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) `

Using the property of determinants and without expanding prove that {:|( -a^(2) , ab,ac),( ba,-b^(2) , bc) ,( ca, cb, -c^(2)) |:} =4a^(2) b^(2) c^(2)

Prove that abs{:(a^(2) + 1, ab , ac),(ab, b^(2) + 1, bc),(ca, cb, c^(2) +1):}=1 + a^(2) + b^(2) +c^(2)

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)) |:}

(a) Prove that int_(0)^(2x) f(x) dx = 2int_(0)^(2x) f(x) dx when f(2a-x) =f(x) and hence evaluate int_(0)^(pi) |cos x| dx . (b) Prove that |{:(-a^(2),ab,ac),(bc,-b^(2),bc),(ca,cb,-c^(2)):}|=4a^(2)b^(2)c^(2) .

Prove that |(-a^2,ab,ac),(bc,-b^2,bc),(ca,cb,-c^2)|=4a^(2)b^(2) c^(2) .

|(a^(2)+1,ab,ac),(ab,b^2+1,bc),(ca,cb,c^2+1)|= 1 + a^2 + b^2 + c^2 .

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

|(b^(2)c^(2),bc,b+c),(c^(2)a^(2),ca,c+a),(a^(2)+b^(2),ab,a+b)|=

If a,b,c,d and p are distinct non -zero real numbers such that : ( a^(2) + b^(2) +c^(2))p^(2) - 2 ( ab + bc + cd) p + ( b^(2) + c^(2) + d^(2)) le 0 , then a,b,c,d :