By using properties of determinants. Show that:`|xx^2y z y y^2z x z z^2x y|=(x-y)(y-z)(z-x)(x y+y z+z x)`

Using the properties of determinants in Exercise 1 to 6, evaluate |{:(a+x,y,z),(x,a+y,z),(x,y,a+z):}|

Using the properties of determinants in Exercise 1 to 6, evaluate |{:(3x,-x+y,-x+z),(x-y,3y,z-y),(x-z,y-z,3z):}|

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

Using the properties of determinants in Exercise 1 to 6, evaluate |{:(0,xy^2,xz^2),(x^2y,0,yz^2),(x^2z,y^2z,0):}|

Using properties of determinants in Exercise 11 to 15 prove that |{:(x,x^2,1+px^3),(y,y^2,1+py^3),(z,z^2,1+pz^3):}|=(1+pxyz)(x-y)(y-z)(z-x)

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

If a ,\ b ,\ c >0\ a n d\ x ,\ y ,\ z in R , then the determinant |\ \ (a^x+a^x)^2(a^x-a^(-x))^2 1(b^y+b^(-y))^2(b^y-b^(-y))^2 1(c^z+c^(-z))^2(c^z-c^(-z))^2 1| is equal to- a. a^x b^y c^x b. a^(-x)b^(-y)c^(-z)\ c. a^(2x)b^(2y)c^(2x) d. zero

Find the equation of the plane through the line of intersection of the planes x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane x - y + z = 0.