In the following number pair reasoning question, four pairs are given in the form a – b. In three of the pairs, the second number is equal to the square of the first number plus 1, while one pair does not follow this rule. Select the odd number pair from the given alternatives.

Introduction / Context:
This question tests your ability to detect a simple algebraic relationship between two numbers in a pair. The given pairs are 11 – 120, 17 – 290, 21 – 442 and 12 – 145. In number pair questions of this kind, the second number is often a function of the first number, such as its square plus or minus a constant. Here, three pairs share a common rule: the second number equals the square of the first number plus 1. One pair fails this rule and must be selected as the odd one out.

Given Data / Assumptions:

• Number pairs: 11 – 120, 17 – 290, 21 – 442 and 12 – 145.
• We suspect a relationship of the form b = a^2 ± k for small k.
• We will compute a^2 for each first number and compare with the second number.
• Exactly one pair will not fit the pattern second = first^2 + 1.

Concept / Approach:
For each pair (a, b), we compute a^2 and then calculate a^2 + 1 and a^2 - 1 to see whether b matches one of these simple expressions. If three pairs show that b = a^2 + 1 and one pair shows b = a^2 - 1, or any other deviation, then the deviating pair is the odd one out. This algebraic method is straightforward and common in competitive exams involving number pairs.

Step-by-Step Solution:
Step 1: Examine the pair 11 – 120.Compute 11^2 = 121. Then 11^2 + 1 = 122 and 11^2 - 1 = 120. The second number is 120, which equals 11^2 - 1, not 11^2 + 1.Step 2: Examine the pair 17 – 290.Compute 17^2 = 289. Then 17^2 + 1 = 290. The second number equals 17^2 + 1, matching the pattern.Step 3: Examine the pair 21 – 442.Compute 21^2 = 441. Then 21^2 + 1 = 442. Again, the second number equals the square of the first number plus 1.Step 4: Examine the pair 12 – 145.Compute 12^2 = 144. Then 12^2 + 1 = 145. The second number equals 12^2 + 1, keeping the same rule.Step 5: Identify the odd pair.In three pairs, we have b = a^2 + 1. Only 11 – 120 breaks this pattern with b = a^2 - 1. Hence 11 – 120 is the odd number pair.

Verification / Alternative check:
An alternative is to subtract the square of the first number from the second number for each pair. For 17 – 290, 290 - 289 = 1. For 21 – 442, 442 - 441 = 1. For 12 – 145, 145 - 144 = 1. For 11 – 120, however, 120 - 121 = -1. So three pairs give a difference of +1 and one pair gives a difference of -1. This difference in sign clearly separates 11 – 120 from the rest and confirms the result obtained earlier.

Why Other Options Are Wrong:
17 – 290: Follows b = a^2 + 1, so it belongs with the majority pattern.
21 – 442: Again shows b = a^2 + 1, consistent with 17 – 290 and 12 – 145.
12 – 145: Also satisfies the same rule b = a^2 + 1 and therefore is not the odd pair.

Common Pitfalls:
Some students may stop after checking only one or two pairs and jump to a conclusion. Others may try unrelated operations such as multiplying or adding digits instead of directly checking the square relationship, which is the simplest here. To avoid mistakes, always test your suspected pattern across all options. If three options agree with one clean rule and one does not, you have almost certainly found the intended logic of the question.

Final Answer:
The odd number pair is 11 – 120, because in this pair the second number equals 11^2 - 1, while in all the other pairs the second number equals the square of the first number plus 1.