Show that `(r_(1)+ r _(2))(r _(2)+ r _(3)) (r_(3)+r_(1))=4Rs^(2)`

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

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

Show that |{:(sum_(r=1)^(16)2^r,a,2(2^(16)-1)),(3sum_(r=1)^(16)4^r,b,4(4^(16)-1)),(7sum_(r=1)^(16)8^r,c,8(8^(16)-1)):}|=0

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))

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

A=[{:(l_(1),m_(1),n_(1)),(l_(2),m_(2),n_(2)),(l_(3),m_(3),n_(3)):}] and B=[{:(p_(1),q_(1),r_(1)),(p_(2),q_(2),r_(2)),(p_(3),q_(3),r_(3)):}] Where p_(i), q_(i),r_(i) are the co-factors of the elements l_(i), m_(i), n_(i) for i=1,2,3 . If (l_(1),m_(1),n_(1)),(l_(2),m_(2),n_(2)) and (l_(3),m_(3),n_(3)) are the direction cosines of three mutually perpendicular lines then (p_(1),q_(1), r_(1)),(p_(2),q_(2),r_(2)) and (p_(3),q_(),r_(3)) are

An equilateral triangle has each of its sides of length 4 cm. If (x_(r),y_(r)) (r=1,2,3) are its vertices the value of |{:(x_(1),y_(1),1),(x_(2),y_(2),1),(x_(3),y_(3),1):}|^2

Statement -1 If Delta(r)=|{:(r,r+1),(r+3,r+4):}| then sum_(r=1)^(n) Delta(r)=-3n Satement-2 If Delta(r)=|{:(f_(1)(r),f_(2)(r)),(f_(3)(r),f_(4)(r)):}| Sigma_(r=1)^(n) Delta (r)={:abs((Sigma_(r=1)^(n)f_1(r),Sigma_(r=1)^(n)f_2(r)),(Sigma_(r=1)^(n)f_3(r),Sigma_(r=1)^(n) f_4(r))):}