Q- if A^2+B^2+C^2=1, then what is the range of AB+BC+CA
sol:- A+B+C>=0 (A+B+C)^2>=0 A^2+B^+C^2+2(AB+BC+CA)>=0 1+2(AB+BC+CA)>=0 AB+BC+CA>= -1/2 Now, (A-B)^2>=0 (B-C)^2>=0 (C-A)^2>=0 (A-B)^2+(B-C)^2+(C-A)^2>=0 2(A^2+B^2+C^2)-2(AB+BC+CA)>=0 (A^2+B^2+C^2)-(AB+BC+CA)>=0 1-(AB+BC+CA)>=0 -(AB+BC+CA)>= -1 (AB+BC+CA)<=1 Range of AB+BC+CA=[-1/2,1].