CAT 2008: Quantitative Questions a Day 51-100 -The discussions-Brought to you by SMOT

[themystique]

assume p = 2 + q
because in the given equation we have to maximize (p+q)..

substitute in the equation.

2/(p+q) +q/2 =k

= 2/(2+2q) +q/2 = k
1/(1+q) +q/2 =k

differentiating the above eqn,

-1/(1+q)^2 +1/2 = 0
from this we get
q= sqrt(2) -1

substituting back we get the value of k as sqrt(2) - 1/2.
option 1.

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[Fuzon]

i dont know if i am being too intuitive but heres my line of thought

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Quantitative Question # 051
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Given p and q be positive such that 2 >= p-q, the min value of 2/(p+q) + q/2 is


(1) √2 - 1/2 (2) (√2 + 1)/2 (3) 1 (4) 1/√2 (5) none of these

p-q<=2

For

2/(p+q) + q/2 to be minimum , q has to be minimum so that q/2 is minimum and p has to be maximum so that 2/(p+q) is minimum

if we take p=3 and q=1 , we have value of the expression as 1

if we take a negative -3 and -1 we get the value as -1 which is further more less than 1

so i think that the answer is oprion 5 , none of these (not sure whether my line of thinking is correct)

puys throw some light one the same agar mein bhatak raha hu to ...

[neynetr]

Originally Posted by thebornattitude

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Quantitative Question # 051
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Given p and q be positive such that 2 >= p-q, the min value of 2/(p+q) + q/2 is


(1) √2 - 1/2 (2) (√2 + 1)/2 (3) 1 (4) 1/√2 (5) none of these

Solution-

Let p=q+2-m.......(i)

so equation becomes ,

2/(2+2q-m) +q/2.....(ii)

now we have to find a way of writing q/2 in the form of denominator of 1'st term, then only we can apply any inequality,else by putting values we cant find, as no of dependents are higher..

so q/2 can be written as
(2+2q-m)/4 + (m-2)/4
put this is (ii) to get

2/(2+2q-m) + (2+2q-m)/4 + (m-2)/4.......(iii)

now take 1'st 2 terms,,,and apply AM>GM, we get

2/(2+2q-m) + (2+2q-m)/4 >=2/root(2)= root(2).....(iv)

so, we get
2/(2+2q-m) + (2+2q-m)/4 + (m-2)/4>= root(2) + (m-2)/4

this will be further minimized when m is taken to be minimum ie m=0....hence


2/(2+2q-m) + (2+2q-m)/4 + (m-2)/4>=root(2) -1/2.......

hence the answer- option (1)

ur method is very gud...

but what if i dont consider any m and move directly with the equation 2/(p+q) +q/2

writing this as:

2/(p+q) + (p+q)/2 - p/2

then i get the min value of the preivous two terms as 2.

so 2 - p/2...

now as i have used the condition AM = GM
so (p+q)/2 = 2/(p+q)
=> p+q=2

but p-q <=2 and morover to minimise the value of the xpression i need -p/2 element maximum.
so for that i choose p = 2 and q = 0.

that'll giv min value 2-1 = 1

bit confused right now as by the quadratic approach i got the answer sqrt(2)-1/2 (my previous post)

i think im going wrong somewhere but dont know where...!!!
any suggestions....???

[thebornattitude]

Originally Posted by neynetr

ur method is very gud...

but what if i dont consider any m and move directly with the equation 2/(p+q) +q/2

writing this as:

2/(p+q) + (p+q)/2 - p/2

....


so for that i choose p = 2 and q = 0.

that'll giv min value 2-1 = 1

bit confused right now as by the quadratic approach i got the answer sqrt(2)-1/2 (my previous post)

i think im going wrong somewhere but dont know where...!!!
any suggestions....???

q cant be equal to 0,,it has to be a positive number.....and moreso,,,1 can be a value,,,but its not the minimum value..

[nbangalorekar]

This was a very tricky question.... had solved before and got the answer as 1/2 but then realized my mistake... thanks Srikar...
Aarav i have a doubt in this qn.... many ppl have solved this qn and got answers with approaches that seem to be perfect... however many of the answers are wrong... including mine....
can you please point out why this has happened with this qn? what has gone wrong in my method??
feel like banging my head

[sushantgupta]

hey aarav this request from my side also....please if you can tell why many poeple including me getting 1 as the answer even though the approach seems right...wud be looking forward to learn where we all are doing it wrong

[sanyo]

Quantitative Question # 051

Ans 1) √2 - 1/2
Solution:

p-q <=2 and p & q both are positive intergers
=> p-q = 2 or
p-q = 1 or
p-q = 0

Now 2/(p+q) + q/2 can be written as [4 + q(p-q +2q)]/[ 2 (p-q +2q)].........(1)

now the first case p-q =2, substituting in (1) we get..
[4 + 2q( 1+q)]/ 4(1+q) ...........(2)
differentiating the eqn (2)and equate it to zero, we get q = root2 -1,
substituting the same in eqn (2), we get root2 - 1/2 (approx 0.9)..........(a)

now the second case p-q =1, substituting in (1) we get..
[4+q(1+2q)] / 2(1+2q).........(3)
differentiating the eqn (3)and equate it to zero, we get q =(2root2 -1)/2,
substituing the same in eqn (3) we get (8 - root2)/4root2 (approx 1.1 .....(b)

now the third case p-q =0, => p = q , substituting in (1) we get..
2/2q + q/2 = 1/q + q/2 .........(4)
differentiating the eqn (4)and equate it to zero, we get q = root2
substituing the same in eqn (4) we get root2 (approx 1.4).......(c)

minimum of (a), (b) and (c) is root2 - 1/2 , hence the answer...

[kk_sonam]

my take is..

2>=p-q
=> 2+2q>=p+q
=> 1/(2+2q)<=1/(p+q)
=> q/(2+2q)<=q/(p+q)

now,

we have to find the min of 2/(p+q) + q/2

applying a.m >= g.m.

we get,

2/(p+q) +q/2 >= 2(q/(p+q))^(1/2)
>= (2q/(1+q))^(1/2)
= (2^(1/2) )*(1-1/(1+q))^(1/2)
>=0 for min, q=0

hence option e

doesnt seem the ans is correct
can anyone please point out my mistake

[rahulchauhan03]

Given p and q be positive such that 2 >= p-q, the min value of 2/(p+q) + q/2 is

(1) √2 - 1/2 (2) (√2 + 1)/2 (3) 1 (4) 1/√2 (5) none of these

Answer

AM>=GM
thus (p+q)/2 >= (pq)^1/2
so minimum value of (p+q)/2 is (pq)^1/2
so we can write the expression which needs to be minimum as

1/(pq)^1/2 + q/2

now p-q<=2 , for p-q = 0,1,2
if p=q then p-q =0 ; q shud be minimum to get the min result that is q=1=p
so result is
1/(pq)^1/2 + q/2 = 1+1/2 = 3/2 = 1.5 ------1
for p-q =1 and q =1 ( to make q/2 minimum) ; p can be 2,3
for p=2
1/(pq)^1/2 + q/2 = 1/sqrt(2) + 1/2
= sqrt(2)/2 + 1/2= (sqrt(2)+1)/2 =1.207 ----2

for p =3 and q =1
1/(pq)^1/2 + q/2= 1/ sqrt(3) +1/2 = 0.64+0.5 = 1.14 ---- 3

So it should be none of these.

[sabsebadapaagal]

Originally Posted by Aarav

Can all those who are claiming the answer as 1/√2 tell the corresponding values of p and q?

P=3.33 and Q=0.33