If |r|*s=r^2,which of the following must be

Question

If |r|*s= r^2 , which of the following must be true, if  r
eq{0} ?
I. rs>0
II. r=s
III.  s\geq{r}

A) I
B) II
C) III
D) I and II
E) I, II, and III

New tricky question

chetan2u · Accepted Answer

Hi...
 |r|*s=r^2 ..
Now |r| and r^2 are always positive..
So positive*s=positive
Here s has to be positive
So two points
1) s is positive and r can be anything
2) numeric value of s and r is equal that is |r|=|s|

If r is negative, s>r and if r is positive, s=r
Thus s can never be <r

chetan2u · Answer

try the following question on the same line to judge yourself on inequalities and modulus
 https://gmatclub.com/forum/is-r-s-s-r-269267.html

apurv09 · Answer

|r|*s=r^2

so, s=-r, or s=+r

so, if s=-r, the sr is <0,
and r not equal to s,
but s will be either =r, or >-r.

so swill be always >=r
so ans C

harmanbindra · Answer

can someome explain this pls...
 |r| = always positive
 |r|* s = r^2
 lets assume s = (-2)^2 and r = - 4

is this case 
1)rs = -16 False
2) r=s False
3) s ≥ r True

So option c

gianthoang912 · Answer

How about "r=0" case?

lary301254M7 · Answer

If |r|*s=r^2, which of the following must be true?
I. rs>0
II. r=s
III. s≥r

My take is, since |r| and r^2 will always be >= 0 , s cannot be negative
(I) immediately out
(II) is also out because easy plug-in would show |-2| * 2 = (-2)^2

Vidhi96 · Answer

|r|s = r^2 => r * r

thus s either has to be r ; if r is positive
s>r, if r is negative

allan89 · Answer

|r|s = r^2

=> |r|^2-|r|s =0
=> |r|(|r|-s)=0
=> |r| =0 or |r|=s
=> since r is not equal to 0 hence s=r or s=-r
=> r can be any value +ve or -ve; but s >0 always therefor s>=r

mysterymanrog · Answer

Way I did this was through testing values:

I) 
Consider the case of r=-2, s=2 
In this case, we have that r and s are not the same sign, and yet the equation still holds. 
II) We can use the previous case in I), to determine that they must not be equal

By elimination, we can conclude that C must be the answer.

However:

III)
This must be true. 
We know that s can never be negative - as if it were negative, that would imply that r^2 is negative, which is impossible without complex numbers (which gmat does not use) 
We know that r may be negative - in this case, s would be greater than r. 
We also know that it might just be the case both s and r are the same number. In this case, s=r. 
So we can determine that III is true from the above reasoning.

Carolweszxdrcftvb · Answer

R !=0 according to the question

Demonicgod · Answer

Given:
∣r∣⋅s=r2|r| \cdot s = r^2∣r∣⋅s=r2
and r≠0r \neq 0r=0.
We want to find which statements must be true among:
I. rs>0rs > 0rs>0
II. r=sr = sr=s
III. s≥rs \ge rs≥r
[hr]
Step 1: Analyze by considering sign of rrr
Case 1: r>0r > 0r>0
Then ∣r∣=r|r| = r∣r∣=r.
Substitute into the equation:
r⋅s=r2r \cdot s = r^2r⋅s=r2
Divide both sides by rrr (since r≠0r \ne 0r=0):
s=rs = rs=r
So when r>0r > 0r>0:

rs=r2>0rs = r^2 > 0rs=r2>0 → I true
r=sr = sr=s → II true
s=rs = rs=r → implies s≥rs \ge rs≥r → III true

✅ So for r>0r > 0r>0, all three statements hold.
[hr]
Case 2: r<0r < 0r<0
Then ∣r∣=−r|r| = -r∣r∣=−r.
Substitute:
(−r)⋅s=r2(-r) \cdot s = r^2(−r)⋅s=r2
Simplify:
−rs=r2-rs = r^2−rs=r2
Divide both sides by rrr (still nonzero):
−s=r-s = r−s=r s=−rs = -rs=−r
Now let’s check each statement.

I: rs=r(−r)=−r2<0rs = r(-r) = -r^2 < 0rs=r(−r)=−r2<0 → False
II: r=sr = sr=s? No, since s=−rs = -rs=−r → False
III: s≥rs \ge rs≥r?
For r<0r < 0r<0, s=−r>0s = -r > 0s=−r>0, and rrr is negative, so indeed s>rs > rs>r. → True

✅ So for r<0r < 0r<0, only III is true.
[hr]
Step 2: Which statement is true for all nonzero r?

For r>0r > 0r>0: I, II, and III true.
For r<0r < 0r<0: only III true.

So the only statement that must always hold (no matter the sign of rrr) is:
III. s≥r\boxed{\text{III. } s \ge r}III. s≥r
[hr]
✅ Final Answer: C) III

harmanbindra

can someome explain this pls...
|r| = always positive
|r|* s = r^2
lets assume s = (-2)^2 and r = - 4

is this case
1)rs = -16 False
2) r=s False
3) s ≥ r True

So option c

If |r|*s=r^2,which of the following must be

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GMAT Problem Solving (PS) Questions

If |r|*s=r^2,which of the following must be

Prep Toolkit

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