraction, multiplication, division.
• Powers and roots: squares, cubes, square roots, cube roots.
• Combinations of operations: e.g., $A \times B + C$, $(A+C)^2$, $A^2 + C^2$, $A^3 - C$.
• Prime numbers, composite numbers, odd/even numbers.
• Digit-based operations (though excluded by the specific rule in this question).

It's helpful to systematically test common patterns when trying to decipher the rule in the given sets. Once a pattern is found, verify it with all provided examples before applying it to the options.