`(2^(n).6^(m+1).10^(m-n).15^(m+n-2))/(4^(m).3^(2m+n).25^(m-1))`

Simplify : (2^n.6^(m+1).10^(m-n).15^(m+n-2))/(4^m.3^(m+n).25^(m-1)) .

If y=1/(1+x^(n-m)+x^(p-m))+1/(1+x^(m-n)+x^(p-n))+1/(1+x^(m-p)+x^(n-p)) , then (dy)/(dx) is equal to

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) (a) 1 (b) m^n.n^m (c) (m^m n^n)/((m+n)^(m+n)) (d) ((m n)^(m n))/((m+n)^(m+n))

m^(3)-n^(3)-m(m^(2)-n^(2))+n(m-n)^(2)

If the straigjht line x cos alpha+ y sin alpha=p touches the curve x^(m)y^(n)=a^(m+n) , prove that. p^(m+n)""m^(m)n^(n)=(m+n)^(m+n)a^(m+n)sin^(n) alpha cos^(m) alpha

Simplify : (1)/(1+x^(m-n)+x^(m-p))+(1)/(1+x^(n-p)+x^(n-m))+(1)/(1+x^(p-m)+x^(p-n)) .

IfI_(m , n)=int_0^(pi/2)sin^m xcos^n xdx , Then show that I_(m , n)=(m-1)/(m+n)I_(m-2,n)(m ,n in N) Hence, prove that I_(m , n)=f(x)={((n-1)(n-3)(m-5)(n-1)(n-3)(n-5))/((m+n)(m+n-2)(m+n-4))pi/4 when both m and n are even ((m-1)(m-3)(m-5)(n-1)(n-3)(n-5))/((m+n)(m+n-2)(m+n-4))}

Evaluate: lim_(nrarroo)(1^(m)+2^(m)+3^(m)+...+n^(m))/(n^(m+1))(mgt-1)

If L=lim_(nto oo) (n^(3)(e^(1//n)+e^(2//n)+………+e))/((n+1)^(m)(1^(m)+4^(m)+….+n^(2m))) is non zero finite real, then

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 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)