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28 Feb 2016, 00:07
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C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
75%
(hard)
Question Stats:
43% (02:02) correct
57%
(02:33)
wrong
based on 60
sessions
History
Date
Time
Result
Not Attempted Yet
\((|b| + k)(|b| + l)(|b| + m)(|b| + n) = 0\)⋯⋯ (I)
\((c + k)(c + l)(c + m)(c + n) = 0\)⋯⋯ (II)
Set X only contains all possible solutions of equation (I) above and set Y only contains all possible solutions of equation (II) above. Given that k, l, m and n are unequal numbers, does the number of elements in set X equal the number of elements in set Y?
(1) \(k\) and \(l\) are positive numbers
(2) \(m\) and \(n\) are negative numbers
Source: Optimus Prep
\((c + k)(c + l)(c + m)(c + n) = 0\)⋯⋯ (II)
Set X only contains all possible solutions of equation (I) above and set Y only contains all possible solutions of equation (II) above. Given that k, l, m and n are unequal numbers, does the number of elements in set X equal the number of elements in set Y?
(1) \(k\) and \(l\) are positive numbers
(2) \(m\) and \(n\) are negative numbers
Source: Optimus Prep
28 Feb 2016, 05:21
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nishith17
Hi,
first lets see the info from the Q--
\((|b| + k)(|b| + l)(|b| + m)(|b| + n) = 0\)⋯⋯ (I)
solutions are |b| = -k,-l, -m,, -n means all the -ive values amongst k,l,m, and n are the solutions..
\((c + k)(c + l)(c + m)(c + n) = 0\)⋯⋯ (II)
solutions are c= -k, -l, -m, -n so 4 solutions
k,l,m,and n are unequal..
requirement
if number of solutions in I and II are same..
Inference- if only 2 of k,l,m, and n are of -ive sign then only both sets wll have same number of solutions...
so if we know how many are positive and how many negative, we can answer the Q
lets see teh choices..
(1) \(k\) and \(l\) are positive numbers
since k and l are positive numbers..
so |b| cannot be equal to -ive(positive number)..
so b= k,-k,l,-l
But we do not know of other two values
insuff
(2) \(m\) and \(n\) are negative numbers
we do not know of other two numbers..
INSUFF
combined,
we know the solution is k,-k,l,-l so 4 solutions and for c also we have four solutions
C
[color=#ff0000]note:- a good Q. edited the solution/color]
28 Feb 2016, 05:52
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Original Explanation
Statement (1):
(|b| + k)(|b| + l)(|b| + m)(|b| + n) = 0
Either |b| = −k, or |b| = −l, or |b| = −m or |b| = −n
Since k, l, m and n are unequal numbers. As k and l are positive, Equation (I) implies that |b| = −k or |b| = −l are not possible solutions to the equation as |b| can not be negative. This leaves us with only |b| = −m or |b| = −n as the only possible values of |b|. However, it is not clear whether m and n are positive or negative from statement (I) alone. This means you can not eliminate any of the values –m, m, −n or n from the solution set of the given equation yet, nor can you include them in the solution set with certainty.
Statement (1) ALONE is NOT sufficient.
Statement (2): Given that m and n are negative, |b| may equal −m or −n. Remember that if m < 0, then −m > 0 and likewise, if n < 0, then −n > 0. So |b| may equal −m or −n implying that b may equal m, −m, n or −n. However, statement (2) does not tell you anything about the signs of k and l, so neither of these two numbers can be included nor excluded from the solution set of the equation with certainty.
Statement (2) ALONE is NOT sufficient.
BOTH statements TOGETHER are sufficient: Together, both statements tell you that k and l are positive while m and n are negative. This means, |b| ≠ −k and |b| ≠ −l while |b| may equal –m or |b| may equal −n.If |b| = −m, then b may equal −m or m.If |b| = −n, then b may equal −n or n. This means there are four possible values of b that satisfy equation (I), namely b = −m, b = m, b = −n and b = n. Hence, set X has exactly 4 members. The number of members of set Y equals 4 as the solution set of equation (II) contains 4 members, namely, c = −k, c = −l, c = −m and c = −n. As a result, the number of elements in set X equals the number of elements in set Y.
BOTH statements TOGETHER are sufficient but NEITHER statement ALONE is sufficient.
The correct answer is C.
Statement (1):
(|b| + k)(|b| + l)(|b| + m)(|b| + n) = 0
Either |b| = −k, or |b| = −l, or |b| = −m or |b| = −n
Since k, l, m and n are unequal numbers. As k and l are positive, Equation (I) implies that |b| = −k or |b| = −l are not possible solutions to the equation as |b| can not be negative. This leaves us with only |b| = −m or |b| = −n as the only possible values of |b|. However, it is not clear whether m and n are positive or negative from statement (I) alone. This means you can not eliminate any of the values –m, m, −n or n from the solution set of the given equation yet, nor can you include them in the solution set with certainty.
Statement (1) ALONE is NOT sufficient.
Statement (2): Given that m and n are negative, |b| may equal −m or −n. Remember that if m < 0, then −m > 0 and likewise, if n < 0, then −n > 0. So |b| may equal −m or −n implying that b may equal m, −m, n or −n. However, statement (2) does not tell you anything about the signs of k and l, so neither of these two numbers can be included nor excluded from the solution set of the equation with certainty.
Statement (2) ALONE is NOT sufficient.
BOTH statements TOGETHER are sufficient: Together, both statements tell you that k and l are positive while m and n are negative. This means, |b| ≠ −k and |b| ≠ −l while |b| may equal –m or |b| may equal −n.If |b| = −m, then b may equal −m or m.If |b| = −n, then b may equal −n or n. This means there are four possible values of b that satisfy equation (I), namely b = −m, b = m, b = −n and b = n. Hence, set X has exactly 4 members. The number of members of set Y equals 4 as the solution set of equation (II) contains 4 members, namely, c = −k, c = −l, c = −m and c = −n. As a result, the number of elements in set X equals the number of elements in set Y.
BOTH statements TOGETHER are sufficient but NEITHER statement ALONE is sufficient.
The correct answer is C.
