Question : If $x\sin^{3}\theta +y\cos^{3}\theta=\sin\theta\cos\theta$ and $x\sin\theta-y\cos\theta=0$, then the value of $\left ( x^{2}+y^{2} \right )$ equals:

Option 1: $1$

Option 2: $\frac{1}{2}$

Option 3: $\frac{3}2$

Option 4: $2$

Question : If $x=a(\sin\theta+\cos\theta), y=b(\sin\theta-\cos\theta)$, then the value of $\frac{x^2}{a^2}+\frac{y^2}{b^2}$ is:

Option 1: 0

Option 2: 1

Option 3: 2

Option 4: –2

Question : If $\tan\theta=1$, then the value of $\frac{8\sin\theta\:+\:5\cos\theta}{\sin^{3}\theta\:–\:2\cos^{3}\theta\:+\:7\cos\theta}$ is:

Option 1: $2$

Option 2: $2\frac{1}{2}$

Option 3: $3$

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

Question : If $\sin\theta+\sin^{2}\theta=1$, then the value of $\cos^{12}\theta+3\cos^{10}\theta+3\cos^{8}\theta+\cos^{6}\theta-1$ is:

Option 1: 1

Option 2: 2

Option 3: 3

Option 4: 0

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

Option 1: $\frac{5}{12}$

Option 2: $\frac{12}{13}$

Option 3: $\frac{11}{12}$

Option 4: $\frac{5}{13}$