If a,b,c are in AP, then show that `r_(1), r_(2), r_(3)` are in HP.

If a_1,a_2,a_3….a_r are in G.P then show that |{:(a_(r+1),a_(r+5),a_(r+9)),(a_(r+7),a_(r+11),a_(r+15)),(a_(r+11),a_(r+7),a_(r+21)):}| is independent of r

Statement I In a Delta ABC, if a lt b lt c and ri si inradius and r_(1), r_(2) ,r_(3) are the exradii opposite to angle A,B,C respectively, then r lt r_(1) lt r_(2) lt r_(3). Statement II For, DeltaABC r_(1)r_(2)+r_(2)r_(3)+r_(3)r_(1)=(r_(1)r_(2)r_(3))/(r)

If a,b,c are in AP and a^(2),b^(2),c^(2) are in HP, then

Six resistances each of v ale r= 6Omega are connected between points A,B and C as shown in the figure if R_(1),R_(2) and R_(3) are the net resistances between A and B, between B and C and between A and C respectively, the R_(1):R_(2):R_(3) will be equal to

omega is the cube root of 1 and omega ne 1 . Now r_(1), r_(2) and r_(3) are the number obtained while tossing dice thrice. Then ………… is the probability for omega^(r^(1)) +omega^(r^(2))+omega^(r^(3))=0

If pth,qth and rth terms of an A.P. are a, b, c respectively, then show that a(q-r)+b(r-p)+c(p-q)=0

If r_(1),r_(2) and r_(3) are the position vectors of three collinear points and scalars l and m exists such that r_(3)=lr_(1)+mr_(2) , then show that l+m=1.

If R_1 and R_2 are equivalence relations in a set A, show that R_(1) cap R_2 is also an equivalence relation.

The distance between two particles of masses m_(1)andm_(2) is r. If the distance of these particles from the centre of mass of the system are r_(1)andr_(2) respectively, then show that r_(1)=r((m_(2))/(m_(1)+m_(2)))andr_(2)=r((m_(1))/(m_(1)+m_(2)))