Question 1: Palette 1/83

AP-EAPCET 2025 MCQ

If a complex number $z=x+i y$ represents a point $P$ on the argand plane and $\arg \left(\frac{z-3+2 i}{z+2-3 i}\right)=\frac{\pi}{4}$, then the locus of $P$ is a

A

circle with the line $x+y=12$ as its diameter

B

circle with radius $\sqrt{11}$

C

circle with the line $x-y=6$ as its diameter

D

circle with radius 5

Question 2: Palette 2/83

AP-EAPCET 2025 MCQ

By taking $\sqrt{a \pm i b}=x \pm i y, x>0$, if we get $\frac{\sqrt{21+12 \sqrt{2 i}}}{\sqrt{21-12 \sqrt{2 i}}}=a+i b$, then $\frac{b}{a}=$

A

$\frac{4 \sqrt{2}}{7}$

B

$\frac{12 \sqrt{2}}{17}$

C

$\frac{4 \sqrt{3}}{7}$

D

$\frac{12 \sqrt{3}}{17}$

Question 3: Palette 3/83

AP-EAPCET 2025 MCQ

Two values of $(-8-8 \sqrt{3} i)^{1 / 4}$ are

A

$\sqrt{3}-i,-1-\sqrt{3 i}$

B

$\sqrt{3}+i, 1+\sqrt{3} i$

C

$-\sqrt{3}+i, \sqrt{3}+i$

D

$1-\sqrt{3} i, \sqrt{3}+i$

Question 4: Palette 4/83

AP-EAPCET 2025 MCQ

If $z$ and $w$ are two non-zero complex numbers such that $|z w|=1$ and $\arg z-\arg w=\frac{\pi}{2}$, then $\bar{z} w=$

A

$i$

B

-1

C

1

D

$-i$

Question 5: Palette 5/83

AP-EAPCET 2025 MCQ

Let $z$ satisfy $|z|=1, z=1-\bar{z}$ and $\operatorname{Im}(z)>0$

Statement $\mathbf{I} z$ is a real number

Statement II Principal argument of $z$ is $\frac{\pi}{3}$.

Then,

A

Statement I is true, Statement II is true and Statement II is a correct explanation of Statement I

B

Statement I is true, Statement II is true, but Statement II is not a correct explanation of Statement I

C

Statement I is false, Statement II is true

D

Statement I is true, Statement II is false

Question 6: Palette 6/83

AP-EAPCET 2025 MCQ

If $w_1$ and $w_2$ are two non-zero complex numbers and ${ }a, b$ are non-zero real numbers such that $\left|a w_1+b w_2\right|=\left|a w_1-b w_2\right|$, then $\frac{w_1}{w_2}$ is

A

a positive real number

B

a negative real number

C

zero

D

purely imaginary number

Question 7: Palette 7/83

AP-EAPCET 2025 MCQ

If $\sinh ^{-1}(2)+\sinh ^{-1}(3)=\alpha$, then $\sinh \alpha=$

A

$2 \sqrt{5}+3 \sqrt{10}$

B

$2 \sqrt{10}+4 \sqrt{5}$

C

$3 \sqrt{10}+4 \sqrt{5}$

D

$2 \sqrt{10}+3 \sqrt{5}$

Question 8: Palette 8/83

AP-EAPCET 2025 MCQ

If $x=3-2 \sqrt{3} \mathrm{i}$, then $x^4-12 x^3+54 x^2-108 x-54=$

A

0

B

6

C

-6

D

9

Question 9: Palette 9/83

AP-EAPCET 2025 MCQ

$z_1, z_2, z_3$ represent the vertices $A, B, C$ of a $\triangle A B C$ respectively in the argand plane. If $\left|z_1-z_2\right|=\sqrt{25-12 \sqrt{3}},\left|\frac{z_1-z_3}{z_2-z_3}\right|=\frac{3}{4}$ and $\angle A C B=30^{\circ}$, then the area (in sq units) of that triangle is

A

$\frac{3}{2}$

B

3

C

5

D

$\frac{5}{2}$

Question 10: Palette 10/83

AP-EAPCET 2025 MCQ

The product of the four values of the complex number $(1+i)^{3 / 4}$ is

A

$2(1+i)$

B

$2(1-i)$

C

$2^3(1+i)$

D

$2^3(1-i)$

Question 11: Palette 11/83

AP-EAPCET 2025 MCQ

If the point $P$ denotes the complex number $z=x+i y$ in the argand plane and $\frac{z-(2-i)}{z+(1+2 i)}$ is purely imaginary number, then the locus of $P$ is

A

a hyperbola not containing the point $(-1,-2)$

B

an ellipse not containing the point $(-1,-2)$

C

a parabola not containing the point $(-1,-2)$

D

a circle not containing the point $(-1,-2)$ and having its centre on the line $x+y+1=0$

Question 12: Palette 12/83

AP-EAPCET 2025 MCQ

If $(\sqrt{3}-i)^n=2^n, n \in N$, then the least possible value of $n$ is

A

3

B

4

C

6

D

12

Question 13: Palette 13/83

AP-EAPCET 2025 MCQ

$ (1+\sqrt{5}+i \sqrt{10-2 \sqrt{5}})^5= $

A

1024

B

-1024

C

512

D

-512

Question 14: Palette 14/83

AP-EAPCET 2025 MCQ

If $z$ is a complex number such that $\frac{z-1}{z-i}$ is purely imaginary and locus of $z$ represents a circle with centre $(\alpha, \beta)$ and radius $r$, then $\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=$

A

$4 r$

B

$r^2$

C

$2 r^2$

D

$4 r^2$

Question 15: Palette 15/83

AP-EAPCET 2025 MCQ

If the least positive integer $n$ satisfying the equation $\left(\frac{\sqrt{3}+i}{\sqrt{3}-i}\right)^n=-1$ is $p$ and the least positive integer $m$ satisfying the equation $\left(\frac{1-\sqrt{3 i}}{1+\sqrt{3} i}\right)^m=\operatorname{cis} \frac{2 \pi}{3}$ is $q$, then $\sqrt{p^2+q^2}=$

A

5

B

10

C

$\sqrt{13}$

D

$\sqrt{17}$

Question 16: Palette 16/83

AP-EAPCET 2025 MCQ

Sum of the squares of the imaginary roots of the equation $z^8-20 z^4+64=0$ is

A

0

B

-12

C

-4

D

-16

Question 17: Palette 17/83

AP-EAPCET 2025 MCQ

For any two non-zero complex numbers $z_1$ and $z_2$, if $\left|z_1+z_2\right|^2=\left|z_1\right|^2+\left|z_2\right|^2$, then

A

$\operatorname{Re}\left(\frac{z_1}{z_2}\right)=0$

B

$\operatorname{lm}\left(\frac{z_1}{z_2}\right)=0$

C

$\operatorname{Re}\left(z_1 z_2\right)=0$

D

$\operatorname{lm}\left(z_1 z_2\right)=0$

Question 18: Palette 18/83

AP-EAPCET 2025 MCQ

If $1, \omega, \omega^2$ are the cube roots of unity, then

$ 1\left(2+\frac{1}{\omega}\right)\left(2+\frac{1}{\omega^2}\right)+2\left(3+\frac{1}{\omega}\right)\left(3+\frac{1}{\omega^2}\right) +3\left(4+\frac{1}{\omega}\right)\left(4+\frac{1}{\omega^2}\right)+\ldots 10 \text { terms }= $

A

3080

B

3465

C

3175

D

3715

Question 19: Palette 19/83

AP-EAPCET 2025 MCQ

$ (1+\sqrt{3} i)^6-(\sqrt{3}+i)^6= $

A

0

B

32

C

64

D

128

Question 20: Palette 20/83

AP-EAPCET 2025 MCQ

If $z=x+i y$ and $x^2+y^2=1$, then $\frac{1+x+i y}{1+x-i y}=$

A

$\bar{z}$

B

$z$

C

$z+1$

D

$z-1$

Question 21: Palette 21/83

AP-EAPCET 2025 MCQ

If $x^6=(\sqrt{3}-i)^5$, then the product of all of its roots is

A

$2^5(\sqrt{3}+i)$

B

$\frac{2^6}{\sqrt{3}+i}$

C

$2^6(\sqrt{3}-i)$

D

$\frac{2^6}{\sqrt{3}-i}$

Question 22: Palette 22/83

AP-EAPCET 2025 MCQ

The minimum value of $|z-1|+|z-5|$ is

A

3

B

5

C

4

D

2

Question 23: Palette 23/83

AP-EAPCET 2025 MCQ

If $z=x+i y$ and if the point $P$ in the argand diagram represents $z$, then the locus of the point $P$ satisfying the equation $2|z-2-3 i|=3|z+i-2|$ is a circle with centre

A

$(10,-21)$

B

$\left(2,-\frac{21}{5}\right)$

C

$(-10,21)$

D

$\left(-2, \frac{21}{5}\right)$

Question 24: Palette 24/83

AP-EAPCET 2025 MCQ

If $z$ is a non-real root of $x^7=1$, then $1+3 z+5 z^2+7 z^3+9 z^4+11 z^5+13 z^6=$

A

$\frac{14}{1-z}$

B

$\frac{-14}{1-z}$

C

$\frac{15}{1-z}$

D

$\frac{-15}{1-z}$

Question 25: Palette 25/83

AP-EAPCET 2025 MCQ

If $\cosh 2 x=199$, then $\cot h x=$

A

$\frac{5}{3 \sqrt{11}}$

B

$\frac{5}{6 \sqrt{11}}$

C

$\frac{7}{3 \sqrt{11}}$

D

$\frac{10}{3 \sqrt{11}}$

Question 26: Palette 26/83

AP-EAPCET 2025 MCQ

If $a=\operatorname{Im}\left(\frac{1+z^2}{2 i z}\right)$ and $z$ is any non-zero complex number such that $|z|=1$, then $a=$

A

$\operatorname{Re}(z)$

B

$\operatorname{Re}(z) \operatorname{Im}(z)$

C

$-\operatorname{Re}(z)$

D

$\operatorname{Re}(z)+\operatorname{Im}(z)$

Question 27: Palette 27/83

AP-EAPCET 2025 MCQ

If $(3+4 i)^{2025}=5^{2023}(x+i y)$, then $\sqrt{x^2+y^2}=$

A

5

B

25

C

125

D

625

Question 28: Palette 28/83

AP-EAPCET 2025 MCQ

If $\left(\frac{\cos \theta+i \sin \theta}{\sin \theta+i \cos \theta}\right)^{2024}+\left(\frac{1+\cos \theta+i \sin \theta}{1-\cos \theta+i \sin \theta}\right)^{2025}=x+i y$ then the value of $x+y$ at $\theta=\frac{\pi}{2}$ is

A

1

B

-1

C

2

D

2024

Question 29: Palette 29/83

AP-EAPCET 2025 MCQ

If $a \pm i b$ and $b \pm a i$ are the roots of $x^4-10 x^3+50 x^2-130 x+169=0$, then $\frac{a}{b}+\frac{b}{a}=$

A

$\frac{25}{12}$

B

$\frac{5}{2}$

C

$\frac{13}{6}$

D

$\frac{34}{15}$

Question 30: Palette 30/83

AP-EAPCET 2025 MCQ

If $i=\sqrt{-1}$, then $\sum\limits_{n=2}^{30} i^n+\sum\limits_{n=30}^{65} i^{n+3}=$

A

0

B

-1

C

i

D

-1

Question 31: Palette 31/83

AP-EAPCET 2025 MCQ

If $z_1$ and $z_2$ are two of the $n$th roots of unity such that the line segment joining them subtends at a right angle at the origin, then for a positive integer $k, n$ takes the form

A

$4 k$

B

$4 k+1$

C

$4 k+2$

D

$4 k+3$

Question 32: Palette 32/83

AP-EAPCET 2025 MCQ

$ (\sqrt{\sqrt{2}+1}+i \sqrt{\sqrt{2}-1})^8= $

A

64

B

$64 i$

C

-64

D

$-64 i$

Question 33: Palette 33/83

AP-EAPCET 2024 MCQ

$\omega$ is a complex cube root of unity and if $z$ is a complex number satisfying $|z-1| \leq 2$ and $\left|\omega^2 z-1-\omega\right|=a$, then the set of possible values of $a$ is

A

$0 \leq a \leq 2$

B

$\frac{1}{2} \leq a \leq \frac{\sqrt{3}}{2}$

C

$|\omega| \leq a \leq \frac{\sqrt{3}}{2}+2$

D

$0 \leq a \leq 4$

Question 34: Palette 34/83

AP-EAPCET 2024 MCQ

If the roots of the equation $z^3+i z^2+2 i=0$ are the vertices of a $\triangle A B C$, then that $\triangle A B C$ is

A

a right angled triangle

B

an equilateral triangle

C

an isosceles triangle

D

a right angled isosceles triangle

Question 35: Palette 35/83

AP-EAPCET 2024 MCQ

$(r, \theta)$ denotes $r(\cos \theta+i \sin \theta)$. If $x=(1, \alpha), y=(1, \beta), z=(1, \gamma)$ and $x+y+z=0$, then $\Sigma \cos (2 \alpha-\beta-\gamma)$ is equal to

A

3

B

0

C

1

D

-1

Question 36: Palette 36/83

AP-EAPCET 2024 MCQ

$\arg \left[\frac{(1+i \sqrt{3})(-\sqrt{3}-i)}{(1-i)(-i)}\right]$ is equal to

A

$\frac{5 \pi}{6}$

B

$\frac{\pi}{4}$

C

$\frac{2 \pi}{3}$

D

$\frac{-\pi}{2}$

Question 37: Palette 37/83

AP-EAPCET 2024 MCQ

If $P(x, y)$ represents the complex number $z=x+iy$ in the argand plane and $\arg \left(\frac{z-3 i}{z+4}\right)=\frac{\pi}{2}$, then the equation of the locus of $P$ is

A

$x^2+y^2+4 x-3 y=0$ and $3 x-4 y>0$

B

$x^2+y^2+4 x-3 y+2=0$ and $3 x-4 y>0$

C

$x^2+y^2+4 x-3 y=0$ and $3 x-4 y<0$

D

$x^2+y^2+4 x-3 y+2=0$ and $3 x-4 y<0$

Question 38: Palette 38/83

AP-EAPCET 2024 MCQ

If $\alpha_1, \alpha_2, \alpha_3, \alpha_4$ and $\alpha_5$ are the roots of $x^5-5 x^4+9 x^3-9 x^2+5 x-1=0$, then $\frac{1}{\alpha_1^2}+\frac{1}{\alpha_2^2}+\frac{1}{\alpha_3^2}+\frac{1}{\alpha_4^2}+\frac{1}{\alpha_5^2}$ is equal to

A

15

B

$\frac{1}{7}$

C

7

D

12

Question 39: Palette 39/83

AP-EAPCET 2024 MCQ

If $Z$ is a complex number such that $|Z| \leq 3$ and $\frac{-\pi}{2} \leq \operatorname{amp} Z \leq \frac{\pi}{2}$, then the area of the region formed by locus of $Z$ is

A

$9 \pi$

B

$\frac{9 \pi}{2}$

C

$3 \pi$

D

$\frac{9 \pi}{4}$

Question 40: Palette 40/83

AP-EAPCET 2024 MCQ

The locus of the complex number $Z$ such that $\arg \left(\frac{Z-1}{Z+1}\right)=\frac{\pi}{4}$ is

A

a straight line

B

a circle

C

a parabola

D

an ellipse

Question 41: Palette 41/83

AP-EAPCET 2024 MCQ

All the values of $(8 i)^{\frac{1}{3}}$ are

A

$\pm(\sqrt{3}+i),-2 i$

B

$\pm \sqrt{3}+i,-2 i$

C

$\pm(\sqrt{3}-i), 2 i$

D

$\pm(2+i), i$

Question 42: Palette 42/83

AP-EAPCET 2024 MCQ

If the number of real roots of $x^9-x^5+x^4-1=0$ is $n$, the number of complex roots having argument on imaginary axis is $m$ and the number of complex roots having argument in 2nd quadrant is $K, m \cdot n \cdot k=$

A

6

B

9

C

12

D

24

Question 43: Palette 43/83

AP-EAPCET 2024 MCQ

Imaginary part of $\frac{(1-i)^3}{(2-i)(3-2 i)}$ is

A

$\frac{22}{65}$

B

$\frac{6}{65}$

C

$-\frac{6}{65}$

D

$-\frac{22}{65}$

Question 44: Palette 44/83

AP-EAPCET 2024 MCQ

The square root of $7+24 i$

A

$4-3 i$

B

$3+4 i$

C

$3-4 i$

D

$4+3 i$

Question 45: Palette 45/83

AP-EAPCET 2024 MCQ

If $n$ is an integer and $Z=\cos \theta+i \sin \theta, \theta \neq(2 n+1) \frac{\pi}{2}$, then $\frac{1+Z^{2 n}}{1-Z^{2 n}}=$

A

$i \tan n \theta$

B

$i \cot n \theta$

C

$-i \tan n \theta$

D

$-i \cot n \theta$

Question 46: Palette 46/83

AP-EAPCET 2024 MCQ

The complex conjugate of $(4-3 i)(2+3 i)(1+4 i)$ is.

A

$7+74 i$

B

$-7+74 i$

C

$-7-74 i$

D

$7-74 i$

Question 47: Palette 47/83

AP-EAPCET 2024 MCQ

If the amplitude of $(z-2)$ is $\frac{\pi}{2}$, then the locus of $z$ is

A

$x=0, y>0$

B

$x=2, y>0$

C

$x>0, y=2$

D

$x>0, y=0$

Question 48: Palette 48/83

AP-EAPCET 2024 MCQ

If $\omega$ is the cube root of unity, $ \frac{a+b \omega+c \omega^2}{c+a \omega+b \omega^2}+\frac{a+b \omega+c \omega^2}{b+c \omega+b \omega^2}= $

A

2

B

-2

C

1

D

-1

Question 49: Palette 49/83

AP-EAPCET 2024 MCQ

If $(3+i)$ is a root of $x^2+a x+b=0$, then $a=$

A

3

B

-3

C

6

D

-6

Question 50: Palette 50/83

AP-EAPCET 2024 MCQ

If $z_1=10+6 i, z_2=4+6 i$ and $z$ is any complex number such that the argument of $\frac{\left(z-z_1\right)}{\left(z-z_2\right)}$ is $\frac{\pi}{4}$,

A

$|z-7-9 i|=3 \sqrt{2}$

B

$|z-7-9 i|=2 \sqrt{2}$

C

$|z-3+9 i|=3 \sqrt{2}$

D

$|z+3-9 i|=2 \sqrt{2}$

Question 51: Palette 51/83

AP-EAPCET 2024 MCQ

If $\frac{3-2 i \sin \theta}{1+2 i \sin \theta}$ is purely imaginary number, then $\theta=$

A

$2 n \pi \pm \frac{\pi}{4}$

B

$2 n \pi \pm \frac{\pi}{2}$

C

$n \pi \pm \frac{\pi}{3}$

D

$n \pi \pm \frac{\pi}{6}$

Question 52: Palette 52/83

AP-EAPCET 2024 MCQ

If $z=x+i y, x^2+y^2=1$ and $z_1=z e^{i \theta}$, then $\frac{z_1^{2 n}-1}{z_1^{2 n}+1}=$

A

$-i \tan \left(n\left(\theta+\tan ^{-1}\left(\frac{y}{x}\right)\right)\right)$

B

$i \cot \left(n\left(\theta+\tan ^{-1} \frac{y}{x}\right)\right)$

C

$i \tan \left(n\left(\theta+\tan ^{-1} \frac{x}{u}\right)\right)$

D

$i \tan \left(n\left(\theta+\tan ^{-1} \frac{y}{x}\right)\right)$

Question 53: Palette 53/83

AP-EAPCET 2024 MCQ

If the point $P$ represents the complex number $z=x+i y$ in the argand plane and if $\frac{z+i}{z-i}$ is a purely imaginary number, then the locus of $P$ is

A

$x^2+y^2+x-y=0$ and $(x, y) \neq(1,0)$

B

$x^2+y^2-x+y=0$ and $(x, y) \neq(1,0)$

C

$x^2+y^2-x+y=0$ and $(x, y)=(1,0)$

D

$x^2+y^2+x+y=0$

Question 54: Palette 54/83

AP-EAPCET 2024 MCQ

$S=\{z \in C /|z+1-i|=1\}$ represents

A

the circle with centre at $(-1,1)$ and radius 1 unit

B

the circle with cente at $(1,-1)$ and radius 1 unit

C

the closed circular disc with centre at $(1,-1)$ and radius 1 unt

D

the closed circular disc with centre at( $-1,1$ ) and radius 1 unt

Question 55: Palette 55/83

AP-EAPCET 2024 MCQ

If $m, n$ are respectively the least positive and greatest negative integer value of $k$ such that $\left(\frac{1-i}{1+i}\right)^k=-i$, then $m-n=$

A

4

B

0

C

6

D

2

Question 56: Palette 56/83

AP-EAPCET 2024 MCQ

If a complex number $z$ is such that $\frac{z-2 i}{z-2}$ is purely imaginary number and the locus of $z$ is a closed curve, then the area of the region bounded by that closed curve and lying in the first quadrant is $\frac{z-2 i}{z-2}$

A

$2 \pi$

B

$\frac{\pi}{2}$

C

$\pi$

D

$\frac{\pi}{4}$

Question 57: Palette 57/83

AP-EAPCET 2024 MCQ

Real part of $\frac{(\cos a+i \sin a)^6}{(\sin b+i \cos b)^8}$ is

A

$\sin (6 a-8 b)$

B

$\cos (6 a-8 b)$

C

$\sin (6 a+8 b)$

D

$\cos (6 a+8 b)$

Question 58: Palette 58/83

AP-EAPCET 2024 MCQ

If real parts of $\sqrt{-5-12 i}, \sqrt{5+12 i}$ are positive values, the real part of $\sqrt{-8-6 i}$ is a negative value and $a+i b=\frac{\sqrt{-5-12 i}+\sqrt{5+12 i}}{\sqrt{-8-6 i}}$, then $2 a+b=$

A

3

B

2

C

-3

D

-2

Question 59: Palette 59/83

AP-EAPCET 2024 MCQ

The set of all real values of $ c $ for which the equation $ z\overline{z} + (4 - 3i)z + (4 + 3i)\overline{z} + c = 0 $ represents a circle, is

A

[25, 50]

B

[-5, 5]

C

$[-20, -5] \cup [5, 20]$

D

[-25]

Question 60: Palette 60/83

AP-EAPCET 2024 MCQ

If $ z = x + iy $ is a complex number, then the number of distinct solutions of the equation $ z^3 + \overline{z} = 0 $ is

A

1

B

3

C

Infinite

D

5

Question 61: Palette 61/83

AP-EAPCET 2022 MCQ

By simplifying $i^{18}-3 i^7+i^2\left(1+i^4\right)(i)^{22}$, we get

A

$-1+3 i$

B

$1-3 i$

C

$1+3 i$

D

$-1-3 i$

Question 62: Palette 62/83

AP-EAPCET 2022 MCQ

The values of $x$ for which $\sin x+i \cos 2 x$ and $\cos x-i \sin 2 x$ are conjugate to each other are

A

$x=n \pi \pm \frac{\pi}{6}$

B

None

C

$x=n \pi \pm \frac{\pi}{3}$

D

$x=\left(n+\frac{1}{2}\right) \pi$

Question 63: Palette 63/83

AP-EAPCET 2022 MCQ

The locus of a point $z$ satisfying $|z|^2=\operatorname{Re}(z)$ is a circle with centre

A

$\left(0, \frac{1}{2}\right)$

B

$\left(-\frac{1}{2}, 0\right)$

C

$\left(\frac{1}{2}, 0\right)$

D

$\left(0,-\frac{1}{2}\right)$

Question 64: Palette 64/83

AP-EAPCET 2022 MCQ

Multiplicative inverse of the complex number $(\sin \theta, \cos \theta)$ is

A

$(\sin \theta, \cos \theta)$

B

$(\sin \theta,-\cos \theta)$

C

$(\cos \theta,-\sin \theta)$

D

$(-\cos \theta, \sin \theta)$

Question 65: Palette 65/83

AP-EAPCET 2022 MCQ

$\sum_\limits{k=0}^{440} i^k=x+i y \Rightarrow x^{100}+x^{99} y+x^{242} y^2+x^{97} y^3=$

A

0

B

$-$4

C

4

D

1

Question 66: Palette 66/83

AP-EAPCET 2022 MCQ

If $e^{i \theta}=\operatorname{cis} \theta$, then $\sum_\limits{n=0}^{\infty} \frac{\cos (n \theta)}{2^n}=$

A

$(4+2 \cos \theta) /(5-4 \cos \theta)$

B

$(4-2 \cos \theta) /(5+4 \cos \theta)$

C

$(4-2 \cos \theta) /(5-4 \cos \theta)$

D

$(4+2 \cos \theta) /(5+4 \cos \theta)$

Question 67: Palette 67/83

AP-EAPCET 2022 MCQ

$i z^3+z^2-z+i=0 \Rightarrow|z|=$

A

1/2

B

2

C

3/2

D

1

Question 68: Palette 68/83

AP-EAPCET 2022 MCQ

If $\frac{x-1}{3+i}+\frac{y-1}{3-i}=i$, then the true statement among the following is

A

$x<0, y<0$

B

$x<0, y>0$

C

$x>0, y<0$

D

$x>0, y>0$

Question 69: Palette 69/83

AP-EAPCET 2022 MCQ

The number of integer solutions of the equation $|1-i|^x=2^x$ is

A

1

B

0

C

2

D

3

Question 70: Palette 70/83

AP-EAPCET 2021 MCQ

Let $Z_1, Z_2$ and $Z_3$ be three non zero complex numbers such that $a=\left|Z_1\right|, b=\left|Z_2\right|$ and $c=\left|Z_3\right|$, if the determinant $\left|\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right|=0$, then

A

$\left|Z_1\right|=\left|Z_2\right|=\left|Z_3\right|=a b c$

B

$\left|Z_1\right|+\left|Z_2\right|+\left|Z_3\right|=0$

C

$\left|Z_1\right|+\left|Z_2\right|+\left|Z_3\right|=a b c$

D

$\left|Z_1-Z_2\right|=\left|Z_2-Z_3\right|$

Question 71: Palette 71/83

AP-EAPCET 2021 MCQ

If $\left|z_1+z_2\right|^2=\left|z_1\right|^2+\left|z_2\right|^2$, where $z_1$ and $z_2$ are two complex numbers, then

A

$\frac{z_1}{z_2}$ is purely real

B

$\frac{z_1}{z_2}$ is purely imaginary

C

$\arg \left(\frac{z_1}{z_2}\right)=\frac{\pi}{4}$

D

$\left|\frac{z_1}{z_2}\right|=1$

Question 72: Palette 72/83

AP-EAPCET 2021 MCQ

A real value of $x$ will satisfy the equation, $\left(\frac{3-4 i x}{3+4 i x}\right)=\alpha-i \beta,(\alpha, \beta$ are real $)$, if

A

$\alpha^2-\beta^2=-1$

B

$\alpha^2-\beta^2=1$

C

$\alpha^2+\beta^2=1$

D

$\alpha^2-\beta^2=2$

Question 73: Palette 73/83

AP-EAPCET 2021 MCQ

What is the value of $(1-i \sqrt{3})^9$ is equal to

A

$2^9$

B

$-2^9$

C

$2^9 i$

D

$-2^9 i$

Question 74: Palette 74/83

AP-EAPCET 2021 MCQ

$\left(\frac{\sqrt{6}-\sqrt{2}}{4}+\frac{\sqrt{6}+\sqrt{2}}{4} i\right)^{2020}$ is equal to

A

$\frac{1}{2}+\frac{\sqrt{3}}{2} i$

B

$\frac{-1}{2}+\frac{\sqrt{3}}{2} i$

C

$\frac{-1}{2}-\frac{\sqrt{3}}{2} i$

D

$\frac{1}{2}-\frac{\sqrt{3}}{2} i$

Question 75: Palette 75/83

AP-EAPCET 2021 MCQ

If $z_1=2+3 i$ and $z_2=3+2 i$, where $i=\sqrt{-1}$, then $\left[\begin{array}{cc}z_1 & z_2 \\ -\bar{z}_2 & \bar{z}_1\end{array}\right]\left[\begin{array}{cc}\bar{z}_1 & -z_2 \\ \bar{z}_2 & z_1\end{array}\right]$ is equal to

A

$13 I$

B

$I$

C

$26 I$

D

Zero matrix

Question 76: Palette 76/83

AP-EAPCET 2021 MCQ

The radius of the circle represented by $(1+i)(1+3i)(1+7i)=x+iy$ is $(i=\sqrt{-1})$.

A

1000

B

10$\sqrt{10}$

C

10000

D

100

Question 77: Palette 77/83

AP-EAPCET 2021 MCQ

If $1, \alpha_1, \alpha_2, \alpha_3$ and $\alpha_4$ are the roots of $z^5-1=0$ and $\omega$ is a cube root of units, then $(\omega-1)\left(\omega-\alpha_1\right)\left(\omega-\alpha_2\right)\left(\omega-\alpha_3\right)\left(\omega-\alpha_4\right)+\omega$ is equal to

A

0

B

$-$1

C

$-$2

D

1

Question 78: Palette 78/83

AP-EAPCET 2021 MCQ

If $a > 0$ and $z=x+i y$, then $\log _{\cos ^2 \theta}|z-a|>\log _{\cos ^2 \theta}|z-a i|,(\theta \in R)$ implies

A

$x>y$

B

$x < y$

C

$x+y=\cos \theta$

D

$x+y<0$

Question 79: Palette 79/83

AP-EAPCET 2021 MCQ

If one root of the equation $i x^2-2(i+1) x+(2-i)=0$ is $(2-i)$, then the other root is

A

$-i$

B

$2+i$

C

$i$

D

$2-i$

Question 80: Palette 80/83

AP-EAPCET 2021 MCQ

If $|z-2|=|z-1|$, where $z$ is a complex number, then locus $z$ is a straight line

A

Parallel to $X$ - axis

B

Parallel to $Y$-axis

C

Parallel to $y=x$

D

Parallel to $y=-x$

Question 81: Palette 81/83

AP-EAPCET 2021 MCQ

If ${\left( {{{1 + i} \over {1 - i}}} \right)^m} = 1$, then m cannot be equal to

A

1934

B

2024

C

2172

D

10¹⁰⁰

Question 82: Palette 82/83

AP-EAPCET 2021 MCQ

$(\sin \theta-i \cos \theta)^3$ is equal to

A

$i^3(\cos 3 \theta+i \sin 3 \theta)$

B

$\cos 3 \theta+i \sin 3 \theta$

C

$\sin 3 \theta-i \cos 3 \theta$

D

$(-i)^3(\cos 3 \theta+i \sin 3 \theta)$

Question 83: Palette 83/83

AP-EAPCET 2021 MCQ

Real part of $(\cos 4+i \sin 4+1)^{2020}$ is

A

$2^{2020} \cos ^{2020} 2 \cos 2020$

B

$2^{2020} \cos ^{2020} 2 \cos 4040$

C

$2^{1020} \cos ^{2020} 2 \cos 4040$

D

$2^{2020} \cos ^{2020} 1 \cos 2020$