Question : If $x^{2} + y^{2} + z^{2} = 2 (x + z - 1)$ , then the value of $x^{3} + y^{3} + z^{3}$ is equal to:
Option 1: 6
Option 2: 1
Option 3: 2
Option 4: 8

Question

Question : If $x$ = $y$ = $z$ , then $\frac{{(x + y + z)}^{2}}{x^{2} + y^{2} + z^{2}}$ is equal to:
Option 1: 4
Option 2: 2
Option 3: 3
Option 4: 1

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Answer 1

Correct Answer: 2

Solution : Given, $x^{2} + y^{2} + z^{2} = 2 (x + z - 1)$
$x^{2} + y^{2} + z^{2} = 2 x + 2 z - 2$
⇒ $x^{2} - 2 x + 1 + y^{2} + z^{2} - 2 z + 1 = 0$
⇒ $(x - 1)^{2} + y^{2} + (z - 1)^{2} = 0$
⇒ $(x - 1)^{2} = 0,$ $y^{2} = 0$ , and $(z - 1)^{2} = 0$
⇒ $x - 1 = 0,$ $y = 0$ , and $z - 1 = 0$
⇒ $x = 1,$ $y = 0$ , and $z = 1$
Now, $x^{3} + y^{3} + z^{3} = 1^{3} + 0 + 1^{3} = 1 + 1 = 2$
Hence, the correct answer is 2.

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Question : If $x^{2} + y^{2} + z^{2} = 2 (x + z - 1)$ , then the value of $x^{3} + y^{3} + z^{3}$ is equal to:
Option 1: 6
Option 2: 1
Option 3: 2
Option 4: 8