Answer:
1. (b)
2. (b)
Step-by-step explanation:
1.
We have,
[tex]x-\frac{1}{x}=5[/tex]
Now, squaring both sides, we get
[tex](x-\frac{1}{x}) ^{2}=5^{2}[/tex] .......(1)
Now, using the identity [tex](a-b)^2=a^2+b^2-2ab[/tex] in the LHS of the equation (1), we get
[tex]x^{2} +\frac{1}{x^2}-2\times x \times\frac{1}{x}=25[/tex]
⇒[tex]x^2+\frac{1}{x^2}-2=25[/tex]
⇒[tex]x^{2} +\frac{1}{x^2}=25+2=27[/tex]
∴ The correct answer is option (b).
2.
We have,
[tex]x-\frac{1}{x}=5[/tex]
Now, squaring both sides, we get
[tex](x-\frac{1}{x}) ^{2}=5^{2}[/tex] .......(1)
Now, using the identity [tex](a-b)^2=a^2+b^2-2ab[/tex] in the LHS of the equation (1), we get
[tex]x^{2} +\frac{1}{x^2}-2\times x \times\frac{1}{x}=25[/tex]
⇒[tex]x^2+\frac{1}{x^2}-2=25[/tex]
⇒[tex]x^{2} +\frac{1}{x^2}=25+2=27[/tex] .......(2)
Again, squaring both sides of equation (2), we get
[tex](x^2+\frac{1}{x^2})^2=(27)^2[/tex] .......(3)
Now, using the identity [tex](a+b)^2=a^2+b^2+2ab[/tex] in the LHS of the
equation (3), we get
[tex](x^2)^2+(\frac{1}{x^2})^2+2\times x^2\times \frac{1}{x^2}=729[/tex]
⇒[tex]x^4+\frac{1}{x^4}+2=729[/tex]
⇒[tex]x^4+\frac{1}{x^4}=729-2=727[/tex]
∴ The correct answer is option (b).
