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

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

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

Using the property of determinants and without expanding {:|( b+c,q+r,y+z),( c+a,r+p,z+x),( a+b,p+q,x+y) |:}=2 {:|(a,q,x),( b,q,y),(c,r,z)|:}

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)

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)

Without expanding prove that Delta ={:|( x+y,y+z,z+x) ,( z,x,y),( 1,1,1) |:} =0

Prove that the value of each the following determinants is zero: |[a-b,b-c,c-a],[ x-y, y-z, z-x],[ p-q, q-r ,r-p]|

Show that abs((a,b,c),(a+2x,b+2y,c+2z),(x,y,z)) =0

Prove that the value of each the following determinants is zero: |[(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]|

If x , y , z are in A.P. , then the value of the determinant |[a+2,a+3,a+2x],[ a+3,a+4,a+2y],[ a+4,a+5,a+2z]| is a. 1 b. 0 c. 2a d. a