If `x = 1 + a + a^2 +`to`infty, y = 1 + b + b^2+ `to.`infty, |a| <1,|b|<1`. Find the sum of the series. `1+ab+a^2b^2+`…….to`infty`in terms of x and y.

Let x=1+a +a^(2) + ….. And y= 1 + b + b^(2)+ …… where |a| lt 1 and |b| lt 1 . Prove that: 1+ ab + a^(2)b^(2) +…… =(xy)/(x+y-1) .

Consider the curve y = tan^(-1) x " and a point "A ( 1 , pi/4) on it. If the variable point P_(i)(x_(i) , y_(i)) moves on the curve for i = 1,2,3 , …. N ( n in N) " such that " y_(i) = sum_(m=1)^(i) tan ^(-1) ( 1/( 2 m^(2))) and B ( x,y) be the limiting position of variable point P_(n) " as " n to infty , then the value of reciprocal of the slope of AB will be ____

Find the domain of the following y = tan^(-1) ( sqrt (x^(2) - 1)) a. (-infty,-2]cup[1, infty) b. (-infty,-1] c. (-infty,-1]cup[2, infty) d. (-infty,-1]cup[1, infty)

The complete set of values of a for which the function f(x)=tan^(-1)(x^(2)-18x +a)gt 0 AA x in R is a. (81, infty) b. [81,infty) c. (-infty,81) d. (-infty,81]

If f(x)=tan^2((pix)/(n^2-5n+8))+cot(n+m)pix;(n in N, m in Q) is a periodic function with 2 as its fundamental period, then m can't belong to a. (-infty,-2) cup (-1, infty) b. (-infty,-3) cup (-2, infty) c. (-2,-1) cup (-3, -2) d. (-3,-5/2) cup (-5/2, -2)

Find the area bounded by the ellipse x^2/a^2 + y^2/b^2 = 1 and the ordinates x=0 and x=ae , where b^2 = a^2 (1-e^2) and e<1 .

Let A and B be two sets such that n(A) = 3 and n(B) = 2. If (x, 1), (y, 2),(z, 1) are in A xx B , find A and B. where x, y and z are distinct elements.

If the curves ax^2 + by^2 =1 and a'x^2 +b'y^2 =1 intersect orthogonally, prove that: 1/a-1/a'=1/b-1/b'

If f:[1, infty) rarr [2, infty) is given by f(x)=x+1/x," find " f^(-1)(x) (assume bijection).