Question : If $\mathrm{k}+\frac{1}{\mathrm{k}}=4$, then what is the value of $\mathrm{k}^4+\frac{1}{\mathrm{k}^4}$ ?
Option 1: 410
Option 2: 192
Option 3: 212
Option 4: 194
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Correct Answer: 194
Solution :
Given, $\mathrm{k}+\frac{1}{\mathrm{k}}=4$,
Squaring both sides, we get
$\mathrm{k}^2+\frac{1}{\mathrm{k^2}}+2(k)(\frac{1}{k})=16$
⇒ $\mathrm{k}^2+\frac{1}{\mathrm{k^2}}+2=16$
⇒ $\mathrm{k}^2+\frac{1}{\mathrm{k^2}}=14$
Again squaring both sides, we get
$\mathrm{k}^4+\frac{1}{\mathrm{k^4}}+2(k^2)(\frac{1}{k^2})=196$
⇒ $\mathrm{k}^4+\frac{1}{\mathrm{k^4}}+2=196$
⇒ $\mathrm{k}^4+\frac{1}{\mathrm{k^4}}=194$
Hence, the correct answer is 194.
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