1)Two real non negative numbers satisfy that ab>=a^3+b^3, find the maximum value of a+b
a) 1/2 b) 1 c) 3/2 d) 2 e) none of these
2) Let x(n) be a sequence of real numbers such that x(1)=2 and x(n+1)=2x(n)/3+1/(3x(n))
then for all n>1 which is always true
a) x(n) >1 b) x(n) <2 c) 1<x(n) <3/2 d) 1<x(n)<2 e) 3/2<x(n) <2
3) if p and q are real positive numbers such that p+q=1 then fidn the minimum value of (p+1/p)^2+(q+1/q)^2
a) 5/2 b) 25/2 c) 15/2 d) 6 e) none of these
p+q =1
ReplyDelete(p+1/p)^2 + (q+1/q)^2 = (p^2+q^2)+ (1/p^2+1/q^2) + 4
= (p+q)^2 - 2pq + (p+q)^2/(pq)^2 - 2/pq + 4
= (1-1/pq)^2 - 2pq + 4
now pq is max for p=q=1/2
= 25/2
correct but we can also do like this
ReplyDelete(p+1/p)^2 + (q+1/q)^2>=2[(p+1/p+q+1/q)/2]^2 using AM of mth power is greater than m the power of AM
=2[(1+1/p+1/q)/2]^2
using AM>=HM on p,q
1/p+1/q>=4
putting this we get
=2[5/2]^2=2.25/4=25/2
[...] Ahmedabad Study Group for CAT'09 - 27-08-2009, 07:50 PM Problems based on concept 2 1)Two real non negative numbers satisfy that ab>=a^3+b^3, find the maximum value of a+b a) 1/2 [...]
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