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Bunuel
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If ab^2 > 0 and ac < 0, which of the following must be true? 23 Jan 2019, 02:52
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D
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83% (00:58) correct 18% (01:04) wrong based on 280 sessions

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If \(ab^2 > 0\) and \(ac < 0\), which of the following must be true?


I. \(ab >0\)

II. \(b^2c < 0\)

III. \(a*c^2 > 0\)

A. I only
B. II only
C. III only
D. II and III only
E. I, II, and III

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Bunuel
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If ab^2 > 0 and ac < 0, which of the following must be true? 04 Jul 2019, 08:23
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If \(ab^2 > 0\) and \(ac < 0\), which of the following must be true?

I. \(ab >0\)

II. \(b^2c < 0\)

III. \(a*c^2 > 0\)


A. I only
B. II only
C. III only
D. II and III only
E. I, II, and III


For the product of two numbers to be positive, they must have the same sign. Since the square of a number is always greater than or equal to 0, then \(ab^2 > 0\) implies that \(a>0\) and \(b \neq 0\).

Since from the above \(a>0\), then \(ac < 0\) implies that \(c < 0\).

So, we have: \(a > 0\), \(c < 0\), and \(b \neq 0\).

Let's check the options:

I. \(ab >0\). Not true if \(b < 0\).

II. \(b^2c < 0\). Always true since \(b^2 > 0\) and \(c < 0\).

III. \(a*c^2 > 0\). Always true since \(a > 0\) and \(c^2 > 0\).


Answer: D
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If ab^2 > 0 and ac < 0, which of the following must be true? 23 Jan 2019, 04:02
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Bunuel
If \(ab^2 > 0\) and \(ac < 0\), then which of the following must be true?
I. \(ab >0\)

II. \(b^2c < 0\)

III. \(a*c^2 > 0\)

A. I only
B. II only
C. III only
D. II and III only
E. I, II, and III
Show more

IMO D

If \(ab^2 > 0\) and \(ac < 0\)

From above you can imply the following
a is +ive, b can be +ive or -ive and c is -ive

I. \(ab >0\), True or False , by using the above

II. \(b^2c < 0\), Always true

III. \(a*c^2 > 0\), Always true
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If ab^2 > 0 and ac < 0, which of the following must be true? 23 Jan 2019, 04:27
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Bunuel
If \(ab^2 > 0\) and \(ac < 0\), then which of the following must be true?


I. \(ab >0\)

II. \(b^2c < 0\)

III. \(a*c^2 > 0\)

A. I only
B. II only
C. III only
D. II and III only
E. I, II, and III
Show more

Given:
\(ab^2 > 0\) - Since b^2 must be positive only, a would be positive too (to get positive product)

and \(ac < 0\) - Since a is positive (obtained above), c must be negative to get negative product

So we have 'a' positive, 'c' negative and no idea about 'b'.

I. \(ab >0\)
Since we have no idea about b, we don't know whether ab will be positive or not.

II. \(b^2c < 0\)
b^2 will be positive (note that none of these is 0) and c is negative so product will be negative. This is true.

III. \(a*c^2 > 0\)
a is positive and c^2 will be positive too. So product is positive. This is true too.

Answer (D)
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If ab^2 > 0 and ac < 0, which of the following must be true? 23 Jan 2019, 05:41
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Bunuel
If \(ab^2 > 0\) and \(ac < 0\), then which of the following must be true?


I. \(ab >0\)

II. \(b^2c < 0\)

III. \(a*c^2 > 0\)

A. I only
B. II only
C. III only
D. II and III only
E. I, II, and III
Show more


Given,

\(ab^2>0.\)

a must be positive. a>0. It is not possible to tell whether b is positive or negative as \(b^2>0\). don't know anything about b.

ac<0. we know that a>0. So, c<0.

I. \(ab >0\)

a>0. don't know anything about b. could be true. MUST not one.

II. \(b^2c < 0\)

b>0 and c<0. Must be true.

III. \(a*c^2 > 0\)

a>0 and \(c^2>0\). Must be true.

D is the correct answer.
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If ab^2 > 0 and ac < 0, which of the following must be true? 23 Jan 2019, 06:40
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Bunuel
If \(ab^2 > 0\) and \(ac < 0\), then which of the following must be true?


I. \(ab >0\)

II. \(b^2c < 0\)

III. \(a*c^2 > 0\)

A. I only
B. II only
C. III only
D. II and III only
E. I, II, and III
Show more


for expression ab^2>0 & ac<0 to be valid
a +ve , b -ve or +ve and c -ve

in that case only
case
II. \(b^2c < 0\)

III. \(a*c^2 > 0\)
would be valid
IMO D
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If ab^2 > 0 and ac < 0, which of the following must be true? 01 Nov 2024, 23:19
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