15 .. elements
Find the no. of distinct elements in
{(1+ω+ω2+ω3+....+ωn)m :m,n=1,2,3,...}
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14 Answers
13yup...but watz the method??....only trial??
at first try i missed -ω,-ω2...:(
1simplifying the bracket u will get 3 options
....(1)m
....(1 + ω)m
... (0)m
so u get 2 ans 0 n 1
now (1 + ω)m = (-ω2)m = -ω2 , ω2 , -ω , ω , -1 , 1 for diff. values of m
13easy one....
\lim_{x\rightarrow \frac{\pi }{2}}sinx^{tanx}
1tan x ln sinx ... write it as ln sinx / cot x ... L hospital ... cot x / - cosec2 x .. .this quantity is 0 as sin x cos x = 0 ....... so limit = ε0 = 1
13thanks!
this one....
a,b,c real and distinct
no. of real soln of
(x-a)3+(x-b)3+(x-c)3=0
1let F(x)=(x-a)3+(x-b)3+(x-c)3
lim x->+∞ F(x)-> +∞
lim x->-∞ F(x)-> -∞
F'(x)=3(x-a)2+3(x-b)2+3(x-c)2
we see that F'(x)>0 for all real x in the domain which means that
F(x) is increasing in the domain (-∞,+∞)
and F(x) is differentiable and continuous in its domain
=> F(x) has one and only one real root in the domain..
cheers!!!
13a,b,c real and positive
prove that
\[ \frac{1}{b(a+b)}+\frac{1}{c(b+c)}+\frac{1}{a(c+a)}\geq\frac{27}{2(a+b+c)^{2}}. \]