Complex Numbers PYQs - Last 5 Years

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All Complex Numbers PYQs Last 10 Years Create chapter test

Practice 100 TS EAMCET Mathematics questions from Complex Numbers. Use the year-wise and type-wise breakdown to prioritize recent PYQs, then continue into the question list below.

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Last 5 Years Complex Numbers Questions

Showing 50 of 100 filtered questions.

1Complex Numbers

If $z=\frac{3+2 i \cos \theta}{1-2 i \sin \theta}$ is a purely imaginary number, then
\(\sin ^2 \theta+\cos ^2 3 \theta=\)

2Complex Numbers

The least positive integral value of $n$ such that $\left[\frac{1+\sin \frac{2 \pi}{9}+i \cos \frac{2 \pi}{9}}{1+\sin \frac{2 \pi}{9}-i \cos \frac{2 \pi}{9}}\right]^n=1$ is

3Complex Numbers

If $x=a+b, y=a \alpha+b \beta, z=a \beta+b \alpha$ and $\alpha, \beta$ are the complex cube roots of unity, then $x^3+y^3+z^3=$

4Complex Numbers

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3+x^2+x+1=0$, then match the items of List I with those of List II

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5Complex Numbers

If $z=x+i y$ is a complex number such that $z \bar{z}^3+\bar{z} z^3=350$ and $x, y$ are integers, then $|z|=$

6Complex Numbers

If a polynomial $P(x)$ given by
$P(x)=2 x^4+a x^3+b x^2+c x+d$ is such that $P(1)=4$,
$P(2)=7, P(3)=12$ and $P(4)=19$, then $P(5)=$

7Complex Numbers

If $\alpha$ and $\beta$ are the roots of the equation $x^2+x+1=0$, then $(\alpha+\beta)^2+\left(\alpha^2+\beta^2\right)^2+\left(\alpha^3+\beta^3\right)^2+\ldots+\left(\alpha^{12}+\beta^{12}\right)^2=$

8Complex Numbers

Let $z=x+i y$ be a point in the argand plane. If the amplitude of $\left(\frac{z-3}{z+2 i}\right)$ is $\frac{\pi}{2}$, then the locus of $z$ is

9Complex Numbers

If $a x^2-x y-3 y^2-5 x+20 y+c=0$ represents a pair of lines passing through the point $(2,3)$, then $a-c=$

10Complex Numbers

One of the values of $(\sqrt{3}-i)^{\frac{1}{6}}$ is

11Complex Numbers

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

12Complex Numbers

If $i$ is the root of the equation $x^2+1=0$, then
\((1+\sqrt{3} i)^{2023}+(1-\sqrt{3} i)^{2023}=\)

13Complex Numbers

If the value of $\sqrt{-5-12 i}+\sqrt{7+24 i}$ is a negative real number $k$, then $k=$

14Complex Numbers

If $\alpha, \beta, \gamma$ are the roots of the equation $2 x^3+x^2-13 x+6=0$, then $\alpha^3+\beta^3+\gamma^3=$

15Complex Numbers

One of the values of $(\sqrt{3}-i)^{2 / 5}$ is

16Complex Numbers

When $3^{2023}$ is divided by 16 , the remainder obtained is

17Complex Numbers

If $z=x+i y$ and the point $P$ in the argand plane represents $z$, then the locus of $z$ satisfying the equation $|z-2|+|z-2 i|=4$ is

18Complex Numbers

If $\left(\frac{\sqrt{3}+i}{\sqrt{3}-i}\right)^4+\left(\frac{\sqrt{3}-i}{\sqrt{3}+i}\right)^4=r$ cis $\theta$, then one of the values of $\sqrt{r \operatorname{cis} \theta}$ is

19Complex Numbers

If $\alpha, \beta, \gamma$ are the real roots of the equation $18 x^3-15 x^2-4 x+4=0$ such that $\alpha=\beta$ and $\alpha>\gamma$, then $\alpha+\beta^2+\gamma^3=$

20Complex Numbers

If $i^2=-1$, then $(1+\sqrt{3} i)^{2022}-(\sqrt{3}-i)^{2022}=$

21Complex Numbers

If $\alpha$ is a multiple root of the equation $x^5-6 x^4+11 x^3-2 x^2-12 x+8=0$, then $3 \alpha^2-2 \alpha+1=$

22Complex Numbers

If $\alpha, \beta, \gamma$ and $\delta$ are the roots of the equation $x^4+x^2+1=0$ such that $\alpha+\beta=-1, \gamma+\delta=1, \alpha^2=\beta$ and $\gamma^2=-\delta$, then $\alpha^{2023}+\beta^{2023}+\gamma^{2022}+\delta^{2022}=$

23Complex Numbers

If $\alpha$ and $\beta$ are non-zero integers and $z=(\alpha+i \beta)(2+7 i)$ is a purely imaginary number, then minimum value of $|z|^2$ is

24Complex Numbers

If $\theta=\frac{\pi}{6}$, then the 10 th term of the series $1+(\cos \theta+i \sin \theta)^1+(\cos \theta+i \sin \theta)^2+\ldots$. is

25Complex Numbers

If $z_1$ and $z_2$ are complex numbers such that $\left|z_1+z_2\right|=\left|z_1\right|+\left|z_2\right|$, then the difference in the amplitude of $z_1$ and $z_2$ is

26Complex Numbers

If $\frac{1+i \cos \theta}{1-2 i \cos \theta}$ is purely real, then $\cos ^3 \theta+\sin ^2 \theta+\cos \theta+1=$

27Complex Numbers

If $i=\sqrt{-1}$, then $1+i^2+i^4+i^6+\ldots \ldots+i^{2024}=$

28Complex Numbers

If $i=\sqrt{-1}$, then $(1+i)^{10}+(1-i)^{10}=$

29Complex Numbers

$\operatorname{Arg}\left(\sin \frac{6 \pi}{5}+i\left(1+\cos \frac{6 \pi}{5}\right)\right)=$

30Complex Numbers

\(\text { If } x+i y=\sqrt{\frac{3+i}{1+3 i}}, \text { then }\left(x^2+y^2\right)^2=\)

31Complex Numbers

If the imaginary part of $\frac{2 z+1}{i z+1}$ is -2, then the locus of the point representing $z$ in the Argand plane is

32Complex Numbers

If $1, \omega, \omega^2$ are the cube roots of unity and $1, \alpha, \alpha^2, \alpha^3$ are the fourth roots of unity in usual notation, then $\alpha+\alpha \omega-\alpha^3 \omega^2=$

33Complex Numbers

$\{x \in[0,2 \pi] / \sin x+i \cos 2 x$ and $\cos x-i \sin 2 x$ are conjugate to each other} $=$

34Complex Numbers

If $|x+i y|=\sqrt{x^2+y^2}$, then $\left|(1-\sqrt{3} i)^9+(\sqrt{3}+i)^9\right|=$

35Complex Numbers

If $\alpha$ and $\beta$ are the roots of the equation $x^2-2 x+2=0$, then $\alpha^{2020}+\beta^{2020}=$

36Complex Numbers

If $z=\frac{-1-i \sqrt{3}}{2}$, then $\sum_{k=1}^{2022}\left(z^k+\frac{1}{z^k}\right)^2=$

37Complex Numbers

The equation of lowest degree with rational coefficients having roots $\sqrt{3}+\sqrt{2} i$ and $\sqrt{3}-\sqrt{2}$ is

38Complex Numbers

If $\cos \alpha$ is the common value of $(-1)^{\frac{1}{4}}$ and $(-i)^{\frac{1}{2}}$ then $\tan \alpha=$

39Complex Numbers

If $(2 x-y+1)+i(x-2 y-1)=2-3 i$, then the multiplicative inverse of $(x-i y)$ is

40Complex Numbers

If $(2-i)$ is one of the roots of the equation $x^4-9 x^3+31 x^2-49 x+30=0$ and $\alpha, \beta(\alpha<\beta)$ are its real roots, then $2 \alpha-\beta=$

41Complex Numbers

If $1, \alpha_1, \alpha_2, \alpha_3, \ldots \alpha_{n-1}$ are $n$th roots of unity then $\sum\limits_{1 \le i < f \le n - 1}^{} {} {a_i}{a_j} = $

42Complex Numbers

If $z=\alpha+i \beta$ satisfies the equation $|z|-z=1+2 i$ and $|z|=\sqrt{\alpha^2+\beta^2}$, then $z \bar{z}=$

43Complex Numbers

If $e^{i t}=\cos t+i \sin t$ and $e^{-i t}=\cos t-i \sin t$, then $\cosh (x+i y)-\cosh (x-i y)=$

44Complex Numbers

If $-i$ and $\alpha$ are the roots of the equation $i z^2-2(i+1) z+(2-i)=0, \tan \theta=\frac{-1}{2}$ and $\theta \in 4$ th quadrant, then $5^3 \cos 6 \theta=$

45Complex Numbers

If $\left(\frac{\sqrt{3}+i}{\sqrt{3}-i}\right)^m=1,2022 < m < 2029$, then $m=$

46Complex Numbers

\(\sqrt{(-3+4 i)(8+6 i)}=\)

47Complex Numbers

If $1, \omega, \omega^2$ are the cube roots of unity, $n \in N$ and $n>2$ then the least value of $n$ such that $1+\omega$ is a root of $x^n-x=0$ is

48Complex Numbers

If the point $(x, y)$ satisfies the equation $\frac{x+i(x-2)}{3+i}-i =\frac{2 y+i(1-3 y)}{i-3}$, then $x+y=$

49Complex Numbers

If $\cos \alpha+\cos \beta+\cos \gamma=0$ and $\sin \alpha+\sin \beta+\sin \gamma=0$ then $\cos 2 \alpha+\cos 2 \beta+\cos 2 \gamma=$

50Complex Numbers

One of the values of $(-32 i)^{\frac{2}{5}}$ is