For natural numbers `m ,n ,if(1-y)^m(1+y)^n=1+a_1y+a_2y^2+ ,a n da_1=a_2=10 ,t h e n` `m n` c. `m+n=80` d. `m-n=20`

Let T_r be the rth term of an A.P., for r=1,2,3, If for some positive integers m ,n , we have T_m=1/na n dT_n=1/m ,t h e nT_(m n) equals 1/(m n)

Let T_r a n dS_r be the rth term and sum up to rth term of a series, respectively. If for an odd number n ,S_n=na n dT_n=(T_n-1)/(n^2),t h e nT_m ( m being even)is a. 2/(1+m^2) b. (2m^2)/(1+m^2) c. ((m+1)^2)/(2+(m+1)^2) d. (2(m+1)^2)/(1+(m+1)^2)

Let (x , y) be such that sin^(-1)(a x)+cos^(-1)(y)+cos^(-1)(b x y)=pi/2dot Match the statements in column I with statements in column II Ifa=1a n db=0,t h e n(x , y) , (p) lies on the circle x^2+y^2=1 Ifa=1a n db=1,t h e n(x , y) , (q) l i e son(x^2-1)(y^2-1)=0 Ifa=1a n db=2,t h e n(x , y) , (r) lies on y=x Ifa=2a n db=2,t h e n(x , y) , (s) lies on (4x^2-1)(y^2-1)=0

If f(x) satisfies the relation f(x+y)=f(x)+f(y) for all x , y in Ra n df(1)=5,t h e n (a) f(x)i sa nod dfu n c t ion (b) f(x) is an even function (c) sum_(n=1)^mf(r)=5^(m+1)C_2 (d) sum_(n=1)^mf(r)=(5m(m+2))/3

If f and g are two functions defined on N , such that f(n)= {{:(2n-1if n is even), (2n+2 if n is odd):} and g(n)=f(n)+f(n+1) Then range of g is (A) {m in N : m= multiple of 4 } (B) { set of even natural numbers } (C) {m in N : m=4k+3,k is a natural number (D) {m in N : m= multiple of 3 or multiple of 4 }

Find the acute angle between the lines (x-1)/l=(y+1)/m=1/na n d=(x+1)/m=(y-3)/n=(z-1)/lw h e r el > m > n ,a n dl ,m ,n are the roots of the cubic equation x^3+x^2-4x=4.

Let T_r denote the rth term of a G.P. for r=1,2,3, If for some positive integers ma n dn , we have T_m=1//n^2 and T_n=1//m^2 , then find the value of T_(m+n//2.)

If tan x = (n)/(n + 1) and tan y = (1)/(2n + 1) , find tan (x + y).

If x and y are positive real numbers and m, n are any positive integers, then Prove that (x^n y^m)/((1+x^(2n))(1+y^(2m))) lt =1/4

In X-Y plane, the path defined by the equation (1)/(x^(m))+(1)/(y^(m)) +(k)/((x+y)^(n)) =0 , is a hyperbola if m =1, k =- 1, n=0