Complex Numbers · Mathematics · AP EAPCET

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MCQ (Single Correct Answer)

1
$\omega$ 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
AP EAPCET 2024 - 23th May Morning Shift
2
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
AP EAPCET 2024 - 23th May Morning Shift
3

$(r, \theta)$ 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

AP EAPCET 2024 - 23th May Morning Shift
4
$\arg \left[\frac{(1+i \sqrt{3})(-\sqrt{3}-i)}{(1-i)(-i)}\right]$ is equal to
AP EAPCET 2024 - 22th May Evening Shift
5

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

AP EAPCET 2024 - 22th May Evening Shift
6

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

AP EAPCET 2024 - 22th May Evening Shift
7

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

AP EAPCET 2024 - 22th May Morning Shift
8
The locus of the complex number $Z$ such that $\arg \left(\frac{Z-1}{Z+1}\right)=\frac{\pi}{4}$ is
AP EAPCET 2024 - 22th May Morning Shift
9
All the values of $(8 i)^{\frac{1}{3}}$ are
AP EAPCET 2024 - 22th May Morning Shift
10
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 $K, m \cdot n \cdot k=$
AP EAPCET 2024 - 22th May Morning Shift
11
Imaginary part of $\frac{(1-i)^3}{(2-i)(3-2 i)}$ is
AP EAPCET 2024 - 21th May Evening Shift
12
The square root of $7+24 i$
AP EAPCET 2024 - 21th May Evening Shift
13
If $n$ is an integer and $Z=\cos \theta+i \sin \theta, \theta \neq(2 n+1) \frac{\pi}{2}$, then $\frac{1+Z^{2 n}}{1-Z^{2 n}}=$
AP EAPCET 2024 - 21th May Evening Shift
14
The complex conjugate of $(4-3 i)(2+3 i)(1+4 i)$ is.
AP EAPCET 2024 - 21th May Morning Shift
15
If the amplitude of $(z-2)$ is $\frac{\pi}{2}$, then the locus of $z$ is
AP EAPCET 2024 - 21th May Morning Shift
16
If $\omega$ is the cube root of unity, $$ \frac{a+b \omega+c \omega^2}{c+a \omega+b \omega^2}+\frac{a+b \omega+c \omega^2}{b+c \omega+b \omega^2}= $$
AP EAPCET 2024 - 21th May Morning Shift
17
If $(3+i)$ is a root of $x^2+a x+b=0$, then $a=$
AP EAPCET 2024 - 21th May Morning Shift
18
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 $\frac{\pi}{4}$,
AP EAPCET 2024 - 20th May Evening Shift
19
If $\frac{3-2 i \sin \theta}{1+2 i \sin \theta}$ is purely imaginary number, then $\theta=$
AP EAPCET 2024 - 20th May Evening Shift
20
If $z=x+i y, x^2+y^2=1$ and $z_1=z e^{i \theta}$, then $\frac{z_1^{2 n}-1}{z_1^{2 n}+1}=$
AP EAPCET 2024 - 20th May Evening Shift
21
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
AP EAPCET 2024 - 20th May Morning Shift
22
$S=\{z \in C /|z+1-i|=1\}$ represents
AP EAPCET 2024 - 20th May Morning Shift
23
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 $m-n=$
AP EAPCET 2024 - 19th May Evening Shift
24
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 $\frac{z-2 i}{z-2}$
AP EAPCET 2024 - 19th May Evening Shift
25
Real part of $\frac{(\cos a+i \sin a)^6}{(\sin b+i \cos b)^8}$ is
AP EAPCET 2024 - 19th May Evening Shift
26
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 $2 a+b=$
AP EAPCET 2024 - 18th May Morning Shift
27
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
AP EAPCET 2024 - 18th May Morning Shift
28
If $ z = x + iy $ is a complex number, then the number of distinct solutions of the equation $ z^3 + \overline{z} = 0 $ is
AP EAPCET 2024 - 18th May Morning Shift
29

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

AP EAPCET 2022 - 5th July Morning Shift
30

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

AP EAPCET 2022 - 5th July Morning Shift
31

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

AP EAPCET 2022 - 5th July Morning Shift
32

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

AP EAPCET 2022 - 4th July Evening Shift
33

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

AP EAPCET 2022 - 4th July Evening Shift
34

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

AP EAPCET 2022 - 4th July Evening Shift
35

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

AP EAPCET 2022 - 4th July Morning Shift
36

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

AP EAPCET 2022 - 4th July Morning Shift
37

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

AP EAPCET 2022 - 4th July Morning Shift
38

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

AP EAPCET 2021 - 20th August Morning Shift
39

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

AP EAPCET 2021 - 20th August Morning Shift
40

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

AP EAPCET 2021 - 20th August Morning Shift
41

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

AP EAPCET 2021 - 20th August Morning Shift
42

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

AP EAPCET 2021 - 20th August Morning Shift
43

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

AP EAPCET 2021 - 19th August Evening Shift
44

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

AP EAPCET 2021 - 19th August Evening Shift
45

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

AP EAPCET 2021 - 19th August Morning Shift
46

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

AP EAPCET 2021 - 19th August Morning Shift
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