Prove that :
(i) `|{:(a,c,a+c),(a+b,b,a),(b,b+c,c):}|=2 abc`
(ii) Prove that : `|{:(a^(2),bc,ac+c^(2)),(a^(2)+ab,b^(2),ac),(ab,b^(2)+bc,c^(2)):}|=4a^(2)b^(2)c^(2)`

Prove that |{:(a^2,bc,ac+c^2),(a^2+ab,b^2,ac),(ab,b^2+bc,c^2):}|=4a^2b^2c^2

Prove that: {:|(1,a, a^2-bc), (1,b,b^2-ca),(1,c,c^2-ab)|:}=0

Find the value of the determinant |{:(a^2,a b, a c),( a b,b^2,b c), (a c, b c,c^2):}|

For any triangle ABC, prove that : (b^(2)-c^(2))cotA+(c^(2)-a^(2))cotB+(a^(2)-b^(2))cotC=0 .

Prove that |{:(a,a+b,a=b+c),(2a,3a+2b,4a+3b+2c),(3a,6a+3b,10a+6b+3c):}|=a^3

a,b,c,d are in G.P. Prove that a^(2)-b^(2),b^(2)-c^(2), c^(2)-d^(2) are also in G.P.

Without expanding the determinant, prove that {:[( a, a ^(2), bc ),( b ,b ^(2) , ca),( c, c ^(2) , ab ) ]:} ={:[( 1, a^(2) , a^(3) ),( 1,b^(2) , b^(3) ),( 1, c^(2),c^(3)) ]:}

For any triangle ABC, prove that : ((b^(2)-c^(2))/(a^(2)))sin2A+((c^(2)-a^(2))/(b^(2)))sin2B+((a^(2)-b^(2))/(c^(2)))sin2C=0

Verify : a^(3)+b^(3)+c^(3)-3abc=(1)/(2)(a+b+c)[(a-b)^(2)+(b-c)^(2)+(c-a)^(2)]

Compute the following : (i) [{:(a,b),(-b,a):}]+[{:(a,b),(b,a):}] (ii) [{:(a^(2)+b^(2),b^(2)+c^(2)),(a^(2)+c^(2),a^(2)+b^(2)):}]+[{:(2ab,2bc),(-2ac,-2ab):}] (iii) [{:(-1,4,-6),(8,5,16),(2,8,5):}]+[{:(12,7,6),(8,0,5),(3,2,4):}] (iv) [{:(cos^(2)x,sin^(2)x),(sin^(2)x,cos^(2)x):}]+[{:(sin^(2)x,cos^(2)x),(cos^(2)x,sin^(2)x):}]