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22 Mar 2011, 11:05
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B
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Difficulty:
95%
(hard)
Question Stats:
32% (01:42) correct
68%
(01:58)
wrong
based on 412
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If abc ≠ 0, what is the value of \(\frac{a^3 + b^3 + c^3}{abc}\) ?
(1) \(|a|=1, |b|=2, |c|=3\)
(2) \(a + b + c = 0\)
(1) \(|a|=1, |b|=2, |c|=3\)
(2) \(a + b + c = 0\)
22 Mar 2011, 11:28
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Statement 1:
|a|=1, |b|=2, |c|=3
So a, b, c can be positive or negative. Thus, the value of the expression cannot be uniquely determined.
NS
Statement 2:
a + b + c = 0
(a + b + c)^3 = a^3 + b^3 + c^3 + (3ab + 3bc + 3ca) (a + b + c) – 3abc
Substitute for a + b + c = 0 in the above equation:
0 = a^3 + b^3 + c^3 - 3abc
=> 3abc = a^3 + b^3 + c^3
=> (a^3 + b^3 + c^3)/abc = 3
Sufficient
|a|=1, |b|=2, |c|=3
So a, b, c can be positive or negative. Thus, the value of the expression cannot be uniquely determined.
NS
Statement 2:
a + b + c = 0
(a + b + c)^3 = a^3 + b^3 + c^3 + (3ab + 3bc + 3ca) (a + b + c) – 3abc
Substitute for a + b + c = 0 in the above equation:
0 = a^3 + b^3 + c^3 - 3abc
=> 3abc = a^3 + b^3 + c^3
=> (a^3 + b^3 + c^3)/abc = 3
Sufficient
General Discussion
22 Mar 2011, 11:29
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since abc ≠ 0 , we know that either all numbers can be +ve or any two can be -ve
Option A does not help to solve the problem since we can not make out that which numbers can be -ve and value changes with every trial and error method. Therefore, A can not be the answer.
Option B tells us that A+B+C=0, therefore we can use the formula that
\(a^3 + b^3 + C^3 - 3abc = (a+b+c) (a^2 + b^2 + c^2 - ab - bc - ac)\)
and since we know that a+b+c = 0 then \(a^3 + b^3 + c^3 = 3abc\)
And thus putting "3abc" at the place of \(a^3 + b^3 + c^3\), will give us the answer.
Therefore answer is "B"
Only thing that since I know the formula thats why I could solve this problem within seconds ..... how to solve if we do not know the formula .... please someone explain.
Option A does not help to solve the problem since we can not make out that which numbers can be -ve and value changes with every trial and error method. Therefore, A can not be the answer.
Option B tells us that A+B+C=0, therefore we can use the formula that
\(a^3 + b^3 + C^3 - 3abc = (a+b+c) (a^2 + b^2 + c^2 - ab - bc - ac)\)
and since we know that a+b+c = 0 then \(a^3 + b^3 + c^3 = 3abc\)
And thus putting "3abc" at the place of \(a^3 + b^3 + c^3\), will give us the answer.
Therefore answer is "B"
Only thing that since I know the formula thats why I could solve this problem within seconds ..... how to solve if we do not know the formula .... please someone explain.
