If x=997 , y=998 z=999 then the value of `x^(2)+y^(2)+z^(2)-xy-yz-zx` will be

Options

A. 3
B. 9
C. 16
D. 4

Answer

To solve the expression \( x^2 + y^2 + z^2 - xy - yz - zx \) given \( x = 997 \), \( y = 998 \), and \( z = 999 \), we can follow these steps:

### Step 1: Write down the expression
We start with the expression:
\[ x^2 + y^2 + z^2 - xy - yz - zx \]

### Step 2: Substitute the values of x, y, and z
Substituting the values of \( x \), \( y \), and \( z \):
\[ 997^2 + 998^2 + 999^2 - (997 \cdot 998) - (998 \cdot 999) - (999 \cdot 997) \]

### Step 3: Calculate each term
Now we calculate each term:
- \( 997^2 = 994009 \)
- \( 998^2 = 996004 \)
- \( 999^2 = 998001 \)

Now, calculate the products:
- \( 997 \cdot 998 = 995006 \)
- \( 998 \cdot 999 = 997002 \)
- \( 999 \cdot 997 = 996003 \)

### Step 4: Substitute the calculated values back into the expression
Now substitute these values back into the expression:
\[ 994009 + 996004 + 998001 - 995006 - 997002 - 996003 \]

### Step 5: Simplify the expression
Now, simplify the expression step by step:
1. Calculate the sum of squares:
\[ 994009 + 996004 + 998001 = 2984004 \]
2. Calculate the sum of products:
\[ 995006 + 997002 + 996003 = 2984001 \]
3. Now substitute these sums into the expression:
\[ 2984004 - 2984001 = 3 \]

### Final Answer
Thus, the value of the expression \( x^2 + y^2 + z^2 - xy - yz - zx \) is:
\[ \boxed{3} \]