Correct Option
The matrix follows a specific pattern where the value in the second row of each column is derived from the value in the first row of that same column. The relationship is defined by the formula: y = x² - x, where x is the number in the first row and y is the number in the second row.
Applying this pattern to the known columns:
- For column 1: 3² - 3 = 9 - 3 = 6
- For column 2: 8² - 8 = 64 - 8 = 56
- For column 3: 10² - 10 = 100 - 10 = 90
- For column 4: 2² - 2 = 4 - 2 = 2
- For column 6: 1² - 1 = 1 - 1 = 0
To find the missing number at '?' in column 5, let it be x. The corresponding value in the second row is 20.
Therefore, x² - x = 20.
Rearranging the equation: x² - x - 20 = 0.
Factoring the quadratic equation:
x² - 5x + 4x - 20 = 0
x(x - 5) + 4(x - 5) = 0
(x - 5)(x + 4) = 0
This yields two possible solutions for x: x = 5 or x = -4.
Given the context of the matrix containing positive integers in the first row, x = 5 is the valid solution.
Thus, the missing number is 5.
Incorrect Options
Options 2 (0), 3 (7), and 4 (3) are incorrect because they do not satisfy the established pattern x² - x = y for the corresponding second row value of 20.
- If x = 0, then 0² - 0 = 0 ≠ 20.
- If x = 7, then 7² - 7 = 49 - 7 = 42 ≠ 20.
- If x = 3, then 3² - 3 = 9 - 3 = 6 ≠ 20.