Skip to main content
Yukti Vault

Algebra

Complex Numbers: JEE Main & Advanced PYQs

309 previous year questions on Complex Numbers from JEE Main and JEE Advanced papers, each with its exam and year, an answer and a solution.

Practice this chapter

Questions 1-50 of 309

  1. The least value of $|z|$ where $z$ is complex number which satisfies the inequality …
  2. Let the circles $C_1:|z|=r$ and $C_2:|z-3-4i|=5$, $z\in\mathbb{C}$, be such that $C_2$ lies within $C_1$. If $z_1$ moves on $C_1$, $z_2$ moves on $C_2$ and …
  3. If the equation $a|z|^{2}+\bar{\alpha}z+\alpha\bar{z}+d=0$ represents a circle where $a,d$ are real constants, then which of the following condition is correct ?
  4. Let $a\neq b$ be two non-zero real numbers. Then the number of elements in the set …
  5. Let the lines $(2-i)z=(2+i)\bar{z}$ and $(2+i)z+(i-2)\bar{z}-4i=0$ (here $i^{2}=-1$) be normal to a circle $C$. If the line $iz+\bar{z}+1+i=0$ is tangent to this circle …
  6. If $a_r=\cos\frac{2r\pi}{9}+i\sin\frac{2r\pi}{9}$, $r=1,2,3,\ldots$, $i=\sqrt{-1}$, then the determinant …
  7. If $S=\left\{z\in\mathbb{C}:\frac{z-i}{z+2i}\in\mathbf{R}\right\}$, then :
  8. Let $S_1,S_2$ and $S_3$ be three sets defined as $S_1=\{z\in\mathbb{C}:|z-1|\le\sqrt{2}\}$ $S_2=\{z\in\mathbb{C}:\operatorname{Re}((1-i)z)\ge1\}$ …
  9. The equation $\arg\left(\frac{z-1}{z+1}\right)=\frac{\pi}{4}$ represents a circle with :
  10. If $(\sqrt{3}+i)^{100}=2^{99}(p+iq)$, then $p$ and $q$ are roots of the equation :
  11. The number of values of $z\in\mathbb{C}$, satisfying the equations $|z-(4+8i)|=\sqrt{10}$ and $|z-(3+5i)|+|z-(5+11i)|=4\sqrt{5}$, is :
  12. Let a complex number be $w=1-\sqrt{3}\,i$. Let another complex number $z$ be such that $|zw|=1$ and $\arg(z)-\arg(w)=\frac{\pi}{2}$. Then the area of the triangle with …
  13. Let $A=\left\{\theta\in(0,2\pi):\frac{1+2i\sin\theta}{1-i\sin\theta}\text{ is purely imaginary}\right\}$. Then the sum of the elements in $A$ is
  14. If $\alpha,\beta\in\mathbf{R}$ are such that $1-2i$ (here $i^{2}=-1$) is a root of $z^{2}+\alpha z+\beta=0$, then $(\alpha-\beta)$ is equal to :
  15. Let $z_1,z_2\in\mathbb{C}$ be the distinct solutions of the equation $z^{2}+4z-(1+12i)=0$. Then $|z_1|^{2}+|z_2|^{2}$ is equal to :
  16. If $z$ is a complex number such that $\frac{z-i}{z-1}$ is purely imaginary, then the minimum value of $|z-(3+3i)|$ is :
  17. Let $n$ denote the number of solutions of the equation $z^{2}+3\bar{z}=0$, where $z$ is a complex number. Then the value of $\sum_{k=0}^{\infty}\frac{1}{n^{k}}$ is equal …
  18. If $z=x+iy$ and $\omega=\frac{z-1}{3(z-2)}$, then the set of points $z$ satisfying $|z|=1=|\omega|$ contains
  19. Let $S=\{z\in\mathbb{C}:z^{2}+4z+16=0\}$. Then $\sum_{z\in S}|z+\sqrt{3}i|^{2}$ is equal to:
  20. Let $S=\{z\in\mathbb{C}:z^{2}+\sqrt{6}\,iz-3=0\}$. Then $\sum_{z\in S}z^{8}$ is equal to :
  21. If for $z=\alpha+i\beta$, $|z+2|=z+4(1+i)$, then $\alpha+\beta$ and $\alpha\beta$ are the roots of the equation
  22. The area of the triangle with vertices $A(z)$, $B(iz)$ and $C(z+iz)$ is :
  23. Let $S$ be the set of all $(\alpha,\beta)$, $\pi<\alpha,\beta<2\pi$, for which the complex number $\frac{1-i\sin\alpha}{1+2i\sin\alpha}$ is purely imaginary and …
  24. If $z$ is a complex number such that $|z|\le1$, then the minimum value of $\left|z+\frac12(3+4i)\right|$ is :
  25. Let $C$ be the circle in the complex plane with centre $z_0=\frac{1}{2}(1+3i)$ and radius $r=1$. Let $z_1=1+i$ and the complex number $z_2$ be outside the circle $C$ …
  26. If the set $\left\{\operatorname{Re}\left(\frac{z-\bar{z}+z\bar{z}}{2-3z+5\bar{z}}\right):z\in\mathbb{C},\ \operatorname{Re}(z)=3\right\}$ is equal to the interval …
  27. Let $\arg(z)$ represent the principal argument of the complex number $z$. Then, $|z|=3$ and $\arg(z-1)-\arg(z+1)=\frac{\pi}{4}$ intersect
  28. If $\alpha,\beta,\gamma,\delta$ are the roots of the equation $x^4+x^3+x^2+x+1=0$, then $\alpha^{2021}+\beta^{2021}+\gamma^{2021}+\delta^{2021}$ is equal to :
  29. If $S=\{z\in\mathbf{C}:|z-i|=|z+i|=|z-1|\}$, then $n(S)$ is :
  30. Let $z$ be a complex number such that $|z|=1$. If $\frac{2+k^{2}z}{k+\bar{z}}=kz$, $k\in\mathbf{R}$, then the maximum distance of $k+ik^{2}$ from the circle …
  31. Let $S=\left\{z=x+iy:\frac{2z-3i}{4z+2i}\text{ is a real number}\right\}$. Then which of the following is NOT correct?
  32. Let $z\in\mathbb{C}$ be such that $\frac{z^{2}+3i}{z-2+i}=2+3i$. Then the sum of all possible values of $z^{2}$ is
  33. If $z_1,z_2,z_3\in\mathbb{C}$ are the vertices of an equilateral triangle, whose centroid is $z_0$, then $\sum_{k=1}^{3}(z_k-z_0)^{2}$ is equal to
  34. Among the statements (S1) : The set $\left\{z\in\mathbb{C}-\{-i\}:|z|=1\text{ and }\frac{z-i}{z+i}\text{ is purely real}\right\}$ contains exactly two elements, and (S2) …
  35. Let $\mathbf{C}$ be the set of all complex numbers. Let $S_1=\{z\in\mathbf{C}\mid|z-3-2i|^2=8\}$, $S_2=\{z\in\mathbf{C}\mid\mathrm{Re}(z)\ge5\}$ and …
  36. If $z=2+3i$, then $z^5+(\bar z)^5$ is equal to :
  37. Let $S=\{z=x+iy:|z-1+i|\ge|z|,\ |z|<2,\ |z+i|=|z-1|\}$. Then the set of all values of $x$, for which $w=2x+iy\in S$ for some $y\in\mathbb{R}$, is
  38. The number of points of intersection of $|z-(4+3i)|=2$ and $|z|+|z-4|=6$, $z\in\mathbf{C}$, is
  39. Let $O$ be the origin and $A$ be the point $z_1=1+2i$. If $B$ is the point $z_2$, $\mathrm{Re}(z_2)<0$, such that $OAB$ is a right angled isosceles triangle with $OB$ as …
  40. For $a\in\mathbb{C}$, let $A=\{z\in\mathbb{C}:\operatorname{Re}(a+\bar{z})>\operatorname{Im}(\bar{a}+z)\}$ and $B=\{z\in\mathbb{C}:\operatorname{Re}(a+\bar{z}) 0$, then …
  41. Let $A=\left\{z\in\mathbf{C}:\left|\frac{z+1}{z-1}\right|<1\right\}$ and $B=\left\{z\in\mathbf{C}:\arg\left(\frac{z-1}{z+1}\right)=\frac{2\pi}{3}\right\}$. Then …
  42. Let $w_1$ be the point obtained by the rotation of $z_1=5+4i$ about the origin through a right angle in the anticlockwise direction, and $w_2$ be the point obtained by …
  43. If $z=x+iy$ satisfies $|z|-2=0$ and $|z-i|-|z+5i|=0$, then
  44. Let the minimum value $v_0$ of $v=|z|^2+|z-3|^2+|z-6i|^2$, $z\in\mathbb{C}$ is attained at $z=z_0$. Then $\left|2z_0^2-\bar z_0^3+3\right|^2+v_0^2$ is equal to
  45. For $z\in\mathbb{C}$ if the minimum value of $\left(|z-3\sqrt2|+|z-p\sqrt2\,i|\right)$ is $5\sqrt2$, then a value of $p$ is
  46. Let $A=\left\{\theta\in[0,2\pi]:1+10\operatorname{Re}\left(\frac{2\cos\theta+i\sin\theta}{\cos\theta-3i\sin\theta}\right)=0\right\}$. Then $\sum_{\theta\in A}\theta^{2}$ …
  47. If $z_1,z_2$ are two distinct complex number such that $\left|\frac{z_1-2z_2}{\frac{1}{2}-z_1\bar{z}_2}\right|=2$, then
  48. If $z=x+iy$, $xy\neq0$, satisfies the equation $z^{2}+i\bar{z}=0$, then $|z^{2}|$ is equal to :
  49. The real part of the complex number $\frac{(1+2i)^{8}\cdot(1-2i)^{2}}{(3+2i)\cdot\overline{(4-6i)}}$ is equal to :
  50. Let $S=\left\{z\in\mathbb{C}:\bar{z}=i\left(z^{2}+\operatorname{Re}(\bar{z})\right)\right\}$. Then $\sum_{z\in S}|z|^{2}$ is equal to

Other chapters