Question

without expanding find thr value of determinant ∆=[2, x , y+z]
2, y , z+x
2, z , x+y

Answer 1

Use AM-GM inequality.

x^2 + y^2 >(or equal) 2xy
y^2 + z^2 > 2yz
z^2 + x^2 > 2xz

Adding all the equations.

x^2+y^2+z^2-xy-yz-xz >(or equal) 0

Since its given that the equation is equal to zero, we conclude that equation has its minimum value. 

So, all the 3 equations should individually attain their minimum values 

So,

x^2+y^2-2xy=0
(x-y)^2=0
So, x=y
Similarly, y=z and x=z.
So, x=y=z