1) If the equation x 4-4x 3+ax 2+bx+1=0 has four positive roots, then show that a=6 and b=-4
2) If S = a 1+a 2+a 3+..........+an then show that s/(s-a1) + s/(s-a2) +s/(s-a3) + ...............+ s/(s-a n) > n2 /( n-1).
3) If p be the first term of the n arithmetic means ; q be the first of the n harmonic means b/w the same two numbers, prove that the value of q can't lie b/w p &[ (n-1) / (n+1)] 2 .
4) If (2n + r ) , where n,r belongs to N, is expressed as the sum of K consecutive odd natural numbers , then show that K is equals to r.
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2 Answers
62Question 1
the sum of roots is 4
the product of roots is 1
let the roots be x1, x2, x3, x4
observe that AM=1 and GM =1
hence we have that AM=GM.. so all four roots should be equal. and they should be 1 each.
a=Σxixj = 6
b=-Σxixjxk = -4
622) If S = a 1+a 2+a 3+..........+an then show that s/(s-a1) + s/(s-a2) +s/(s-a3) + ...............+ s/(s-a n) > n2 /( n-1).
Take AM HM inequality on s/(s-a1) , s/(s-a2) and so on..
thus,
\frac{\frac{s}{s-a_1}+\frac{s}{s-a_2}+...+\frac{s}{s-a_n}}{n}\geq \frac{n}{\frac{s-a_1}{s}+\frac{s-a_2}{s}+...+\frac{s-a_n}{s}}
Thus, \large \frac{s}{s-a_1}+\frac{s}{s-a_2}+...+\frac{s}{s-a_n}\geq \frac{n^2}{\frac{s-a_1+s-a_2+...+s-a_n}{s}}
Thus, \large \frac{s}{s-a_1}+\frac{s}{s-a_2}+...+\frac{s}{s-a_n}\geq \frac{n^2}{\frac{(n-1)s}{s}}
hence proved :)
