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Problems based on concept 2

with 3 comments


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

Written by Implex

August 27, 2008 at 8:08 am

Posted in Inequalities, Problems

Tagged with ,

3 Responses

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  1. p+q =1

    (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

    prade

    August 27, 2008 at 2:33 pm

  2. correct but we can also do like this
    (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

    outtimed

    August 27, 2008 at 3:23 pm

  3. […] 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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