CAT 2008: Quantitative Questions a Day 1 to 50 - The discussions

[sabsebadapaagal]

Originally Posted by srikar2097

I didn't get two things in your approach -
1) How did you declare that p,q,r,s is an increasing GP?

---it is given that ps=qr => p/q=r/s (common ratio... since the ratios are equal therefore we can tell that it is a GP ... now as p<q<r<s... there fore it is an incresing GP)

2) Initially, you had a^1/2(r^3/2 - 1). What happened to 'a^1/2'

We have
a^1/2(r^3/2 - 1)<1

for this to hold true at least one quantity should be less than 1... a^1/2 cant be less than 1(a is a positive integer... square root of all postive integers are greater than 1 thus (r^3/2 - 1) should be less than 1 )

Pls find my comment embedded above

[srikar2097]

Originally Posted by sabsebadapaagal

Pls find my comment embedded above

I see what you are saying...
But the inequality given is q less than or equal to r. How does this fit into your picture of increasing GP?

[sabsebadapaagal]

Originally Posted by srikar2097

I see what you are saying...
But the inequality given is q less than or equal to r. How does this fit into your picture of increasing GP?

I guess this is the mistake ... I didnt see equal to

anyways thanks alot

[phoenix19]

Originally Posted by Aarav

Here is the official solution to 001 - we might change this if by end of the day someone comes with a better approach

Solution:

Given a(a+1) + b(b+1) + c(c+1) + d(d+1) = 312 => (a-1)^2 + (b-1)^2 + (c-1)^2 + (d-1)^2 = 312 - 3S + 4 = 256. Let a-1 = A, b-1 = B, c-1 = C, d-1 = D => we have non-negative integers A, B, C, D such that A^2 + B^2 + C^2 + D^2 = (A+B+C+D)^2 (the LHS <= RHS always) and can only be true if all but one number is zero => three among a, b, c, d are 1 and fourth number is 17.

Thus (3) holds true - (2) can guarantee the values for c and d. For a=b we have exactly 2 determinable values - thus (1) is not true.

=> choice (5) is the right answer

HEY am new and following the thread only now... if its not a problem can someone explain the line i highlighted!!

[slam]

Originally Posted by sabsebadapaagal

It will be clear that p,q,r,s form an increasing GP.

Let a be the starting term of this GP and r be the common ratio.. now the terms come out to be a,ar,ar^2,ar^3

Now plugging these values in √s - √p <=1
√ar^3 - √a <=1

=> a^1/2(r^3/2 - 1) <=1

r^3/2 -1 <= 1

r^3/2 <=2

r<= 4^1/3

Now, applying this to both the conditions(X and Y) above we are not able to find the value of p

thus option 5

How did you decide on a GP? p/q = r/s, we do not know the value of q/r.

----------
slam.

[implex]

@srikar why haven't you considered p=2,3 and other integers which are not perfect squares??

[srikar2097]

Originally Posted by implex

@srikar why haven't you considered p=2,3 and other integers which are not perfect squares??

Because it's stated that s,p are positive integers and also that s^1/2 - p^1/2 =< 1. Which seems to be satisfied if s,p are perfect squares.

Although I must confess I don't understand the reason for "

[IIM maniac]

The question is followed by two statements X and Y. Answer each question using the following instruction:Choose 1 if the question can be answered by X onlyChoose 2 if the question can be answered by Y onlyChoose 3 if the question can be answered by either X or Y Choose 4 if the question can be answered by both X and YChoose 5 if the question can be answered by neither X nor Y
The positive integers are such that p < q = r < s < 100, ps = qr and vs - vp <= 1. What is the value of p?
(X) The last digit of s is either 1, 2 or 3
(Y) 50 < p and r < 90


Well, if we consider the first statement, i.e. X) The last digit of s is either 1, 2 or 3
the possible numbers would be
93,92,91
83, 82, 81
73, 72, 71
63,62, 61
53,52,51,
43,42,41
so on...
But it is given that Sqrt s - Sqrt p <= 1, then for S only choice left is 81 !! and in this case P would be 64, 49, 36, 25,16.. But if we consider the second condition, i.e.
Y) 50 < p and r < 90
then only chioce for P is 64 !

So Sqrt 81 - Sqrt64 = -1 (It satisfies the condition)

Now P*S = 5184,

from this we can find r & q,
we consider r=q, then only 72 satisfies the condition.

So Answer could be found out from two statesment together. Even thoguht you cant only assume r=q and if you dont find r & s still you will get the value of P as 64 from both statements.

Hence answer is Option 4

Correct me if i am wrong !!

[implex]

Originally Posted by srikar2097

Because it's stated that s,p are positive integers and also that s^1/2 - p^1/2 =< 1. Which seems to be satisfied if s,p are perfect squares.

Although I must confess I don't understand the reason for "

what ? I can't understand how are people assuming perfect squares?
sqrt(3)-sqrt(2)<=1
but neither oof them are perfect squares..

we can't assume perfect squares !!

[Aarav]

I was fearing putting as DS problem as last year we couldn't construct these with aplomb and people ended up getting the answers in 2 minutes. With 3 different answers already, I think I have done a fair job this time around