Question: The operation ⊙ is defined for all real numbers x and y by the equation x⊙y = \(x^2+y^2-xy\). Each of the following must be true EXCEPT

1. x⊙x = \(x^2\)
2. x⊙y = y⊙x
3. x⊙1 =x
4. x⊙0 =\(x^2\)
5. x⊙y = (-x) ⊙(-y)

Answer:

Approach Solution (1):

It is an operation and what it means is given, so check all values by substituting the values

It is given that x⊙y= \(x^2+y^2-xy\)

1. x⊙x =\(x^2\)… =\(x^2+x^2-xx=2x^2-x^2=x^2\)… True
2. x⊙y = y⊙x… In x⊙y = \(x^2+y^2-xy\), x and y are interchangeable and still the equation remains the same. So true
3. x⊙1 = x….\(x^2+1^2-x*1=x^2+1-x\) This is not equal to x. Hence false
4. x⊙0 =\(x^2\)...=\(x^2+0^2-x*0=x^2\)… True
5. x⊙y = (-x) ⊙(-y)… -x⊙y= \((-x)^2+(-y)^2-(-x)(-y)=x^2+y^2-xy\)

Correct Option: C

“The operation ⊙ is defined for all real numbers x and y by the equation x⊙y = \(x^2+y^2-xy\). Each of the following must be true EXCEPT?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

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