Show that `((1)/(r _(1))+ (1)/(r_(2)))((1)/(r_(2))+(1)/(r _(1)))((1)/(r _(3))+(1)/(r_(1)))=(64 R^(3))/(a ^(2)b^(2) c^(2))`

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

((n),(r))+((n),(r-1))=((n+1),(r-1))

Prove that ((n),(r))+2((n),(r-1))+((n),(r-2))=((n+2),(r))

((n),(r))+2.((n),(r-1))+((n),(r-2))=((n+2),(r))

The differintial eauation of all circle of radius r, is given by (a) {1+(y_(1))^(2)}^(2)=r^(2)y_(2)^(3) (b) {1+(y_(1))^(2)}^(3)=r^(2)y_(2)^(3) ( c ) {1+(y_(1))^(2)}^(3)=r^(2)y_(2)^(2) (d) None of these

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))

Sum of the series sum_(r=1)^(n) (r^(2)+1)r! is

If sum_(r=1)^(n)T_(r)=(n)/(8)(n+1)(n+2)(n+3)," find "sum_(r=1)^(n)(1)/(T_(r)) .

Let R_(t) represents activity of a sample at an insant and N_(t) represent number of active nuclei in the sample at the instant. T_(1//2) represents the half life. {:(,"Column I",,"Column II"),((A),t=T_(1//2),(p),R_(t)=(R_(0))/(2)),((B),t=(T_(1//2))/(ln2),(q),N_(0)-N_(t)=(N_(0))/(2)),((C),t=(3)/(2)T_(1//2),(r),(R_(t)-R_(0))/(R_(0)) = (1-e)/(e)),(,,(s),N_(t)=(N_(0))/(2sqrt(2))):}

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