Arithmetic Mean Zero: Five Numbers MCQ Analysis

We are given five numbers whose arithmetic mean is zero. Let the numbers be $n_1, n_2, n_3, n_4, n_5$. The condition for the arithmetic mean is:

Since the mean is zero:

This implies that the sum of the five numbers must be zero:

We need to determine which of the given statements must be true based on this condition.

Evaluating Each Option

• Option 1: The product of the numbers is zero

  This statement is not necessarily true. The product is zero only if at least one number is zero. However, the sum can be zero without any number being zero. Example: $\{1, 2, 3, -1, -5\}$. Sum = $1+2+3-1-5 = 0$. Product = $1 \times 2 \times 3 \times (-1) \times (-5) = 30 \neq 0$. Thus, this option is incorrect.

• Option 2: At most two of these numbers are positive

  This statement is not necessarily true. We can have more than two positive numbers. Example: $\{1, 1, 1, -1, -2\}$. Sum = $1+1+1-1-2 = 0$. In this case, three numbers are positive. Thus, this option is incorrect.

• Option 3: There cannot be exactly one zero

  This statement is false. It is possible to have exactly one zero. Example: $\{5, 5, -5, -5, 0\}$. Sum = $5+5-5-5+0 = 0$. This set has exactly one zero. Thus, this option is incorrect.

• Option 4: There cannot be exactly one non-zero number

  This statement must be true. Let's assume there is exactly one non-zero number, say $x$. Then the other four numbers must be zero. The set would be $\{x, 0, 0, 0, 0\}$. The sum is $x + 0 + 0 + 0 + 0 = x$. For the sum to be zero (as required by the zero mean condition), $x$ must be 0. But if $x=0$, then all numbers are zero, meaning there are *zero* non-zero numbers, contradicting our assumption of exactly one non-zero number. Therefore, it is impossible to have exactly one non-zero number when the sum is zero. This statement must be true.

Conclusion: The only statement that must hold true given the condition is that there cannot be exactly one non-zero number among the five.