Question : If $(x+y)^2=xy+1$ and $x^3-y^3=1$, what is the value of $(x-y)$?
Option 1: 1
Option 2: 0
Option 3: –1
Option 4: 2
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Correct Answer: 1
Solution : Given: $(x+y)^2=xy+1$ and $x^3-y^3=1$ Consider, $(x+y)^2=xy+1$ ⇒ $x^2+y^2+2xy-xy=1$ ⇒ $x^2+y^2+xy=1$ Now, we know, $x^3-y^3=(x-y)(x^2+xy+y^2)$ ⇒ $1=(x-y)×1$ $\therefore(x-y)=1$ Hence, the correct answer is 1.
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Question : If $xy = -6$ and $x^3+ y^3= 19$ ($x$ and $y$ are integers), then what is the value of $\frac{1}{x^{–1}}+\frac{1}{y^{–1}}$?
Question : If $x^2-xy+y^2=2$ and $x^4+x^2y^2+y^4=6$, then the value of $(x^2+xy+y^2)$ is:
Question : If $x+y+z=0$ and $x^2+y^2+z^2=40$, then what is the value of $x y+y z+z x?$
Question : $x,y,$ and $z$ are real numbers. If $x^3+y^3+z^3 = 13, x+y+z = 1$ and $xyz=1$, then what is the value of $xy+yz+zx$?
Question : If $x=1-y$ and $x^2=2-y^2$, then what is the value of $xy$?
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