If (x^2+8)yz < 0, wz > 0, and xyz < 0, the which of the following mus

Given,

\((x^2 + 8)yz< 0\) , which means \(yz<0\)

\(wz > 0\), which means i) \(w>0, z>0\) or ii) \(w<0, z<0\)

\(xyz < 0\), since we know, \(yz<0\), hence \(x>0\).
Therefore, i) \(y>0, z<0\) or ii) \(y<0, z>0\)

Now since z is common in all three expressions, lets use cases based around sign of z.

i) if \(z>0\), then \(w>0, y<0, x>0\)
ii) if \(z<0\), then \(w<0, y>0, x>0\)


Lets check the given expressions:
I. \(x < 0\) - False
II. \(wy < 0\) - True, since \(w\) & \(y\) always have opposite signs
III. \(yz < 0\) - True, since \(y\) & \(z\) always have opposite signs.

Answer D.

Thanks,
GyM

Ps - Thanks Manish, its a good question!!

This question needs part to part treatment.

part 1:
(\(x^2\)+8)yz<0

From this part we understand that x^2 will always be positive. When we add the value of x^2 with 8 ultimate result will be positive. Thus, yz has to be negative as (\(x^2\)+8)yz<0.
So, iii is one of the options.

part 2:

xyz<0

we know yz <0 , thus x has to be positive. We can eliminate option containing i.

part 3:

wz>0
yz<0...........................we have figure it out earlier.

we can multiply them to prove ii option.

after multiply we will get: (\(z^2\))wy<0..............................................wz>0 and yz<0 . one is positive and another is negative. ultimately, we will get negative answer.

now, \(z^2\) is always positive. Thus, wy <0. This is a must.

So , the best answer is II and III.

(x^2 + 8)yz < 0; as x^2 + 8 >0 so yz<0
this means either y>0, z<0 or y<0, z>0

wz >0
case 1: w>0 and z>0
from above statement, if z>0 then y<0 hence wy<0

case 2: w<0 and z<0
from above statement, if z<0 then y>0 hence wy<0

xyz<0
as yz<0 so x>0

Ans D

I made a small table with columns x y z w and wrote the possibilities:
x-----y-----z-----w
+ve +ve -ve -ve
+ve +ve +ve +ve
-ve +ve -ve -ve ---- NOT POSSIBLE
-ve -ve +ve ----NOT POSSIBLE

As we can see II and III are always True.!

(x^2+8) is always >0, so y*z<0
-> y or z has to be negative

III. This statement is needed to fulfill the equation (see above)


II. We know that y or z has to be negative
Two possible combinations that w*y<0:
-w*y<0 then -w*-z>0 OR
w*-y<0 then w*z>0
-> y and z have in both cases the "opposite signs" -> y*z<0 -> is needed to fulfill the equation

I. This statement is wrong because x cannot be negative. x has to be positive as z or y has to be negative.
Possible combinations:
x*-y*z<0 OR
x*y*-z<0

Solution: III. and II. are right -> D

Tough question. Have to start by making deductions with the easiest inequality and work your way through possible cases.


WZ > 0

W and Z must have the same sign (1)


(X^2 + 8) YZ < 0

(X’2 + 8) as a factor will always be positive. Thus in order for the expression to be negative —— YZ < 0

Y and Z must have opposite signs (2)


XYZ < 0

We know from (2) that YZ must be negative and have opposite signs.

Thus: X must be positive (3)



Case 1: W = +pos.

Then this means: From (1) —- Z = +

And this means: From (2) ——- Y = (-)neg.

And finally: from (3) ——— X = +

Or


Case 2: W = (-)neg.

This means from (1): Z = (-)neg.

And this means from (2): Y = (+)

And from (3): X = (+)


I. X < 0

False, X is positive as shown above.

II. WY < 0

Either in case 1 or case 2, W and Y always have opposite signs. must be TRUE.

III. YZ < 0

Either in case 1 or case 2, Y and Z always have opposite signs. Must be TRUE.


II AND III Only

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