For \(x^2*y^5*z>0\) to hold true:
1. \(x\) must not be zero;
and
2. \(y\) and \(z\) must be either both positive or both negative.

(1) \(\frac{xz}{y}>0\) --> first condition is satisfied: \(x\neq{0}\), but we don't know aout the second one: \(\frac{xz}{y}>0\) means that either all of them are positive (answer YES) or ANY two are negative and the third one is positive, so it's possible \(y\) and \(z\) to have opposite signs (answer NO). Not sufficient.

(2) \(\frac{y}{z}<0\) --> \(y\) and \(z\) have opposite signs --> second condition is already violated, so the answer to the question is NO. Sufficient.

Answer: B.

Side note for (2): \(\frac{y}{z}<0\) does not mean that \(x^2*y^5*z<0\), it means that \(x^2*y^5*z\leq{0}\) because it's possible \(x\) to be equal to zero and in this case \(x^2*y^5*z=0\). But in any case \(x^2*y^5*z\) is not MORE than zero, so we can answer NO to the question.

Hoe it's clear.
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Hi,

To have expression > 0 , there are two cases -
1) both y and z is positive or
2) both y and z is negative ( because x2 is alwasy positive, irrespecitve of sign of x )

Option 1 is not sufficient .

Option 2 is sufficient as it tells that y and z is of opposite sign. So expression is less than zero.

s1: --> x,y, z not = 0.
---> if y = -ve, then xz must be -ve so either x or z is -ve.
if y = +ve, then xz must be +ve x and z must be same sign.

consider y = -ve and x=-ve and z = +ve , evaluate question, answer is false.
consider y= -ve and x=+ve and z = -ve, evaluation question, answer is true.

Therefore: s1 not sufficient.

s2: --> either y=-ve and z=+ve OR y=+ve and z=-ve

Ignore x=+ve or x=-ve since there is a x^2 in question
consider y=-ve and z=+ve, evaluation question, answer is false.
consider y=+ve and z=-ve, evaluation question, answer is false.

s2 alone sufficient.

How to quickly see s1 is insufficient?

1. xz/y>0
we need y & z to have similar signs.
but we can't be sure from the first statement, z>0, y<0, x<0, the statement is true
z>0, y>0, x>0, true
z<0, y>0, x>0 true.
Thus, 1) unsuff.

what if X=0? Why are we not considering that?

@Bibha

We are not concidering X = 0 because 2) makes it clear that x^2 * y^5 * z <= 0

< because y & z have different signs

= When X = 0. ( also Y & Z cannot be zero as per statement 2 )
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My approach to see St:1 is insufficient is
xz/y>0 mean both are of same sign.
x^2*y^5*z => x*((x*z)*y^5) here ((x*z)*y^5) is positive since y^5 does not change y sign and xz is same so both give positive, here we dont know remaining x therefore insufficient.

Bumping for review and further discussion.
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