Question : If $\frac{x^{2}-x+1}{x^{2}+x+1}=\frac{2}{3}$, then the value of $\left (x+\frac{1}{x} \right)$ is:
Option 1: 4
Option 2: 5
Option 3: 6
Option 4: 8
Correct Answer: 5
Solution : Let $( x+\frac{1}{x})$ be $y$. $\frac{x^{2}-x+1}{x^{2}+x+1}=\frac{2}{3}$ Taking $x$ common in both numerator and denominator, we get, $⇒\frac{x( x+\frac{1}{x}-1)}{x( x+\frac{1}{x}+1)}=\frac{2}{3}$ $⇒\frac{y-1}{y+1}=\frac{2}{3}$ $⇒3y-3=2y+2$ $\therefore y = x+\frac{1}{x} = 5$ Hence, the correct answer is 5.
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Question : If $x^2-3 x+1=0$, then the value of $\left(x^4+\frac{1}{x^2}\right) \div\left(x^2+1\right)$ is:
Option 1: 5
Option 2: 6
Option 3: 9
Option 4: 7
Question : Find the value of the following expression. $\frac{\left[\frac{5}{8}-\left\{\frac{3}{8}-\left(\frac{5}{8}-\frac{3}{8}\right)\right\}\right] \text { of } 8.8-1.2}{4 \frac{1}{6} \div 2.5 \times 2 \div \frac{1}{6} \text { of } 60+\left(\frac{3}{4}-\frac{3}{8}\right)}$
Option 1: $5 \frac{22}{43}$
Option 2: $3 \frac{23}{67}$
Option 3: $4 \frac{44}{85}$
Option 4: $4 \frac{4}{5}$
Question : If $x\left(5-\frac{2}{x}\right)=\frac{5}{x}$, then the value of $x^2+\frac{1}{x^2}$ is:
Option 1: $\frac{54}{25}$
Option 2: $\frac{53}{28}$
Option 3: $\frac{53}{27}$
Option 4: $\frac{54}{23}$
Question : If $x^2-5 x+1=0$, then the value of $\left(x^4+\frac{1}{x^2}\right) \div\left(x^2+1\right)$ is:
Option 1: 21
Option 2: 22
Option 3: 25
Option 4: 24
Question : What is the simplified value of: $\frac{1}{8}\left\{\left(x+\frac{1}{y}\right)^2-\left(x-\frac{1}{y}\right)^2\right\}$
Option 1: $\frac{x}{y}$
Option 2: $\frac{2x}{y}$
Option 3: $\frac{x}{2y}$
Option 4: $\frac{4x}{y}$
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