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

[mehrozkhn]

sol:
let A = [2/(p+q) + q/2]
we have to find minimum value of A
now p-q <= 2
we can see
p+q = p - q + 2q
since q is positive
p + q >= p-q
=> p + q >= 2
=> 2/(p+q) >= 1 ........(1)
now minimum value of q is equal to zero
hence A(min) = 1 + 0
= 1
hence answer is (3)

[swati1606]

p-q<=2
to find: 2/p+q +q/2
writing it in the form:
(4+pq+q^2)/2(p+q)

to find the minimum of this expression, the denominator should be maximizes and the numerator minimized

tackling the denominator first,
2(p+q)=2(p-q)+4q
max value of p-q=2
hence denominator reduces to 4(q+1)--------- 1.

now going for the numerator

4+q(p+q)= 4+q(p-q+2q)
since p-q has been taken=2 to maximize denominator, this numerator expression reduces to:
4+2q(q+1)---------2.


the overall (numerator/denominator) form becomes(dividing 2. by 1.)
= 1/(q+1) + q/2

now apply maxima minima method to obtain the minimum value of this expression.

it comes out to be option (a) and thats my take to this problem.

[nbangalorekar]

Originally Posted by sabsebadapaagal

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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

q>=p-2
substitute in required eq: {since our main concern shud be reducing 'q' as it is in numerator}
2/(2p-2) + (p-2)/2 = k {'k' is a variable}
p^2 -p(3+2k)+(3+2k)=0
apply b^2>=4ac
(3+2k)^2>=4(3+2k)
3+2k>=4
min value of k=1/2
so answer is (5) none

[srikar2097]

q>=p-2
substitute in required eq: {since our main concern shud be reducing 'q' as it is in numerator}
2/(2p-2) + (p-2)/2 = k {'k' is a variable}
p^2 -p(3+2k)+(3+2k)=0
apply b^2>=4ac
(3+2k)^2>=4(3+2k)
3+2k>=4
min value of k=1/2
so answer is (5) none

If we get k=1/2, then value of p=2 and q is either 0 or 1. So we actually do get some values i.e. 1 or 7/6. Right. So acc. to your method we should get choice (3) as the answer?

But, I didn't understand how taking the discriminant will give the least value? Can you explain?

[shivam_01]

Originally Posted by mehrozkhn

sol:
let A = [2/(p+q) + q/2]
we have to find minimum value of A
now p-q <= 2
=> (p+q)^2 - 2pq <= 4
=> p+q <= sqrt(2pq+4)
=> 2/(p+q) >= 2 /(2pq+4)^2
=> 2/(p+q) >= sqrt[2/(pq+2)] ....(1)

now using AM-GM inequality we get
(p+q)/2 >=sqrt (pq)
=> 1/sqrt(pq) >= 2/(p+q)
=> 1/sqrt(pq) >= sqrt[2/(pq+2)] using (1)
=> (2pq + 4) >= 4pq
=> pq<= 2 ....(2)
now using (1) and (2)
A(min) = sqrt[2/(2+2)] + 0 (since minimum value of q is zero)

= 1/sqrt 2

hey u have taken that one wrong
p-q <= 2
=> (p+q)^2 - 2pq <= 4
this should be written as
(p+q)^2-4pq <= 4

[nbangalorekar]

Originally Posted by srikar2097

If we get k=1/2, then value of p=2 and q is either 0 or 1. So we actually do get some values i.e. 1 or 7/6. Right. So acc. to your method we should get choice (3) as the answer?

But, I didn't understand how taking the discriminant will give the least value? Can you explain?

why cant we use the discriminant to arrive at the answer? 'p' has to be a real valued. so naturally when we use the discriminant we get a real value for 'p'.
the least value for the expression can thus be arrived at.
Aarav is my approach flawed?
my only concern is that iv assumed that to arrive at the least value for the expression we will have to minimize 'q'.... thats bcos the first part [2/(p+q)] wont affect so much as th second part {q/2} will as 'q' appears in numerator..

[itsmyturn]

P+q should be maximum to get the minimum of the given expression.. so p has to be q+2;
substituting this & making the derivative d/dq(given expression) = 0 gives q = sqrt(2) - 1;
substituting this value v get the ans as sqrt(2) - 1/2 which is option(a)
option(A)

[ankaj]

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 theseP-q

[dewan_iitr]

taking the extreme condition - p-q =2

f = 2/(p+q) + q/2
= 1/(p-1) + p/2 -1
Differnetiate

so p = 1+ root2
f= root2 - 1/2

Option (1)

is this correct??.. i am not very sure..but will go with this for now

[slam]

Originally Posted by sabsebadapaagal

------------------------------------------------------
Quantitative Question # 051
------------------------------------------------------

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
=> q = p + x - 2 , x > 0
= p + K, K = x-2

Substitute this in the expression,
exp = [ (2)p^2 + (3K)p + (K^2 + 4) ] / (4p + 2K)

Differentiate this for max/min, we get:
4p^2 + (4K)p + (K^2 - = 0
This gives:
p = (-K + 2sqrt(2))/2 (the other root is rejected to keep p and q positive)
=> 2p + K = 2 sqrt(2) = p + q

Put this back in the expression,
exp = 2/(2sqrt(2)) + (p + K)/2
= 1/sqrt(2) + (p + K)/2

To keep this value minimum, I would get exp = 1/sqrt(2).
(Question here is, is it valid to take p tending to zero, given that p is positive. Comments?)