Question

Q.4. If x + 1 / x = 2 find the value of $x^{99}+1/x^{100}$
Q.5. If x / y + y / x = - 1 (x, y $\ne$ 0), find the value of $x^3-y^3$.

Answer 1

Dear student,
Here are the solutions of your asked queries:
$$\begin{gathered} \mathrm{x}+\frac{1}{\mathrm{x}}=2 \\ \Rightarrow \mathrm{x}+\frac{1}{\mathrm{x}}-2=0 \\ \Rightarrow \frac{{\mathrm{x}}^2+1-2\mathrm{x}}{\mathrm{x}}=0 \\ \Rightarrow {\mathrm{x}}^2+1-2\mathrm{x}=0 \\ \Rightarrow {\left(\mathrm{x}-1\right)}^2=0 \\ \Rightarrow \mathrm{x}-1=0 \\ \Rightarrow \mathrm{x}=1 \\ \mathrm{So}\mathrm{we}\mathrm{have}; \\ {\mathrm{x}}^{99}+\frac{1}{{\mathrm{x}}^{100}}=1^{99}+\frac{1}{1^{100}}=1+1=2 \\ \mathrm{Q}.5.\frac{\mathrm{x}}{\mathrm{y}}+\frac{\mathrm{y}}{\mathrm{x}}=-1 \\ \Rightarrow {\mathrm{x}}^2+{\mathrm{y}}^2=-\mathrm{xy} \\ \Rightarrow {\mathrm{x}}^2+{\mathrm{y}}^2+\mathrm{xy}=0...\left(\mathrm{i}\right) \\ \mathrm{So}\mathrm{we}\mathrm{have}; \\ {\mathrm{x}}^3-{\mathrm{y}}^3 \\ =\left(\mathrm{x}-\mathrm{y}\right)\left({\mathrm{x}}^2+{\mathrm{y}}^2+\mathrm{xy}\right)\{\mathrm{since}{\mathrm{a}}^3-{\mathrm{b}}^3=\left(\mathrm{a}-\mathrm{b}\right)\left({\mathrm{a}}^2+\mathrm{ab}+{\mathrm{b}}^2\right)\} \\ =\left(\mathrm{x}-\mathrm{y}\right)\times 0\{\mathrm{using}(\mathrm{i}\left)\right\} \\ =0 \end{gathered}$$
Regards