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)|:}`

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

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)

|{:(a,b,c),(x,y,z),(p,q,r):}|=|{:(x,y,z),(p,q,r),(a,b,c):}|=|{:(y,b,q),(x,a,p),(z,c,r):}|

|{:(" "a," "b," "c),(" "a+2x," "b+2y," "c+2z),(" "x," "y," "z):}|

If p+q+r=0=a+b+c , then the value of the determinant |[p a, q b, r c],[ q c ,r a, p b],[ r b, p c, q a]|i s (a) 0 (b) p a+q b+r c (c) 1 (d) none of these

If x+y+z=0 prove that |[a x, b y, c z],[ c y, a z ,b x],[ b z ,c x ,a y]|=x y z|[a, b, c],[c ,a ,b],[b ,c ,a]|

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]|

Let's find the L.C.M of the following algebraic expressions. (p+q) (q+r), (q+r)(r+p), (r+p)(p+q)

If the p-th, q-th and the r-th terms of a H.P. be x, y, z respectively, prove that, yz (q-r) + zx(r-p)+xy(p-q) = 0

If pth, qth and rth terms of on AP are a. b and c, then show that a(q - r) + b(r - p) + c(p - q) = 0.