Amit M Agarwal Solutions for Chapter: Complex Numbers, Exercise 3: Work Book Exercise 4.3

If $\cos \theta +i\sin \theta$ is a root of the equation $x^n+a_1{x}^{n-1}+a_2{x}^{n-2}+\dots +a_{n-1}x+a_n=0$, then the value of $\sum _{r=1}^n{a}_r\cos r\theta$ is

If $\omega$ is a complex $n^{th}$ root of unity, then $\sum _{r=1}^n\left(ar+b\right){\omega }^{r-1}$ is equal to

If $\alpha , \beta$ respectively are the fifth and fourth non-real roots of unity, then the value of $\left(1+\alpha \right)\left(1+\beta \right)\left(1+{\alpha }^2\right)\left(1+{\beta }^2\right)\left(1+{\beta }^3\right)\left(1+{\alpha }^3\right)$ is

When the polynomial $5x^3+Mx+N$ is divided by $x^2+x+1$, the remainder is $0$. The value of $\left(M+N\right)$ is equal to

If $z$ and $w$ are two complex numbers simultaneously satisfying the equations, $z^3+w^5=0$ and $z^2·{\omega }^{-4}=1,$ then

Number of ordered pairs $\left(z,\omega \right)$ of the complex numbers $z$ and $\omega$, satisfying the system of equations, $z^3+(\overline{\omega }{)}^7=0$ and $z^5·{\omega }^{11}=1$ is

If $1, z_1, z_2, z_3, \dots , z_{n-1}$ is the $n^{\mathrm{th}}$ root of unity and $w$ is a non-real complex cube root of unity, then the product of $\prod _{r=1}^{n-1}\left(\omega -z_r\right)$ cannot be equal to

If $Z_r, r=1, 2, 3,\dots , 50$ are the roots of the equation $\sum _{r=0}^{50}{\left(Z\right)}^r=0$, then the value of $\sum _{r=1}^{50}\frac{1}{Z_r-1}$ is