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 step-by-step, we need to analyze the given cubic equation and the determinant involving the roots \( x, y, z \).

### Step 1: Understand the given cubic equation
The equation is:
\[ 2t^3 - \tan[x+y+z]\pi t^2 - 11t + 2020 = 0 \]
From the properties of cubic equations, we know that the sum of the roots \( x + y + z \) can be expressed in terms of the coefficients.

### Step 2: Identify the sum of the roots
The sum of the roots \( x + y + z \) for the cubic equation \( at^3 + bt^2 + ct + d = 0 \) is given by:
\[ x + y + z = -\frac{b}{a} \]
Here, \( a = 2 \) and \( b = -\tan[x+y+z]\pi \). Thus:
\[ x + y + z = \frac{\tan[x+y+z]\pi}{2} \]

### Step 3: Set up the equation for the greatest integer function
Since \( [x+y+z] \) denotes the greatest integral value less than or equal to \( x+y+z \), we can denote \( n = [x+y+z] \). Therefore:
\[ x + y + z = 10n\frac{\pi}{2} \]

### Step 4: Determine the value of \( n \)
To satisfy \( \tan[n\pi] = 0 \), we need \( n \) to be an integer. The simplest case is when \( n = 0 \), which leads to:
\[ x + y + z = 0 \]

### Step 5: Analyze the determinant
The determinant we need to evaluate is:
\[ D = \begin{vmatrix} x & y & z \\ y & z & x \\ z & x & y \end{vmatrix} \]

### Step 6: Simplify the determinant
We can simplify the determinant by adding all rows:
\[ D = \begin{vmatrix} x + y + z & x + y + z & x + y + z \\ y & z & x \\ z & x & y \end{vmatrix} \]
Substituting \( x + y + z = 0 \):
\[ D = \begin{vmatrix} 0 & 0 & 0 \\ y & z & x \\ z & x & y \end{vmatrix} \]

### Step 7: Conclude the value of the determinant
Since the first row consists entirely of zeros, the determinant \( D \) is equal to zero:
\[ D = 0 \]

### Final Answer
Thus, the value of the determinant \( |(x,y,z),(y,z,x),(z,x,y)| \) is:
\[ \boxed{0} \]