AkshdeepS wrote:
If \(abc > 0\), then which of the following must be true?
a. \(\frac{a}{b} < 0\)
b. \(a > 0\)
c. \(\frac{ab}{c} > 0\)
d. \(bc < 0\)
e. \(a > bc\)
Let us first analysis all the possible cases of "\(abc > 0\)".
1) \(a > 0 , b > 0 , c > 0\), then \(abc > 0\) (always)
\(+ + +\)
2) \(a < 0 , b < 0 , c > 0\), then also \(abc > 0\)
\(- - +\)
Case 2 has two more variations as any 2 of the 3 variables can be positive and one negative
2.1) \(a < 0 , b > 0 , c < 0\)
\(- + -\)
2.2) \(a > 0 , b < 0 , c < 0\)
\(+ - -\)
All the above cases are those which are true as per given condition in the question.
Now we can check options which are 100% true according to above given information.
Remember we are not looking for could be true options. Let us try to prove the options false and eliminate one by one
a. \(\frac{a}{b} < 0\)
Not true for case (2)
As per case (2) a and b both are negative, and if we divide -a by -b, we well get positive result. Eliminate
b. \(a > 0\)
As per case (2) and (2.2) this information is not true. Eliminate
c. \(\frac{ab}{c} > 0\)
1) \((+)*(+)/(+) > 0\)
2)\((-) * (-)/ (+) > 0\)
2.1 \((-)* (+) / (-) > 0\)
2.2) \((+) *(-)/ (-) > 0\)
This information is true for every case. Correct.
d. \(bc < 0\)
As per case (1) and (2.2) this information is not true. Eliminate
e. \(a > bc\)
This information may or may not be true as no real values are known.
If a = 10 , b = 2, c = 1 (True)
If a = -5 , b = 2, c = -1
\(abc > 0\), but a is not greater than bc. (False)Hope this helps.
As there are lot of signs there may be some typos