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11 Mar 2023, 03:39
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Question 14 - Official Solution:
\(x_1+x_2+...+x_{100} = 1\);
\(x_1+x_2+...+x_{99} = 2\);
\(x_1+x_2+...+x_{98} = 3\);
\(...\)
\(x_1= 100\).
What is the value of \(x_1*x_2*...*x_{100}\)?
A. \(-100\)
B. \(-1\)
C. \(0\)
D. \(1\)
E. \(100\)
By subtracting the second equation from the first, we obtain \((x_1+x_2+...+x_{100})-(x_1+x_2+...+x_{99}) = 1-2\), which simplifies to \(x_{100}= -1\).
Similarly, subtracting the third equation from the second yields \((x_1+x_2+...+x_{99}) -(x_1+x_2+...+x_{98}) = 2-3\), which simplifies to \(x_{99}= -1\).
By repeating this process for each successive pair of equations, we can determine that every term up to \(x_1\) is also equal to -1.
Therefore, \(x_1*x_2*...*x_{100}=100(-1)(-1)(-1)...(-1)=100(-1)^{99}=-100\).
Answer: A
\(x_1+x_2+...+x_{100} = 1\);
\(x_1+x_2+...+x_{99} = 2\);
\(x_1+x_2+...+x_{98} = 3\);
\(...\)
\(x_1= 100\).
What is the value of \(x_1*x_2*...*x_{100}\)?
A. \(-100\)
B. \(-1\)
C. \(0\)
D. \(1\)
E. \(100\)
By subtracting the second equation from the first, we obtain \((x_1+x_2+...+x_{100})-(x_1+x_2+...+x_{99}) = 1-2\), which simplifies to \(x_{100}= -1\).
Similarly, subtracting the third equation from the second yields \((x_1+x_2+...+x_{99}) -(x_1+x_2+...+x_{98}) = 2-3\), which simplifies to \(x_{99}= -1\).
By repeating this process for each successive pair of equations, we can determine that every term up to \(x_1\) is also equal to -1.
Therefore, \(x_1*x_2*...*x_{100}=100(-1)(-1)(-1)...(-1)=100(-1)^{99}=-100\).
Answer: A
30 Aug 2024, 09:00
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Question 21 - Official Solution:
If \(a > b > c > d > e\) and \(abcde > 0\), then which of the following must be true ?
I. \(ab > 0\)
II. \(bc > 0\)
III. \(de > 0\)
A. I only
B. II only
C. III only
D. II and III only
E. I, II, and III
Given that the product of five numbers, \(a\), \(b\), \(c\), \(d\), and \(e\), is positive, there must be an even number of negative numbers among them (0, 2, or 4 negative numbers). Since it is also given that \(a > b > c > d > e\), we can have the following three cases:
\(a\; |\; b\; |\; c\; |\; d\; |\; e\)
\(+ | + | + | + | +\) (no negative numbers)
\(+ | + | + | - | -\) (two negative numbers)
\(+ | - | - | - | -\) (four negative numbers)
Let's analyze each option, taking into consideration that the question asks which of them MUST be true, not COULD be true.
I. \(ab > 0\)
If we have the third case, then this option is not true. Eliminate.
II. \(bc > 0\)
This option is true for each of the three cases. Therefore, this option is always true.
III. \(de > 0\)
This option is true for each of the three cases. Therefore, this option is always true.
Consequently, only options II and III are always true, and thus the answer is D.
Answer: D
If \(a > b > c > d > e\) and \(abcde > 0\), then which of the following must be true ?
I. \(ab > 0\)
II. \(bc > 0\)
III. \(de > 0\)
A. I only
B. II only
C. III only
D. II and III only
E. I, II, and III
Given that the product of five numbers, \(a\), \(b\), \(c\), \(d\), and \(e\), is positive, there must be an even number of negative numbers among them (0, 2, or 4 negative numbers). Since it is also given that \(a > b > c > d > e\), we can have the following three cases:
\(a\; |\; b\; |\; c\; |\; d\; |\; e\)
\(+ | + | + | + | +\) (no negative numbers)
\(+ | + | + | - | -\) (two negative numbers)
\(+ | - | - | - | -\) (four negative numbers)
Let's analyze each option, taking into consideration that the question asks which of them MUST be true, not COULD be true.
I. \(ab > 0\)
If we have the third case, then this option is not true. Eliminate.
II. \(bc > 0\)
This option is true for each of the three cases. Therefore, this option is always true.
III. \(de > 0\)
This option is true for each of the three cases. Therefore, this option is always true.
Consequently, only options II and III are always true, and thus the answer is D.
Answer: D
11 Mar 2023, 06:30
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Question 10 - Official Solution:
If \(m\) is a positive number and \(n\) is a negative number, and \(|m| > |n|\), then which of the following has the greatest value ?
A. \(|\frac{m - n}{n}|\)
B. \(|\frac{m - n}{m}|\)
C. \(|\frac{m + n}{m - n}|\)
D. \(|\frac{m + n}{n}|\)
E. \(|\frac{m + n}{m}|\)
Probably the easiest wat to solve this problem would be plug the values. Let \(m=2\) and \(n=-1\) (this satisfies conditions given in the stem), then:
A. \(|\frac{m - n}{n}|=|\frac{2 - (-1)}{-1}|=|-3|=3\).
B. \(|\frac{m - n}{m}|=|\frac{2 - (-1)}{2}|=|1.5|=1.5\).
C. \(|\frac{m + n}{m - n}|=|\frac{2 + (-1}{2 - (-1)}|=|\frac{1}{3}|=\frac{1}{3}\)
D. \(|\frac{m + n}{n}|=|\frac{2 + (-1)}{-1}|=|-1|=1\).
E. \(|\frac{m + n}{m}|=|\frac{2 + (-1)}{2}|=|\frac{1}{2}|=\frac{1}{2}\)
As we can see the answer is option A.
Still, to test our absolute value skills, let's also solve the question using absolute value properties. Let's establish two points:
(i) \(|m - n| > |m + n|\)
(this is because it's given that \(m\) is a positive number and \(n\) is a negative number. For example, \((|5 - (-1)| = 6) > (|5 + (-1)| = 4)\)).
(ii) \(|\frac{a}{b}|=\frac{|a|}{|b|}|\). So:
A. \(|\frac{m - n}{n}|=\frac{|m - n|}{|n|}\)
B. \(|\frac{m - n}{m}|=\frac{|m - n|}{|m|}\)
C. \(|\frac{m + n}{m - n}|=\frac{|m + n|}{|m - n|}\)
D. \(|\frac{m + n}{n}|=\frac{|m + n|}{|n|}\)
E. \(|\frac{m + n}{m}|=\frac{|m + n|}{|m|}\)
A and B have the same numerator so let's compare these two options first (both the denominator and numerator are positive, so the one with smaller denominator will have the larger value). Since given that \(|m| > |n|\), the A > B.
B, C and E have the same numerator, so let's compare these three options next (again, both the denominator and numerator are positive, so the one with smaller denominator will have the larger value). The denominator of E is less then that of D (it's given that \(|m| > |n|\)) and since also given that \(m\) is a positive number and \(n\) is a negative number, the denominator of E is also less then that of C.
So, we are left to compare A and E. The numerator of A is greater than that of E (check (i) above) plus the denominator of A is less than that of E (it's given that \(|m| > |n|\)), so A > E. Therefore, A has the greatest value.
Answer: A
If \(m\) is a positive number and \(n\) is a negative number, and \(|m| > |n|\), then which of the following has the greatest value ?
A. \(|\frac{m - n}{n}|\)
B. \(|\frac{m - n}{m}|\)
C. \(|\frac{m + n}{m - n}|\)
D. \(|\frac{m + n}{n}|\)
E. \(|\frac{m + n}{m}|\)
Probably the easiest wat to solve this problem would be plug the values. Let \(m=2\) and \(n=-1\) (this satisfies conditions given in the stem), then:
A. \(|\frac{m - n}{n}|=|\frac{2 - (-1)}{-1}|=|-3|=3\).
B. \(|\frac{m - n}{m}|=|\frac{2 - (-1)}{2}|=|1.5|=1.5\).
C. \(|\frac{m + n}{m - n}|=|\frac{2 + (-1}{2 - (-1)}|=|\frac{1}{3}|=\frac{1}{3}\)
D. \(|\frac{m + n}{n}|=|\frac{2 + (-1)}{-1}|=|-1|=1\).
E. \(|\frac{m + n}{m}|=|\frac{2 + (-1)}{2}|=|\frac{1}{2}|=\frac{1}{2}\)
As we can see the answer is option A.
Still, to test our absolute value skills, let's also solve the question using absolute value properties. Let's establish two points:
(i) \(|m - n| > |m + n|\)
(this is because it's given that \(m\) is a positive number and \(n\) is a negative number. For example, \((|5 - (-1)| = 6) > (|5 + (-1)| = 4)\)).
(ii) \(|\frac{a}{b}|=\frac{|a|}{|b|}|\). So:
A. \(|\frac{m - n}{n}|=\frac{|m - n|}{|n|}\)
B. \(|\frac{m - n}{m}|=\frac{|m - n|}{|m|}\)
C. \(|\frac{m + n}{m - n}|=\frac{|m + n|}{|m - n|}\)
D. \(|\frac{m + n}{n}|=\frac{|m + n|}{|n|}\)
E. \(|\frac{m + n}{m}|=\frac{|m + n|}{|m|}\)
A and B have the same numerator so let's compare these two options first (both the denominator and numerator are positive, so the one with smaller denominator will have the larger value). Since given that \(|m| > |n|\), the A > B.
B, C and E have the same numerator, so let's compare these three options next (again, both the denominator and numerator are positive, so the one with smaller denominator will have the larger value). The denominator of E is less then that of D (it's given that \(|m| > |n|\)) and since also given that \(m\) is a positive number and \(n\) is a negative number, the denominator of E is also less then that of C.
So, we are left to compare A and E. The numerator of A is greater than that of E (check (i) above) plus the denominator of A is less than that of E (it's given that \(|m| > |n|\)), so A > E. Therefore, A has the greatest value.
Answer: A
