If `a>b` and `c < d` prove that `a+c < b+d`

If each of a, b and c is a non-zero number and (a)/(b)= (b)/(c ) , show that: (a + b+c) (a-b + c) = a^(2) + b^(2) + c^(2)

For any three vectors a,b and c prove that ["a b c"]^(2)=|(a*a,a*b,a*c),(b*a,b*b,b*c),(c*a,c*b,c*c)|

In A B C , ray A D bisects /_A and intersects B C in D . If B C=a , A C=b and A B=c , prove that B D=(a c)/(b+c) (ii) D C = (a b)/(b+c)

If a+b+c!= and |(a,b,c),(b,c,a),(c,a,b)|=0 then prove that a=b=c

If a!=b!=c\ a n d\ |{:(a, b, c), (a^2,b^2,c^2), (b+c, c+a, a+b):}|=0 then a+b+c=0 b. a b+b c+c a=0 c. a^2+b^2+c^2=a b+b c+c a d. a b c=0

The value of the determinant |-a b c a-b c a b-c| is equal to- 0 b. (a-b)(b-c)(c-a) c. (a+b)(b+c)(c+a) d. 4a b c

If |b+c c+a a+b a+b b+c c+a c+a a+b b+c|=k|ab c c a b b c a| , then value of k is 1 b. 2 c. 3 d. 4

If a ,\ b ,\ c are real numbers such that |(b+c,c+a ,a+b),( c+a,a+b,b+c),(a+b,b+c,c+a)|=0 , then show that either a+b+c=0 or, a=b=c .

Using the factor theorem it is found that a+b , b+c and c+a are three factors of the determinant |[-2a ,a+b, a+c],[ b+a,-2b,b+c],[c+a ,c+b,-2c]| . The other factor in the value of the determinant is

If a,b,c are in G.P. then the value of |(a,b,a+b),(b,c,b+c),(a+b,b+c,0)|= (A) 1 (B) -1 (C) a+b+c (D) 0