`("lim")_(xto0)((2^m+x)^(1/m)-(2^n+x)^(1/n))/x is equal t o` (a) 2`(1/(m2^m)-1/(n2^n))' (b) `(1/(m2^m)+1/(n2^n))` (c)`1/(m2^(-m))-1/(n2^(-n))` (d) `1/(m2^(-m))+1/(n2^(-n))`

lim_(xto1)(x^(m)-1)/(x^(n)-1) is

Compute lim_(xto1)(x^(m)-1)/(x^(n)-1) .

sum_(m-1)^ntan^(-1)((2m)/(m^4+m^2+2)) is equal to (a) tan^(-1)((n^2+n)/(n^2+n+2)) (b) tan^(-1)((n^2-n)/(n^2-n+2)) (c) tan^(-1)((n^2+n+2)/(n^2+n)) (d) none of these

The value of ^n C_1+^(n+1)C_2+^(n+2)C_3++^(n+m-1)C_m is equal to (a) ^m+n C_(n-1) (b) ^m+n C_(n-1) (c) ^mC_(1)+^(m+1)C_2+^(m+2)C_3++^(m+n-1) (d) ^m+1C_(m-1)

lim_(xto oo)((1)/(1-n^(2))+(2)/(1-n^(2))+ . . .+(n)/(1-n^(2))) is

Prove that tan^(-1)((m)/(n))-tan^(-1)((m-n)/(m+n))=(pi)/(4).

A function f is defined by f(x)=|x|^m|x-1|^nAAx in Rdot The local maximum value of the function is (m ,n in N), 1 (b) m^nn^m (m^m n^n)/((m+n)^(m+n)) (d) ((m n)^(m n))/((m+n)^(m+n))

If 2 cos alpha = x + (1)/(x) and 2 cos beta = y + (1)/(y) , show that (x^(m))/(y^(n))- (y^(n))/(x^(m))= 2i sin (m alpha - n beta)

Let m, in N and C_(r) = ""^(n)C_(r) , for 0 le r len Statement-1: (1)/(m!)C_(0) + (n)/((m +1)!) C_(1) + (n(n-1))/((m +2)!) C_(2) +… + (n(n-1)(n-2)….2.1)/((m+n)!) C_(n) = ((m + n + 1 )(m+n +2)…(m +2n))/((m +n)!) Statement-2: For r le 0 ""^(m)C_(r)""^(n)C_(0)+""^(m)C_(r-1)""^(n)C_(1) + ""^(m)C_(r-2) ""^(n)C_(2) +...+ ""^(m)C_(0)""^(n)C_(r) = ""^(m+n)C_(r) .

In a game a coin is tossed 2n+m times and a player wins if he does not get any two consecutive outcomes same for at least 2n times in a row. The probability that player wins the game is a. (m+2)/(2^(2n)+1) b. (2n+2)/(2^(2n)) c. (2n+2)/(2^(2n+1)) d. (m+2)/(2^(2n))