If x, y and z are the roots of the equation `2t^(3)-(tan[x+y+z]pi)t^(2)-11t+2020=0`, then `|(x,y,z),(y,z,x),(z,x,y)|` is equal to (where, `[x]` denotes the greatest integral value less than or equal to x)

Options

A. 20
B. `-10`
C. 0
D. 1

Answer

To solve the problem, we need to find the value of the determinant given the roots \( x, y, z \) of the cubic equation:

\[ 2t^3 - \tan[x+y+z] \pi t^2 - 11t + 2020 = 0 \]

### Step 1: Identify the sum of the roots
From Vieta's formulas, we know that the sum of the roots \( x + y + z \) can be expressed as:

\[ x + y + z = -\frac{\text{coefficient of } t^2}{\text{coefficient of } t^3} = \frac{\tan[x+y+z] \pi}{2} \]

### Step 2: Set up the equation
Thus, we have:

\[ x + y + z = \frac{\tan[x+y+z] \pi}{2} \]

### Step 3: Apply the greatest integer function
Let \( n = \lfloor x + y + z \rfloor \). Then we can say:

\[ x + y + z = \tan(n \pi / 2) \]

### Step 4: Analyze the tangent function
The tangent function \( \tan(n \pi / 2) \) is defined as follows:
- It is 0 when \( n \) is even (0, ±2, ±4, ...).
- It is undefined when \( n \) is odd (±1, ±3, ...).

Since \( x + y + z \) must be a real number, we conclude that \( n \) must be even. Therefore, we can set \( x + y + z = 0 \).

### Step 5: Set up the determinant
We need to evaluate the determinant:

\[ \begin{vmatrix} x & y & z \\ y & z & x \\ z & x & y \end{vmatrix} \]

### Step 6: Perform row operations
We can perform row operations to simplify the determinant. Let's add all three rows together:

\[ R_1 \rightarrow R_1 + R_2 + R_3 \]

This gives us:

\[ \begin{vmatrix} x+y+z & x+y+z & x+y+z \\ y & z & x \\ z & x & y \end{vmatrix} \]

### Step 7: Substitute the value of \( x+y+z \)
Since \( x + y + z = 0 \):

\[ \begin{vmatrix} 0 & 0 & 0 \\ y & z & x \\ z & x & y \end{vmatrix} \]

### Step 8: Evaluate the determinant
If any row or column of a determinant is all zeros, the determinant itself is zero. Thus:

\[ \text{Determinant} = 0 \]

### Final Answer
The value of the determinant \( |(x,y,z),(y,z,x),(z,x,y)| \) is:

\[ \boxed{0} \]

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