TS EAMCET 2020 (Online) 14th September Evening Shift | Complex Numbers Question 103 | Mathematics

For $n>1$ and $n \in \mathbf{N}$, if $z_1, z_2, \ldots, z_n$ are the roots of the equation $(z+1)^n=z^n$, then $\sum_{i=1}^n \frac{\cot ^{-1}\left(2\left|\operatorname{Im} z_i\right|\right)-1}{2 \operatorname{Re} z_i}=$

$i$

$\frac{1}{2}[\pi-(\pi-2) n]$

$\frac{1}{2}[\pi+(\pi+2) n]$

TS EAMCET 2020 (Online) 14th September Morning Shift

MCQ (Single Correct Answer)

-0

$z_1, z_2$ are two fixed points on the Argand plane. If $z$ is a complex number such that $\left|z-z_1\right|+\left|z-z_2\right|=\lambda$, then the locus of $z$ is

a circle when $\left|z_1-z_2\right|<\lambda$

a parabola when $\left|z_1+z_2\right|=\lambda$

an ellipse when $\left|z_1-z_2\right|<\lambda$

a straight line when $\left|z_1\right|=\left|z_2\right|=\lambda$

TS EAMCET 2020 (Online) 14th September Morning Shift

MCQ (Single Correct Answer)

-0

The roots of the equation $(x-1)^5=32(x+1)^5$ are

$\frac{1+2 e^{\frac{2 k \pi i}{5}}}{1-2 e^{\frac{2 k \pi i}{5}}}, k=1,2,3,4,5$

$\frac{1-2 e^{\frac{2 k \pi i}{5}}}{1+2 e^{\frac{2 k \pi i}{5}}}, k=0,1,2,3,4$

$1,2 \omega, 3 \omega^2, 2 \omega+3 \omega^2, 5 \omega^2+7$

$\frac{3+2 e^{\frac{2(k+1) \pi i}{5}}}{3-2 e^{\frac{2(k+1) \pi i}{5}}}, k=0,1,2,3,4$

TS EAMCET 2020 (Online) 14th September Morning Shift

MCQ (Single Correct Answer)

-0

If $\omega$ is a non-real cube root of unity and $x=\omega^2-\omega-3$, then the value of $x^4+6 x^3+10 x^2-12 x-19$ is

-19

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