Let `1+r+r^(2)+...=A` Where `|r|<1` then `1+2r+3r^(2)+4r^(3)+......=`

Let f:r rarr R, where f(x)=2^(|x|)-2^(-x) then f(x) is

Two circles of radii r_(1) and r_(2), are both touching the coordinate axes and intersecting each other orthogonally.The value of (r_(1))/(r_(2)) (where r_(1)>r_(2)) equals -

Let t_r=2^(r/2)+2^(-r/2) then sum_(r=1)^10 t_r^2=

Let S={(x,y) in R^(2):(y^(2))/(1+r)-(x^(2))/(1-r)=1} , where r ne pm 1 . Then S represents:

Let R_(1) and R_(2) be equivalence relations on a set A, then R_(1)uu R_(2) may or may not be

Let R_(1) and R_(2) be two relation defined as follows : R_(1) = { (a,b) in R^(2) , a^(2) + b^(2) in Q} and R_(2) = {(a,b) in R^(2) , a^(2) + b^(2) cancel(in)Q) where Q is the set of the rational numbers. Then:

The volume of the frustum of a cone is (1)/(3) pih [r_(1)^(2) +r_(2)^(2)- r_(1)r_(2)] , where h is vertical height of the frustum and r_(1), r_(2) are the radii of the ends.

Let A= {1, 2, 3, . . 20} . Let R_1 and R_2 two relation on A such that R_1 = {(a, b) : b is divisible by a} R_2 = {(a, b) : a is an integral multiple of b}. Then, number of elements in R_1 – R_2 is equal to ________.

Prove that r_(1)^(2)+r_(2)^(2) +r_(3)^(3) +r^(2) =16R^(2) -a^(2) -b^(2) -c^(2). where r= in radius, R = circumradius,, r_(1), r_(2), r_(3) are ex-radii.

Let the rth term t_(r) of a series is given by t_(r)=r/(1+r^(2)+r^(4)) , the value of lim_(ntooo)sum_(r=1)^(n)t_(r) is