Complex Numbers and Quadratic Equations

Let arg(z) represent the principal argument of the complex number z. Then $\left|z\right|=3$ and art(z – 1) – arg(z + 1) = $\frac{\pi }{4}$ intersect

Let α be root of the equation 1 + x² + x⁴ = 0. Then the value of α¹⁰¹¹ + α²⁰²² - α³⁰³³ is equal to

If $A={\displaystyle \sum _{n=1}^{\infty }\frac{1}{{\left(3+{\left(-1\right)}^n\right)}^n}andB={\displaystyle \sum _{n=1}^{\infty }\frac{{\left(-1\right)}^n}{{\left(3+{\left(-1\right)}^n\right)}^n},then\frac{A}{B}}}$ ${\displaystyle \underset{}{\overset{}{}}\frac{}{{{}^{}}^{}}{\displaystyle \underset{}{\overset{}{}}\frac{{}^{}}{{{}^{}}^{}}\frac{}{}}}$ is equal to:

Let A be a 3 × 3 invertible matrix. If $\left|adj\left(24A\right)\right|=\left|adj\left(3adj\left(2A\right)\right)\right|,then{\left|A\right|}^2$ is equal to

Let a and b be the roots of the equation x² + (2i – 1) = 0. Then, the value of <!-- [if gte mso 9]><xml> </xml><![endif]-->  $\left|{\alpha }^8+{\beta }^8\right|$ is equal to:

Let O be the origin and A be the point z₁ = 1 + 2i. If B is the point z₂, Re(z₂) < 0, such that OAB is a right angled isosceles triangle with OB as hypotenuses, then which of the following is NOT true?

If z = x + iy satisfies $\left|z\right|-2=0and\left|z-i\right|-\left|z+5i\right|=0,then$

Let z = a ib, b $\ne$ 0 be complex numbers satisfying z² = $\overline{z}.2^{1-\left|z\right|}.$ Then the least value of n $\in$ N, such that zn = ${\left(z+1\right)}^n,$ is equal to………

Let $\lambda \ne 0$ be in R. If $\alpha$ and $\beta$ are the roots of the equation, $x^2-x+2\lambda =0$ and $\alpha$ and $\gamma$ are the roots of the equation, $3x^2-10x+27\lambda =0$ , then $\frac{\beta \gamma }{\lambda }$ is equal to: