If $$x>y^2>z^4$$ which of the following statements could be true?
$$I.\ \ x>y>z$$
$$II.\ \ z>y>x$$
$$III.\ \ x>z>y$$
A. I only
B. I and II only
C. 1 and III only
D. II and III only
E. I, II and II
The OA is E.
I'm really confused with this PS question. Please, can any expert assist me with it? Thanks in advanced.
If x>y^2>z^4, which of the following statements...
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Hello LUANDATO.
We will solve this PS question taking particular numbers.
The hypothesis is $$x>y^2>z^4.$$
$$I.\ \ x>y>z$$ Let's set x=10, y=3 and z=1. Then $$10>3^2>1^4\ and\ 10>3>1.\ $$ So, option I could be true.
$$II.\ \ z>y>x$$ Let's set $$x=\frac{1}{4},\ y=\frac{1}{3}\ and\ z=\frac{1}{2},$$ then $$\frac{1}{4}>\left(\frac{1}{3}\right)^2\ >\left(\frac{1}{2}\right)^4\ and\ \frac{1}{2}>\frac{1}{3}>\frac{1}{4}.$$ So, option II could be true.
$$III.\ \ x>z>y$$ Let's set $$x=10,\ y=\frac{1}{3}\ and\ z=\frac{1}{2},$$ then $$10>\left(\frac{1}{3}\right)^2\ >\left(\frac{1}{2}\right)^4\ and\ \ 10>\frac{1}{2}>\frac{1}{3}.$$ So, option III could be true.
That is why the correct answer is E .
I hope this examples can help you.
I'm available if you'd like a follow up.
Regards.
We will solve this PS question taking particular numbers.
The hypothesis is $$x>y^2>z^4.$$
$$I.\ \ x>y>z$$ Let's set x=10, y=3 and z=1. Then $$10>3^2>1^4\ and\ 10>3>1.\ $$ So, option I could be true.
$$II.\ \ z>y>x$$ Let's set $$x=\frac{1}{4},\ y=\frac{1}{3}\ and\ z=\frac{1}{2},$$ then $$\frac{1}{4}>\left(\frac{1}{3}\right)^2\ >\left(\frac{1}{2}\right)^4\ and\ \frac{1}{2}>\frac{1}{3}>\frac{1}{4}.$$ So, option II could be true.
$$III.\ \ x>z>y$$ Let's set $$x=10,\ y=\frac{1}{3}\ and\ z=\frac{1}{2},$$ then $$10>\left(\frac{1}{3}\right)^2\ >\left(\frac{1}{2}\right)^4\ and\ \ 10>\frac{1}{2}>\frac{1}{3}.$$ So, option III could be true.
That is why the correct answer is E .
I hope this examples can help you.
I'm available if you'd like a follow up.
Regards.
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We are given that x > y^2 > z^4 and need to determine which statements must be true. Let's test each Roman Numeral.BTGmoderatorLU wrote:If $$x>y^2>z^4$$ which of the following statements could be true?
$$I.\ \ x>y>z$$
$$II.\ \ z>y>x$$
$$III.\ \ x>z>y$$
A. I only
B. I and II only
C. 1 and III only
D. II and III only
E. I, II and II
The OA is E.
I'm really confused with this PS question. Please, can any expert assist me with it? Thanks in advanced.
I. x > y > z
Notice that the order of arrangement of x, y, and z in the inequality x > y > z is the same as the order of arrangement of x, y^2, and z^4 in the inequality x > y^2 > z^4, so we want to test positive integers in this case.
x = 10
y = 3
z = 1
Notice that 10 > 3 > 1 for x > y > z AND 10 > 9 > 1 for x > y^2 > z^4.
We see that I could be true.
II. z > y > x
Notice that the order of arrangement of x, y, and z in the inequality z > y > x differs from the order of arrangement of x, y^2, and z^4 in the inequality x > y^2 > z^4, so we want to test positive proper fractions in this case. This is because we need to decrease the value of y and z to make them work within the given inequality.
x = 1/5
y = 1/3
z = 1/2
Notice that 1/2 > 1/3 > 1/5 for z > y > x AND 1/5 > 1/9 > 1/16 for x > y^2 > z^4.
We see that II could be true.
III. x > z > y
Notice that the order of arrangement of y and z in the inequality x > z > y differs from the order of arrangement of y^2 and z^4 in the inequality x > y^2 > z^4, so we once again want to test positive proper fractions. This is because we need to decrease the value of z to make it work within the given inequality (that is, we want to swap the order of z^4 and y^2 even if z > y).
x = 1/2
y = 1/4
z = 1/3
Notice that 1/2 > 1/3 > 1/4 for x > z > y AND 1/2 > 1/16 > 1/81 for x > y^2 > z^4.
We see that III could be true.
Answer: E
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