The mean of five numbers is 0 and their variance is 2 .If three of those numbers are -1,1 and 2, then the other two numbers are

Options

A. `-5` and 3
B. `-4` and 2
C. `-3` and 1
D. `-2` and 0

Answer

To solve the problem, we need to find the two unknown numbers \( x \) and \( y \) given that the mean of five numbers is 0 and their variance is 2. The three known numbers are -1, 1, and 2.

### Step 1: Use the Mean Condition
The mean of the five numbers is given by:

\[ \text{Mean} = \frac{-1 + 1 + 2 + x + y}{5} = 0 \]

This simplifies to:

\[ \frac{2 + x + y}{5} = 0 \]

Multiplying both sides by 5 gives:

\[ 2 + x + y = 0 \]

Rearranging this, we find:

\[ x + y = -2 \quad \text{(Equation 1)} \]

### Step 2: Use the Variance Condition
The variance of the five numbers is given by:

\[ \text{Variance} = \frac{(-1)^2 + 1^2 + 2^2 + x^2 + y^2}{5} - \left(\frac{0}{5}\right)^2 = 2 \]

Calculating the squares of the known numbers:

\[ (-1)^2 = 1, \quad 1^2 = 1, \quad 2^2 = 4 \]

So, we have:

\[ \frac{1 + 1 + 4 + x^2 + y^2}{5} = 2 \]

This simplifies to:

\[ \frac{6 + x^2 + y^2}{5} = 2 \]

Multiplying both sides by 5 gives:

\[ 6 + x^2 + y^2 = 10 \]

Rearranging this, we find:

\[ x^2 + y^2 = 4 \quad \text{(Equation 2)} \]

### Step 3: Substitute for \( y \)
From Equation 1, we can express \( y \) in terms of \( x \):

\[ y = -2 - x \]

### Step 4: Substitute \( y \) into Equation 2
Now, substitute \( y \) into Equation 2:

\[ x^2 + (-2 - x)^2 = 4 \]

Expanding the equation:

\[ x^2 + (4 + 4x + x^2) = 4 \]

Combining like terms:

\[ 2x^2 + 4x + 4 = 4 \]

Subtracting 4 from both sides:

\[ 2x^2 + 4x = 0 \]

### Step 5: Factor the Equation
Factoring out \( 2x \):

\[ 2x(x + 2) = 0 \]

Setting each factor to zero gives:

\[ 2x = 0 \quad \text{or} \quad x + 2 = 0 \]

Thus, we find:

\[ x = 0 \quad \text{or} \quad x = -2 \]

### Step 6: Find Corresponding \( y \) Values
1. If \( x = 0 \):
\[ y = -2 - 0 = -2 \]
So one pair is \( (0, -2) \).

2. If \( x = -2 \):
\[ y = -2 - (-2) = 0 \]
So the other pair is \( (-2, 0) \).

### Conclusion
The two unknown numbers are \( 0 \) and \( -2 \).