Question : If $x^2- 2xy = 84$ and $x-y=-10$, then the value of $y$ is:

Option 1: 2

Option 2: 1

Option 3: 4

Option 4: 3

Question : If $\cos\theta = \frac{x^2\:-\:y^2}{x^2\:+\:y^2}$, then the value of $\cot\theta$ is equal to:

Option 1: $\frac{2xy}{x^{2}\:-\:y^{2}}$

Option 2:

$\frac{2xy}{x^2\:+\:y^2}$

Option 3:

$\frac{x^{2}\:+\:y^{2}}{2xy}$

Option 4:

$\frac{x^{2}\:-\:y^{2}}{2xy}$

Question : The simplified value of $\small \left (1-\frac{2xy}{x^{2}+y^{2}}\right )\div\left (\frac{x^{3}-y^{3}}{x-y}-3xy\right)$ is:

Option 1: $\frac{1}{x^{2}-y^{3}}$

Option 2: $\frac{1}{x^{2}+y^{2}}$

Option 3: $\frac{1}{x-y}$

Option 4: $\frac{1}{x+y}$

Question : If $x^2+y^2=29$ and $xy=10$, where $x>0,y>0$ and $x>y$. Then the value of $\frac{x+y}{x-y}$ is:

Option 1: $- \frac{7}{3}$

Option 2: $\frac{7}{3}$

Option 3: $\frac{3}{7}$

Option 4: $-\frac{3}{7}$

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 : $\text { If } x^2+y^2+z^2=x y+y z+z x \text { and } x=1 \text {, then find the value of } \frac{10 x^4+5 y^4+7 z^4}{13 x^2 y^2+6 y^2 z^2+3 z^2 x^2}$.

Option 1: 2

Option 2: 0

Option 3: –1

Option 4: 1