Prove that
`|(1+a,1,1),(1,1+b,1),(1,1,1+c)|=abc(1+(1)/(a)+(1)/(b)+(1)/(c)).`

Show that {:[( 1+a,1,1),( 1,1+b,1) ,( 1,1,1+c) ]:}= abc ( 1+(1)/(a) +(1)/(b) +(1)/(c) ) = abc +bc +ca +ab

Prove that |{:(1,a,a^2),(1,b,b^2),(1,c,c^2):}|=(a-b)(b-c)(c-a) .

Prove that |(1,a,a^(2)),(1,b,b^(2)),(1,c,c^(2))|=(a-b)(b-c)(c-a)

Prove that |{:(1,,ab,,(1)/(a)+(1)/(b)),(1 ,,bc,,(1)/(b)+(1)/(c)),( 1,, ca,,(1)/(c)+(1)/(a)):}|=0

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

Prove that (1)/(a) + (1)/(b) + (1)/(c ) ge (1)/(sqrt((bc))) + (1)/(sqrt((ca))) + (1)/(sqrt((ab))) , where a,b,c gt 0

Prove that |1+a 1 1 1 1 1+b 1 1 1 1 1+c 1 1 1 1 1+d|=a b c d(a+1/a+1/b+1/c+1/d)dot Hence find the value of the determinant if a ,b ,c , d are the roots of the equation p x^4+q x^3+r x^2+s x+t=0.

Prove that Delta = |{:(1,,1,,1),(a,,b,,c),(bc+1^(2),,ac+b^(2),,ab+c^(2)):}| = 2(a-b)(b-c)(c-1)

Sove that |(1,a,a^(2)),(1,b,b^(2)),(1,c,c^(2))|^(2)=|(1+a^(2)+a^(4),1+ab+a^(2)b^(2),1+ac+a^(2)c^(2)),(1+ab+a^(2)b^(2),1+b^(2)+b^(4),1+bc+b^(2)c^(2)),(1+ac+a^(2)c^(2),1+ba+b^(2)c^(2),1+c^(2)+c^(4))| and hence show RHS determinent is =(a-b)^(2)(b-c)^(2)(c-a)^(2)

Evaluate Delta = {:[ (1,a,bc),( 1,b,ca),(1,c,ab)]:}