Show that: `((a+1/b)^m xx\ (a-1/b)^n)/((b+1/a)^m\ xx\ (b-1/a)^n)=(a/b)^(m+n)`

Show that : ((a^(m))/(a^(-n)))^(m-n)xx((a^(n))/(a^(-1)))^(n-1)xx((a^(l))/(a^(-m)))^(l-m)=1

If the expansion in powers of x of the function 1//[(1-a x)(1-b x)] is a a_0+a_1x+a_2x^2+a_3x^3+ ,t h e na_n i s a. (b^n-a^n)/(b-a) b. (a^n-b^n)/(b-a) c. (b^(n+1)-a^(n+1))/(b-a) d. (a^(n+1)-b^(n+1))/(b-a)

Simplify: (i)\ ((x^(a+b))/(x^c))^(a-b)\ ((x^(b+c))/(x^a))^(b-c)\ ((x^(c+a))/(x^b))^(c-a) (ii)\ ((x^l)/(x^m))^(1/(lm))\ xx\ ((x^m)/(x^n))^(1/(mn))\ xx\ \ ((x^n)/(x^l))^(1/(ln))

The point of intersection of tangents drawn to the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1 at the points where it is intersected by the line l x+m y+n=0 , is ((-a^2l)/n ,(b^2m)/n) (b) ((-a^2l)/m ,(b^2n)/m) ((a^2l)/m ,(-b^2n)/m) (d) ((a^2l)/m ,(b^2n)/m)

The value of sum_(r=1)^n{(2r-1)a+1/(b^r)} is equal to (a) a n^2+(b^(n-1)-1)/(b^n(b-1)) . (b) a n^2+(b^(n)-1)/(b^n(b-1)) (c) a n^3+(b^(n-1)-1)/(b^n(b-1)) (d) none of these

The value of int_(a)^(b)(x-a)^(3)(b-x)^(4)dx is ((b-a)^(m))/(n). Then (m, n) is

("lim")_(xvec 0)((2^m+x)^(1/m)-(2^n+x)^(1/n))/xi se q u a lto 1/(m2^m)-1/(n2^n) (b) 1/(m2^m)+1/(n2^n) 1/(m2^(-m))-1/(n2^(-n)) (d) 1/(m2^(-m))+1/(n2^(-n))

IfI(m , n)=int_0^1x^(m-1)(1-x)^(n-1)dx ,(m , n in I ,m ,ngeq0),t h e n (a) I(m , n)=int_0^oo(x^(m-1))/((1+x)^(m-n))dx (b) I(m , n)=int_0^oo(x^(m-1))/((1+x)^(m+n))dx (c) I(m , n)=int_0^oo(x^(n-1))/((1+x)^(m+n))dx (d) I(m , n)=int_0^oo(x^n)/((1+x)^(m+n))dx

If a > b >0, with the aid of Lagranges mean value theorem, prove that n b^(n-1)(a-b) < a^n -b^n < n a^(n-1)(a-b) , if n >1. n b^(n-1)(a-b) > a^n-b^n > n a^(n-1)(a-b) , if 0 < n < 1.

If (9^n\ xx\ 3^2\ xx\ (3^(-n/2))^(-2)-\ 27^n)/(3^(3m)\ xx\ 2^3)=1/(27) , prove that m-n=1