Analyzing Number Pairs and Mathematical Operations

The question asks us to identify a number pair that does not follow the same mathematical operation relating the first number to the second number, unlike the other given pairs. We are given four number pairs: (6, 1260), (5, 600), (7, 2050), and (4, 240). We need to find the pattern that applies to most of these pairs and then find the one pair that is an exception.

Finding the Mathematical Pattern in Number Pairs

Let's examine the relationship between the first number (let's call it $x$) and the second number (let's call it $y$) in each pair. We can test various mathematical operations. Let's look at the ratios or powers involved.

• For (6, 1260), $1260 / 6 = 210$. Notice that $6^3 = 216$. $210$ is close to $6^3$. Could the pattern involve $x^3$? Let's check $6^4 = 1296$. $1260$ is close to $6^4$. Specifically, $1260 = 1296 - 36$. $36$ is $6^2$. So, $1260 = 6^4 - 6^2$. This suggests a pattern $y = x^4 - x^2$.

Let's test this potential pattern, $y = x^4 - x^2$, with the other number pairs.

Applying the Pattern to Each Number Pair

We will now apply the mathematical operation $y = x^4 - x^2$ to the first number ($x$) of each pair and see if we get the second number ($y$).

• Pair 1: (6, 1260)
  Here, $x = 6$. Let's calculate $6^4 - 6^2$: $$ 6^4 - 6^2 = (6 \times 6 \times 6 \times 6) - (6 \times 6) $$ $$ = (36 \times 36) - 36 $$ $$ = 1296 - 36 $$ $$ = 1260 $$
  The calculated value (1260) matches the given second number. This pair follows the pattern.
• Pair 2: (5, 600)
  Here, $x = 5$. Let's calculate $5^4 - 5^2$: $$ 5^4 - 5^2 = (5 \times 5 \times 5 \times 5) - (5 \times 5) $$ $$ = (25 \times 25) - 25 $$ $$ = 625 - 25 $$ $$ = 600 $$
  The calculated value (600) matches the given second number. This pair follows the pattern.
• Pair 3: (7, 2050)
  Here, $x = 7$. Let's calculate $7^4 - 7^2$: $$ 7^4 - 7^2 = (7 \times 7 \times 7 \times 7) - (7 \times 7) $$ $$ = (49 \times 49) - 49 $$ $$ = 2401 - 49 $$ $$ = 2352 $$
  The calculated value (2352) does NOT match the given second number (2050). This pair does NOT follow the pattern.
• Pair 4: (4, 240)
  Here, $x = 4$. Let's calculate $4^4 - 4^2$: $$ 4^4 - 4^2 = (4 \times 4 \times 4 \times 4) - (4 \times 4) $$ $$ = (16 \times 16) - 16 $$ $$ = 256 - 16 $$ $$ = 240 $$
  The calculated value (240) matches the given second number. This pair follows the pattern.

Identifying the Odd Number Pair

Based on our analysis, the mathematical operation $y = x^4 - x^2$ successfully describes the relationship in pairs (6, 1260), (5, 600), and (4, 240). However, this pattern does not hold for the pair (7, 2050), where the pattern would predict 2352 instead of 2050.

Therefore, the odd number pair is (7, 2050).

Revision Table: Understanding Number Pair Patterns

This table summarizes how we analyzed each number pair to find the pattern.

Additional Information: Common Mathematical Operations in Number Series

Problems involving number pairs or number series often use simple or complex mathematical operations. Recognizing these patterns is key to solving such questions. Some common patterns include:

• Arithmetic Progression: Adding or subtracting a constant value.
• Geometric Progression: Multiplying or dividing by a constant value.
• Square or Cube Series: $x^2, x^3$, etc., or related values like $x^2 \pm c$, $x^3 \pm c$.
• Combinations of Operations: Patterns like $ax + b$, $ax^2 + bx + c$, $x^n \pm x^m$, etc.
• Factorials: $x!$ or related expressions.
• Differences or Ratios: Looking at the differences between consecutive numbers or the ratios between them to find a pattern in the differences/ratios themselves.

Practicing different types of pattern recognition helps improve logical reasoning and problem-solving skills required for competitive exams.