By using properties of determinants , show that : ` {:|( x,x^(2) , yz) ,( y,y^(2) , zx ) ,( z , z^(2) , xy ) |:} =( x-y)(y-z) (z-x) (xy+yz+ zx) `

By using properties of determinants , show that : (i) {:|( a-b-c, 2a, 2a),( 2b, b-c-a, 2b ) , ( 2c,2c, c-a-b) |:} = ( a+b+c)^(3) (ii) {:|( x+y+2z, x,y ),( z, y+z+2x, y ),( z,x,z+x+2y )|:} =2( x+y+z)^(3)

Using properties of determinants porve 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)

Simplify: x(x-y)+y(y-z)+z(z-x)

Using the property of determinants and without expanding {:|( x,a,x+a),( y,b,y+b),(z,c,z+c)|:} =0

Find the compound ratio of the inverse ratios of the ratios x^(2) : (yz)/x, y^(2) : (zx)/y " and " z^(2) : (xy)/z .

Using Cofactors of elements of third column , evaluate , Delta ={:|( 1,x,yz),(1,y,zx),( 1,z,xy) |:}

Prove that |x^2x^2-(y-z)^2y z y^2y^2-(z-x)^2z x z^2z^2-(x-y)^2x y|=(x-y)(y-z)(z-x)(x+y+z)(x^2+y^2+z^2)dot

|{:(x+y+z," "-z," "-y),(" "-z,x+y+z," "-x),(" "-y," "-x,x+y+z):}|=2(x+y)(y+z)(z+x)

Factorise 4x ^(2) + 9y ^(2) + 25 z ^(2) - 12 xy - 30 yz + 20 zx

Factorise : |{:(x,y,z),(x^2,y^2,z^2),(yz,zx,xy):}|