If x 0, y

Question

If x > 0, y < 0 and z < 0,  (|x|+|y|+|z|)^2  = ?

A.  x^2+y^2+z^2+2xy+2yz+2zx

B.  x^2+y^2+z^2+2xy-2yz+2zx

C.  x^2+y^2+z^2-2xy+2yz-2zx

D.  x^2+y^2+z^2-2xy-2yz-2zx

E.  x^2-y^2-z^2+2xy+2yz+2zx

LogicGuru1 · Accepted Answer

s

Remember that these kind of question can be tricky if you don't apply the last trick after sorting your reasoning.
The reasoning is anything that comes out of mod is always positive . It can not be negative.
But the last part of the trick that many of us forget to apply is to check whether your answers depends on the absolute value (value that comes out of the mod) or on the "raw value" that is the original original value of the integer without the mod (values with original polarity -ve or +ve)

IN THIS QUESTION:- value of y and z are negative
SO the expression  (|x|+|y|+|z|)^2  will always be positive
You don't have to worry about terms with square. because squaring kills negative polarity so x^2, y^2 and z^2 are not our worries. 
Now to get a positive value from the options that contains "Raw values" and not the "absolute values" you must remember that x is +ve and y and z are -ve , so if they are multiplied with x their product will be negative. You have to remove that negative polarity . What is the easiest way of removing a negative polarity ? multiply it with -1 or simply -
Therefore all terms that contains either y or z should be multiplied with -1 to give us terms that are positive,
so our answer should contains -2xy and -2xz 
Also because -y and -z will multiply to give +yz we don't have to multiply it with -1 
finally our expression should look like 
 x^2+y^2+z^2 - 2xy +2yz-2xz

THAT IS OPTION C

C is the answer

chetan2u · Answer

Hi,

We do not require to know the formula for this but realize that there are two -ive qty,z and y..

the mod of x does not have any effect as it is positive..

lets see the eq..
  (|x|+|y|+|z|)^2 ...
all terms in the Eq when expanded should be +ive.. 
When we see the choices all terms are same except change in sign at few places..
 What will be any term with single z and single y, 2yz, it will be +ive as y is -ive and z is -ive.. 
 What will be any term with only one of single z or single y, 2yx or 2xz, it will be -ive as x is +ive and other y or z is -ive.. 
so for 2xz to be positive, it should be -2xz and 2xy should be -2xy..

So, we have to look for a choice where all terms are positive apart from terms containing single power of z or single power of y but not both..
lets see the choices
A.  x^2+y^2+z^2+2xy+2yz+2zx  --- all terms are +ive, making 2xz and 2xy negative.. eliminate
B. x^2+y^2+z^2+2xy-2yz+2zx --------2xy, 2yz, 2zx all are -ive .. eliminate
C. x^2+y^2+z^2-2xy+2yz-2zx  -----------2xy, 2yz, 2zx all are +ive .. CORRECT
D.  x^2+y^2+z^2-2xy-2yz-2zx -------- 2yz is -ive .. eliminate
E. x^2-y^2-z^2+2xy+2yz+2zx --------2xy,y^2,z^2, 2zx all are -ive .. eliminate

ans C

neeraj609 · Answer

Because all its the square of the sum of absolute values, it MUST be >=0. Only C is the option which has outcome of each combination as positive. So i went with C as answer.

MathRevolution · Answer

If x>0, y<0 and z<0, (|x|+|y|+|z|)^2=?

A. x^2+y^2+z^2+2xy+2yz+2zx
B. x^2+y^2+z^2+2xy-2yz+2zx
C. x^2+y^2+z^2-2xy+2yz-2zx
D. x^2+y^2+z^2-2xy-2yz-2zx
E. x^2-y^2-z^2+2xy+2yz+2zx

==> |A|=A when A>0 ,and |A|=-A when A<0.
So, (|x|+|y|+|z|)^2=(x-y-z)^2=x^2+(-y)^2+(-z)^2+2x(-y)+2(-y)(-z)+2(-z)x
=x^2+y^2+z^2-2xy+2yz-2zx.
Therefore, the answer is C.

elenap818 · Answer

Are you sure C is the answer?
The question says that y and z are negative, agree. But here we need to find the square of the sm of three absolute values, not three integers. If the question asked for (x+y+z)^2 I would have agreed. But here we have three absolute values which are always positive or equal to zero by definition.
Probably I'm missing something...could you please explain?
Thanks

chetan2u · Answer

Hi,
you are correct it will not effect the mod values ..
But the choices are not in mod values but integers, and that i swhy we have to see if the choices match the given conditions..

smritidabas · Answer

I am confused too!
How will I know whether or not to take mods seriously when the correct answer is otherwise!
The question says tis asking us to find the square of three absolute values, not three integers. 
Where am i wrong here??

chetan2u · Answer

Hi,
substitute x as 2 and y and z as -1 to check the answer..
If x>0, y<0 and z<0, (|x|+|y|+|z|)^2= (2+1+1)^2 = 16 ..

But lets substitute values in A.
 x^2+y^2+z^2+2xy+2yz+2zx = 2^2+(-1)^2+(-1)^2+2*2*(-1)+2*(-1)*(-1)+2*2*(-1) = 4+1+1-4+2-4=0 ...NOT equal to 16..

lets see C
 x^2+y^2+z^2-2xy+2yz-2zx = 2^2+(-1)^2+(-1)^2-2*2*(-1)+2*(-1)*(-1)-2*2*(-1) = 4+1+1+4+2+4=16 ... equal to 16..

So C is CORRECT

LeoN88 · Answer

Once we remove the mod sign.
{x + (-y) + (-z)}^2, using the formula we get  C.

Concept tested:
lxl = x, x>=0
 = (-x), x<0

Sourav700 · Answer

MathRevolution

(a + b + c) 2 = a2 + b2 + c2 + 2(ab + bc + ca)

now modx. mody = +ve * +ve

shouldn't the answer be A.

Irrespective of the individual signs of x,y,z 2*modx*mody is a positive.

I strongly feel the answer should be A and not C. Could you please clear this for me?

Thanks.

Sourav700 · Answer

i see why one would opt for C.

If you read the question as: which of these would give you value equal to (modx + mody + modz)2 then answer is C.

on the other hand if you expand (modx + mody + modz)2 you'd get A.

Isn't the question asking you what is the value of (modx + mody + modz)2?

Abhishek009 · Answer

Plug in some values and try

x = 1  ;  y = -1  &  z = -2

So,  (|x|+|y|+|z|)= ( 1 + 1 + 2 ) = 4

Now,  (|x|+|y|+|z|)^2  = 16

Now, Plug in the values in the given options you will definitely land up with the correct Answer as (C)

Apeksha2101 · Answer

Help me understand this, if we put all 3 in mod, it clearly means we need to compute all values as +ve, then why the answer is C??

AshwinChadha · Answer

While I get the logic behind the negative sign to make it positive, I still don't understand why we need to do that to absolute values. Because |-2| = 2, so |Y|, Y, which is negative = Y. And then once it is multiplied by 2x, it should remain positive as the absolute value brackets when opened keep it as a positive. So by adding the - (or multiplying by -1), are we not just contradicting ourselves?

Bunuel · Answer

If x > 0, y < 0 and z < 0, (|x|+|y|+|z|)^2 = ? A. x^2+y^2+z^2+2xy+2yz+2zx B. x^2+y^2+z^2+2xy-2yz+2zx C. x^2+y^2+z^2-2xy+2yz-2zx D. x^2+y^2+z^2-2xy-2yz-2zx E. x^2-y^2-z^2+2xy+2yz+2zx (|x|+|y|+|z|)^2 = =|x|^2+|y|^2+|z|^2+2|x|*|y|+2|y|*|z|+2|z|*|x| Since |a|^2 = a^2 , then we'd get: |x|^2+|y|^2+|z|^2+2|x|*|y|+2|y|*|z|+2|z|*|x|= =x^2+y^2+z^2+2|x|*|y|+2|y|*|z|+2|z|*|x| Next, x > 0 implies |x| = x ; y < 0 implies |y| = -y ; and z < 0 implies |z| = -z . Hence, we'd have: x^2+y^2+z^2+2|x|*|y|+2|y|*|z|+2|z|*|x|= =x^2+y^2+z^2+2x*(-y)+2(-y)*(-z)+2(-z)*x = =x^2+y^2+z^2-2xy+2yz-2zx Answer: C.

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If x > 0, y < 0 and z < 0, (|x|+|y|+|z|)^2 = ?

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