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
Correct Answer: 22
Solution : $x^2 - 5x + 1 = 0$ $⇒x(x - 5 + \frac{1}{x}) = 0$ $⇒(x - 5 + \frac{1}{x}) = 0$ $⇒(x + \frac{1}{x}) = 5$ Cubing both sides, we get, $⇒(x + \frac{1}{x})^3 = 5^3$ $\because$ $(a + b)^3 = a^3 + b^3 + 3ab(a + b)$ $⇒x^3 + \frac{1}{x^3} + 3.x.\frac{1}{x} (x + \frac{1}{x}) = 125$ $⇒x^3 + \frac{1}{x^3} + 3(5) = 125$ $⇒x^3 + \frac{1}{x^3} + 15 = 125$ $⇒x^3 + \frac{1}{x^3} = 125 - 15=110$ Now, $\frac{x^4 + \frac{1}{x^2}}{x^2 + 1}$ Dividing the numerator and denominator by $x$, $=\frac{\frac{1}{x}(x^4 + \frac{1}{x^2})}{(x^2 + 1)\frac{1}{x}} $ $= \frac{(x^3 + \frac{1}{x^3})}{(x + \frac{1}{x})}$ $=\frac{110}{5}$ $=22$ Hence, the correct answer is 22.
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Question : The value of $\left(5 \frac{1}{4} \div \frac{3}{7}\right.$ of $\left.\frac{1}{2}\right) \div\left(5 \frac{1}{9}-7 \frac{7}{8} \div 9 \frac{9}{20}\right) \times \frac{11}{21}-\left(5 \div 2\right.$ of $\left.\frac{1}{2}\right)$ is:
Option 1: $\frac{15}{28}$
Option 2: $-2$
Option 3: $\frac{35}{24}$
Option 4: $0$
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 : The value of $\left(5 \frac{1}{4} \div \frac{3}{7}\right.$ of $\left.\frac{1}{2}\right) \div\left(5 \frac{1}{9}-7 \frac{7}{8} \div 9 \frac{9}{20}\right) \times \frac{11}{21}+\left(2 \div 2\right.$ of $\left.\frac{1}{2}\right)$ is:
Option 1: $\frac{7}{2}$
Option 2: $3$
Option 3: $5$
Option 4: $\frac{9}{4}$
Question : The value of $1 \frac{2}{5}-\left[3 \frac{3}{4} \div\left\{1 \frac{1}{4} \div \frac{1}{2}\left(1 \frac{1}{2} \times 3 \frac{1}{3} \div 1 \frac{1}{3}\right)\right\}\right]$ is:
Option 1: 3
Option 2: 0
Option 3: 2
Option 4: 1
Question : The value of $2 \frac{1}{3} \div 2 \frac{1}{2}$ of $1 \frac{3}{5}+\left(\frac{3}{8}+\frac{1}{7} \times 1 \frac{3}{4}\right)$ is:
Option 1: $\frac{25}{24}$
Option 2: $\frac{29}{24}$
Option 4: $\frac{5}{24}$
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