show that `r_(2)r_(3) +r_(3) r_(1)+ r _(1) r_(2)=s^(2)`

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)

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)))

In the circuit shown here, E_(1) = E_(2) = E_(3) = 2 V and R_(1) = R_(2) = 4 ohms . The current flowing between point A and B through battery E_(2) is

A spherical capacitor consists of two concentric spherical conductors, held in position by suitable insulating supports (Fig. 2.34). Show that the capacitance of a spherical capacitor is given by C = (4 pi epsilon_(0)r_(1)r_(0))/(r_(1)-r_(2)) where r_(1) and r_(2) are the radii of outer and inner spheres, respectively.

Show that (b-c)/(r _(1))+ (c-a)/(r _(2))+(a-b)/(r _(3)) =0.

Write ‘True’ or ‘False’ and justify your answer: The volume of the frustum of a cone (1)/(3)pi h(r_(1)^(2)+r_(2)^(2)-r_(1)r_(2)) where h is the vertical height of the frustum and r_(1),r_(2) are the radii of the ends.

Prove that if two bubbles of radii r_(1) and r_(2)(r_(1)ltr_(2)) come in contact with each other then the radius of curvature of the common surface r=(r_(1)r_(2))/(r_(2)-r_(1))

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.

Two resistors of resistances R_(1)=100pm3 ohm and R_(2)=200pm4 ohm are connected (a) series , (b) in parallel. Find the equivalent resistance of the (a) series combination, (b) parallel combination. Use for (a) the relation R = R_(1)+R_(2) and for (b) (1)/(R') = (1)/(R_(1))+(1)/(R_(2)) and (DeltaR')/(R'^(2)) = (DeltaR_(1))/(R_(1)^(2))+(DeltaR_(2))/(R_(2))