Question : If $\theta$ is a positive acute angle and $\tan2\theta \tan3\theta=1$, then the value of $2(\cos^2\frac{5\theta}{2}-1)$ is:

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

Option 2: -1

Option 3: 0

Option 4: $\frac{1}{2}$

Question : If $(2+\sqrt{3})a=(2-\sqrt{3})b=1$, then the value of $\frac{1}{a}+\frac{1}{b}$ is:

Option 1: $1$

Option 2: $2$

Option 3: $2\sqrt{3}$

Option 4: $4$

Question : If $\frac{1}{a}(a^2+1)=3$, then the value of $\frac{a^6+1}{a^3}$ is:

Option 1: 9

Option 2: 18

Option 3: 27

Option 4: 1

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

Option 1: 0

Option 2: 2

Option 3: –2

Option 4: 4

Question : If $\tan\theta-\cot\theta=0$ and $\theta$ is positive acute angle, then the value of $\frac{\tan(\theta+15)}{\tan(\theta-15)}$ is:

Option 1: $3$

Option 2: $\frac{1}{\sqrt{3}}$

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

Option 4: $\sqrt{3}$