If a,b,c,d are integers such that both (-2a) * (−3c) * b and

Hi,
good Q..

\((-2A) * (−3C) * B\) and \(D * (-2B) * 5A\) are positive
so their product should also be positive..
therefore \((-2A) * (−3C) * B * D * (-2B) * 5A\) is positive..
0r \(A^2*B^2*C*D*(-60)\) IS POSITIVE..
so \(C*D*(-60)\) should be positive..
or C*D should be negative
C
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A question that makes you realize the pattern therein.

You are given that \((-2a) * (−3c) * b\) > 0 ----> abc > 0 ....(1)

and \(d * (-2b) * 5a\) ----> abd < 0 ...(2)

From 1 and 2, 2 cases are possible

1. Case 1, a (+), b(+), c(+), d(-) ---> eliminate A,B and E.

2. Case 2, a (-), b(-), c(+), d(-) ---> eliminate D.

Thus C is the only remaining option and is hence the correct answer.

Hope this helps.

Don't pay attention to quotients. They don't have any influence.

To make the expressions easier for understanding rewrite them in the following way:

Expression abc = positive
Expression abd = negative

In order to be positive and negative correspondingly 'abc' and 'abd' components have to have following set of signs:
abc abd
+++ ++-
+-- +-+
-+- -++
--+ ---

So, we can see that for each pair c and d always have different signs. Therefore for all possible options c*d must be negative.

Nice question
Since (-2a)*(-3c)*b> 0 => 6*a*b*c>0 = > a*b*c>0----1
d*(-2b)*(5a)>0 => -10*d*b*a>0- ==> d*b*a<0 ----2

Now let us look at the choices
A and B -- a*b and b*c . Well a*b can be positive in which case c>0 and a*b<0 in which case c<0 . The same thing applies to choice B and hence both are not correct choices
> 0
Let us look at D and E - a*d and a*c respectively. For reasons noted above if b<0 then a*c<0 but if b>0 then a*c >0. Also a*d for the same reason can be +ve and -ve.
Which brings us to C. At this stage we can check this answer and go.. But let us see what is going on

if c<0 then from inequality (1) a*b*c> 0 ==> that a*b <0 . From inequality (2) d*b*a<0 we see that if a*b<0 then d>0 ===> c*d<0
if c>0 then a*b>0 . From (2) d<0 ==> again c*d<0 so answer choice C!

A simple analysis will help you.

The given expressions are (ignoring co-efficients):
abc is positive.
-abd is positive.

Note that both have ab common so whatever sign that brings, will be same in both products. If c and d had the same signs, abc and abd would have the same sign so abc and -abd would have opposite signs.
So c and d must have opposite signs so that both abc and -abd have the same sign (positive).

Answer (C)
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