Question : If $x+\frac{1}{x}=3$, then the value of $\frac{3x^{2}-4x+3}{x^{2}-x+1}$ is:

Option 1: $\frac{4}{3}$

Option 2: $\frac{3}{2}$

Option 3: $\frac{5}{2}$

Option 4: $\frac{5}{3}$

Question : If $\frac{x}{3}+\frac{3}{x}=1$, then the value of $x^3$ is:

Option 1: 1

Option 2: 27

Option 3: 0

Option 4: –27

Question : If $\sin (x - y) = \frac{1}2$ and $\cos (x + y) = \frac{1}2$, then what is the value of $\sin x \cos x + 2\sin^2x + cos^3x \sec x$?

Option 1: $2$

Option 2: $\sqrt{2}+1$

Option 3: $1$

Option 4: $\frac{3}{4}$

Question : If $x+\frac{1}{x}=0$, then the value of $x^{5}+\frac{1}{x^{5}}$ is:

Option 1: 2

Option 2:  –1

Option 3: 1

Option 4: 0

Question : Find the value of the expression $\frac{x^2-1}{x-1}-\frac{x^2-9}{x-3}$.

Option 1: 1

Option 2: –2

Option 3: –1

Option 4: 2