Show that `Delta=|(y+z)^2x y z xx y(x+z)^2y z x z y z(x+y)^2|=2x y z(x+y+z)^3`.

show that [((x+y)^2 , zx , zy),( zx, (z+y)^2 ,xy),(zy,xy,(z+x)^2)]=2xyz (x +y+z)^3

Verify that x ^(3) + y ^(3) + z ^(3) - 3xyz =1/2 (x + y + z) [(x-y)^(2) + (y-z) ^(2) + (z-x) ^(2) ]

If cos^(-1)x+cos^(-1)y+cos^(-1)z=pi, then, a. x^2+y^2+z^2+x y z=0 b. x^2+y^2+z^2+2x y z=0 c. x^2+y^2+z^2+x y z=1 d. x^2+y^2+z^2+2x y z=1

If Delta = abs{:(x+y+z^(2) , x^(2) + y+ z, x+y^(2) + z),(z^2, x^2, y^2),(x+y,y+z,x+z):} , (where ( x ne y ne z) x, y, z in R- {0} ) then Delta= ........

Using the property of determinants andd without expanding in following exercises 1 to 7 prove that |{:(x+y+2z,x,y),(z,y+z+2x,y),(z,x,z+x+2y):}|=2(x+y+z)^3

If m^(th), n^(th) and p^(th) terms of an A.P and G.P be equal and be respectively x,y, z then………...................................................... a ) x^(y).y^(z).z^(x) = x^(z).Y^(x).z^(y) b ) (x-y)^(x).(y-z)^(y)= (z-x)^(z) c ) (x-y)^(z) (y-z)^(x)= (z-x)^(y) d ) None of these

Find the product of (5x - y +z) (5x - y +z)

Using the proprties of determinants in Exercise 7 to 9, prove that |{:(y^2z^2,yz,y+z),(x^2z^2,zx,z+x),(x^2y^2,xy,x+y):}|=0

Show that the plane 2x-2y+z+12=0 touches the sphere x^2+y^2+z^2-2x-4y+2z-3=0.