Question : If $x+\frac{1}{x}=-2$, then the value of $x^p +x^q$ is: (where $p$ is an even number and $q$ is an odd number)
Option 1: –2
Option 2: 2
Option 3: 1
Option 4: 0
Correct Answer: 0
Solution : Given: $x+\frac{1}{x}=-2$ $x^2 + 1 = -2x$ ⇒ $x^2 + 2x + 1 = 0$ ⇒ $(x+1)^2 = 0$ ⇒ $x = -1$ Given expression, $x^p + x^q$, where p is even and q is odd, $(-1)^{even} = 1$ and $(-1)^{odd} = -1$ $\therefore$ The expression becomes 1 + (–1) = 0 Hence, the correct answer is 0.
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