Question : If $x^2-xy+y^2=2$ and $x^4+x^2y^2+y^4=6$, then the value of $(x^2+xy+y^2)$ is:

Option 1: 1

Option 2: 12

Option 3: 3

Option 4: 36

Question : If x + y = 7 and xy = 19, calculate the value of x² + y².

Option 1: 17

Option 2: 12

Option 3: 11

Option 4: 19

Question : If $( x^{3}-y^{3}):( x^{2}+xy+y^{2})=5:1$ and $( x^{2}-y^{2}):(x-y)=7:1$, then the ratio $2x:3y$ equals:

Option 1: $4:1$

Option 2: $2:3$

Option 3: $4:3$

Option 4: $3:2$

Question : If the highest common factor (HCF) of x and y is 15, then the HCF of 36x² – 81y² and 81x² – 9y² is divisible by ________.

Option 1: 135

Option 2: 120

Option 3: 180

Option 4: 90

Question : If $x=\frac{\sqrt{5}+1}{\sqrt{5}-1}$ and $y=\frac{\sqrt{5}-1}{\sqrt{5}+1}$, then the value of $\frac{x^{2}+xy+y^{2}}{x^{2}-xy+y^{2}}$ is:

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

Option 2: $\frac{4}{3}$

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

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