If x, y and z are the roots of the equation 2t3-(tan [x + y + z] π )t2-111t+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 )

Question

If x, y and z are the roots of the equation 2t3-(tan [x + y + z] π )t2-111t+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 ) (A) 20 (B) -10 (C) 0 (D) 1

Tardigrade Admin · Accepted Answer

[x + y + z] is an integer ⇒ tan ([x + y + z] π ) = 0 Hence, the sum of roots =x+y+z=0 ⇒ | x y z y z x z x y |=-(x + y + z)(x2 + y2 + z2 - x y - y z - z x)=0

Q. If $x, y$ and $z$ are the roots of the equation $2 t^{3} - (t an [x + y + z] π) t^{2} - 111 t + 2020 = 0,$ 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 $x$ )

$20$

$- 10$

$0$

$1$

$[x + y + z]$ is an integer
$\Rightarrow t an ([x + y + z] π) = 0$
Hence, the sum of roots $= x + y + z = 0$
$\Rightarrow ∣ ∣ x y z y z x z x y ∣ ∣ = - (x + y + z) (x^{2} + y^{2} + z^{2} - x y - yz - z x) = 0$

Q. If x,y and z are the roots of the equation 2t3−(tan[x+y+z]π)t2−111t+2020=0, then ∣∣​xyz​yzx​zxy​∣∣​ is equal to (where, [x] denotes the greatest integral value less than or equal to x )

20

−10

0

1

[x+y+z] is an integer ⇒tan([x+y+z]π)=0 Hence, the sum of roots =x+y+z=0 ⇒∣∣​xyz​yzx​zxy​∣∣​=−(x+y+z)(x2+y2+z2−xy−yz−zx)=0

Q. If $x, y$ and $z$ are the roots of the equation $2 t^{3} - (t an [x + y + z] π) t^{2} - 111 t + 2020 = 0,$ 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 $x$ )

$20$

$- 10$

$0$

$1$

$[x + y + z]$ is an integer
$\Rightarrow t an ([x + y + z] π) = 0$
Hence, the sum of roots $= x + y + z = 0$
$\Rightarrow ∣ ∣ x y z y z x z x y ∣ ∣ = - (x + y + z) (x^{2} + y^{2} + z^{2} - x y - yz - z x) = 0$