`|(y +k, y , y),(y, y + k, y),(y, y, y+k)|= k^(2)(3y + k)`.

By using properties of determinants , show that : (i) {:[( x+4, 2x, 2x),( 2x,x+4, 2x),( 2x,2x, x+4) ]:}=( 5x +4) (4-x)^(2) ( ii) {:[( y+k , y , y ),( y,y+ k , y ),( y,y , y+k ) ]:} =k^(2) ( 3y +k )

By using properties of determinants. Show that: (i) |(x+4,2x,2x),(2x,x+4,2x),(2x,2x,x+4)|=(5x-4)(4-x)^2 (ii) |(y+k,y,y),(y,y+k,y),(y,y,y+k)|=y^2(3y+k)

Prove that abs[[y+k,y,y],[y,y+k,y],[y,y,y+k]]=k^2(3y+k)

det [[x ^ (k), x ^ (k + 2), x ^ (k + 3) y ^ (k), y ^ (k + 2), y ^ (k + 3) z ^ (k) , z ^ (k + 2), z ^ (k + 3)]] = (xy) (yz) (zx) {(1) / (x) + (1) / (y) + (1) / ( from)}

Using properties of determinants , find the value of k if |{:(x,y,x+y),(y,x+y,x),(x+y,x,y):}|=k(x^(3)+y^(3)) .

( k - 1 ) x - y = 5 , (k + 1 ) x + ( 1 - k ) y = ( 3k + 1 )