Let $z$ be a complex number (not lying on the $X$-axis) of maximum modulus such that $\left| z + \frac{1}{z} \right| = 1$. Then:

In a cubic equation,the coefficient of $x^2$ is $0$ and the remaining coefficients are real. If one root is $\alpha = 3 + 4i$,and the remaining roots are $\beta$ and $\gamma$,then find the value of $\alpha \beta \gamma$.

If $z_1=x_1+i y_1$, $z_2=x_2+i y_2$, $z_3=x_1+\frac{i x_2}{2}$, and $z_4=2 y_1+i y_2$ are complex numbers such that $|z_1|=1$, $|z_2|=2$, and $\operatorname{Re}(z_1 \bar{z}_2)=0$, then:

If $x_r = \cos(\pi/3^r) - i\sin(\pi/3^r)$ (where $i = \sqrt{-1}$),then the value of $x_1 \cdot x_2 \cdot x_3 \cdots \infty$ is:

If $\cos \alpha+3 \cos 3 \beta+5 \cos 5 \gamma=0$,$\sin \alpha+3 \sin 3 \beta+5 \sin 5 \gamma=0$ and $\cos 3 \alpha+27 \cos 9 \beta+125 \cos 15 \gamma=\left(\lambda^2-4\right) \cos (\alpha+3 \beta+5 \gamma)$,then $\lambda$ is equal to

The value of $\left(\frac{1+\sin \frac{2 \pi}{9}+i \cos \frac{2 \pi}{9}}{1+\sin \frac{2 \pi}{9}-i \cos \frac{2 \pi}{9}}\right)^3$ is