Question : If $x=asin\theta-bcos\theta, y=acos\theta+bsin\theta$, then which of the following is true?

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

Option 2: $x^2+y^2=a^2-b^2$

Option 3: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

Option 4: $x^2+y^2=a^2+b^2$

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 $x^4+y^4+x^2 y^2=17 \frac{1}{16}$ and $x^2-x y+y^2=5 \frac{1}{4}$, then one of the values of $(x-y)$ is:

Option 1: $\frac{5}{2}$

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

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

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

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

Option 1: 16

Option 2: 13

Option 3: 12

Option 4: 14

Question : If $x+\frac{1}{x}=2 \cos \theta$, then $x^3+\frac{1}{x^3}=?$

Option 1: $2 \cos 2θ$

Option 2: $\cos 3θ$

Option 3: $2 \cos 3θ$

Option 4: $\cos 2θ$