`(n^a/n^b)^(1/(ab)).(n^b/n^c)^(1/(bc)).(n^c/n^a)^(1/(ca))=?`

If I=int ((((sinx)^n-sinx)^(1/n) cosx)/((sinx)^(n+1) )) dx is eqaul to (a) (n/(n^2-1))(1-1/(sinx^(n-1)))^(1/n+1)+c (b) (n/(n^2+1))(1-1/(sinx^(n-1)))^(1/n+1)+c (c) (n/(n^2+1))(1-1/(sinx^(n-1)))^(1/n)+c (d) (n/(n^2-1))(1-1/(sinx))^(1/n+1)+c

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 sum_(r=0)^n(a+r+a r)(-a)^r is equal to a. (-1)^n[(n+1)a^(n+1)-a] b. (-1)^n(n+1)a^(n+1) c. (-1)^n((n+2)a^(n+1))/2 d. (-1)^n(n a^n)/2

The coefficient of 1//x in the expansion of (1+x)^n(1+1//x)^n is (a). (n !)/((n-1)!(n+1)!) (b). ((2n)!)/((n-1)!(n+1)!) (c). ((2n)!)/((2n-1)!(2n+1)!) (d). none of these

(d^n)/(dx^n)(logx)=? (a) ((n-1)!)/(x^n) (b) (n !)/(x^n) (c) ((n-2)!)/(x^n) (d) (-1)^(n-1)((n-1)!)/(x^n)

If a >0,b >0 then (lim)_(n->oo)((a-1+b^(1/n))/a)^n= b^(1//a) b. a^(1/b) c. a^b d. b^a

if (1+a)^(n)=.^(n)C_(0)+.^(n)C_(1)a++.^(n)C_(2)a^(2)+ . . .+.^(n)C_(n)a^(n) , then prove that (.^(n)C_(1))/(.^(n)C_(0))+(2(.^(n)C_(2)))/(.^(n)C_(1))+(3(.^(n)C_(3)))/(.^(n)C_(2))+. . . +(n(.^(n)C_(n)))/(.^(n)C_(n-1))= Sum of first n natural numbers.

Find .^(n)C_(1)-(1)/(2).^(n)C_(2)+(1)/(3).^(n)C_(3)- . . . +(-1)^(n-1)(1)/(n).^(n)C_(n)

The value of lim_(ntooo)(e^(n))/((1+(1)/(n))^(n^(2))) is (a) -1 (b) 0 (c) 1 (d) ∞

The standard deviation of the data: x : , 1, a , a^2 , ..., a^n f: , ^n C_0 , ^n C_1 , ^n C_2 , ..., ^n C_n is ((1+a^2)/2)^n-((1+a)/2)^(2n) (b) ((1+a^2)/2)^(2n)-((1+a^2)/2)^(2n) (c) ((1+a)/2)^(2n)-((1+a^2)/2)^n (d) none of these