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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.

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Questions 51-100 of 309

  1. Let $z_1$ and $z_2$ be two complex numbers such that $z_1+z_2=5$ and $z_1^{3}+z_2^{3}=20+15i$. Then, $\left|z_1^{4}+z_2^{4}\right|$ equals -
  2. If the locus of $z\in\mathbb{C}$, such that $\operatorname{Re}\left(\frac{z-1}{2z+i}\right)+\operatorname{Re}\left(\frac{\bar{z}-1}{2\bar{z}-i}\right)=2$, is a circle of …
  3. Let $S=\{z\in\mathbf{C}:|z-1|=1$ and $(\sqrt{2}-1)(z+\bar{z})-i(z-\bar{z})=2\sqrt{2}\}$. Let $z_1,z_2\in S$ be such that $|z_1|=\max_{z\in S}|z|$ and …
  4. Let $A=\{z\in\mathbf{C}:1\le|z-(1+i)|\le2\}$ and $B=\{z\in A:|z-(1-i)|=1\}$. Then, $B$ :
  5. Let $\alpha$ be a root of the equation $1+x^{2}+x^{4}=0$. Then the value of $\alpha^{1011}+\alpha^{2022}-\alpha^{3033}$ is equal to :
  6. Let $z$ be a complex number such that the real part of $\frac{z-2i}{z+2i}$ is zero. Then, the maximum value of $|z-(6+8i)|$ is equal to
  7. If $z=\frac{1}{2}-2i$ is such that $|z+1|=\alpha z+\beta(1+i)$, $i=\sqrt{-1}$ and $\alpha,\beta\in\mathbb{R}$, then $\alpha+\beta$ is equal to
  8. The sum of all possible values of $\theta\in[-\pi,2\pi]$, for which $\frac{1+i\cos\theta}{1-2i\cos\theta}$ is purely imaginary, is equal to :
  9. Let $\mathbb{C}$ be the set of all complex numbers. Let $S_1=\{z\in\mathbb{C}:|z-2|\le1\}$ and $S_2=\{z\in\mathbb{C}:z(1+i)+\bar z(1-i)\ge4\}$. Then, the maximum value …
  10. Let $\alpha$ and $\beta$ be the roots of the equation $x^{2}+(2i-1)=0$. Then, the value of $|\alpha^{8}+\beta^{8}|$ is equal to :
  11. If the set $R=\{(a,b):a+5b=42,\ a,b\in\mathbb{N}\}$ has $m$ elements and $\sum_{n=1}^{m}\left(1-i^{n!}\right)=x+iy$, where $i=\sqrt{-1}$, then the value of $m+x+y$ is
  12. Let a circle $C$ in complex plane pass through the points $z_1=3+4i$, $z_2=4+3i$ and $z_3=5i$. If $z(\neq z_1)$ is a point on $C$ such that the line through $z$ and …
  13. Let $z$ be a complex number such that $|z+2|=1$ and $\operatorname{Im}\left(\frac{z+1}{z+2}\right)=\frac{1}{5}$. Then the value of …
  14. The area of the polygon, whose vertices are the non-real roots of the equation $\bar{z}=iz^{2}$ is :
  15. Let the product of $\omega_1=(8+i)\sin\theta+(7+4i)\cos\theta$ and $\omega_2=(1+8i)\sin\theta+(4+7i)\cos\theta$ be $\alpha+i\beta$, $i=\sqrt{-1}$. Let $p$ and $q$ be the …
  16. If $z\ne0$ be a complex number such that $\left|z-\frac1z\right|=2$, then the maximum value of $|z|$ is :
  17. Let $z_1=1+2i$, $z_2=2+i$, $\frac1{z_1}+\frac1{z_2}=\frac6w$, and $z=\frac{iw}{2-\bar w}$. Consider the statements (S1) $|z|=\frac{5}{\sqrt{17}}$ (S2) …
  18. Let the complex number $z=x+iy$ be such that $\frac{2z-3i}{2z+i}$ is purely imaginary. If $x+y^{2}=0$, then $y^{4}+y^{2}-y$ is equal to:
  19. If $z$ is a complex number, then the number of common roots of the equations $z^{1985}+z^{100}+1=0$ and $z^{3}+2z^{2}+2z+1=0$, is equal to
  20. Let $S_1=\left\{z_1\in\mathbf{C}:|z_1-3|=\frac12\right\}$ and $S_2=\left\{z_2\in\mathbf{C}:|z_2-|z_2+1||=|z_2+|z_2-1||\right\}$. Then, for $z_1\in S_1$ and $z_2\in S_2$ …
  21. Let $z_1$ and $z_2$ be two complex numbers such that $\arg(z_1-z_2)=\frac{\pi}{4}$ and $z_1,z_2$ satisfy the equation $|z-3|=\operatorname{Re}(z)$. Then the imaginary …
  22. Let $S=\{z\in\mathbf{C}:|z-3|\le1$ and $z(4+3i)+\bar{z}(4-3i)\le24\}$. If $\alpha+i\beta$ is the point in $S$ which is closest to $4i$, then $25(\alpha+\beta)$ is equal …
  23. The imaginary part of $\left(3+2\sqrt{-54}\right)^{1/2}-\left(3-2\sqrt{-54}\right)^{1/2}$ can be :
  24. Let $\alpha,\beta$ be the roots of the equation $x^{2}-x+2=0$ with $\operatorname{Im}(\alpha)>\operatorname{Im}(\beta)$. Then …
  25. If $\alpha$ satisfies the equation $x^{2}+x+1=0$ and $(1+\alpha)^{7}=A+B\alpha+C\alpha^{2}$, $A,B,C\ge0$, then $5(3A-2B-C)$ is equal to $\_\_\_\_\_\_$ .
  26. If $\alpha$ is a root of the equation $x^{2}+x+1=0$ and $\sum_{k=1}^{n}\left(\alpha^{k}+\frac{1}{\alpha^{k}}\right)^{2}=20$, then $n$ is equal to $\_\_\_\_\_\_$ .
  27. If $z$ be a complex number satisfying $|\mathrm{Re}(z)|+|\mathrm{Im}(z)|=4$, then $|z|$ cannot be :
  28. If the real part of the complex number $z=\frac{3+2i\cos\theta}{1-3i\cos\theta}$, $\theta\in\left(0,\frac\pi2\right)$ is zero, then the value of …
  29. If $z^{2}+z+1=0$, $z\in\mathbf{C}$, then $\left|\sum_{n=1}^{15}\left(z^{n}+(-1)^{n}\frac{1}{z^{n}}\right)^{2}\right|$ is equal to $\_\_\_\_\_\_$ .
  30. If $z_1,z_2$ are complex numbers such that $\mathrm{Re}(z_1)=|z_1-1|$, $\mathrm{Re}(z_2)=|z_2-1|$ and $\arg(z_1-z_2)=\frac\pi6$, then $\mathrm{Im}(z_1+z_2)$ is equal to :
  31. The least positive integer $n$ such that $\frac{(2i)^{n}}{(1-i)^{n-2}}$, $i=\sqrt{-1}$, is a positive integer, is $\_\_\_\_\_\_$ .
  32. Let $w=z\bar{z}+k_1z+k_2iz+\lambda(1+i)$, $k_1,k_2\in\mathbb{R}$. Let $\operatorname{Re}(w)=0$ be the circle $C$ of radius $1$ in the first quadrant touching the line …
  33. If $a$ and $b$ are real numbers such that $(2+\alpha)^4=a+b\alpha$, where $\alpha=\frac{-1+i\sqrt3}2$, then $a+b$ is equal to :
  34. Let $z=a+ib$, $b\ne0$ be complex numbers satisfying $z^2=\bar z\cdot2^{1-|z|}$. Then the least value of $n\in\mathbb{N}$, such that $z^n=(z+1)^n$, is equal to ________.
  35. Let $z=\frac{1-i\sqrt{3}}{2}$, $i=\sqrt{-1}$. Then the value of …
  36. A point $z$ moves in the complex plane such that $\arg\left(\frac{z-2}{z+2}\right)=\frac{\pi}{4}$, then the minimum value of $\left|z-9\sqrt{2}-2i\right|^{2}$ is equal …
  37. Let $z=x+iy$ be a non-zero complex number such that $z^2=i|z|^2$, where $i=\sqrt{-1}$, then $z$ lies on the :
  38. Sum of squares of modulus of all the complex numbers $z$ satisfying $\bar{z}=iz^{2}+z^{2}-z$ is equal to $\_\_\_\_\_\_$ .
  39. Let $z$ be those complex numbers which satisfy $|z+5|\le4$ and $z(1+i)+\bar{z}(1-i)\ge-10$, $i=\sqrt{-1}$. If the maximum value of $|z+1|^{2}$ is $\alpha+\beta\sqrt{2}$ …
  40. If the four complex numbers $z$, $\bar z$, $\bar z-2\mathrm{Re}(\bar z)$ and $z-2\mathrm{Re}(z)$ represent the vertices of a square of side $4$ units in the Argand …
  41. Let $z_1,z_2$ be the roots of the equation $z^{2}+az+12=0$ and $z_1,z_2$ form an equilateral triangle with origin. Then, the value of $|a|$ is $\_\_\_\_\_\_$ .
  42. Let $P=\{z\in\mathbf{C}:|z+2-3i|\le1\}$ and $Q=\{z\in\mathbf{C}:z(1+i)+\bar{z}(1-i)\le-8\}$. Let in $P\cap Q$, $|z-3+2i|$ be maximum and minimum at $z_1$ and $z_2$ …
  43. For $\alpha,\beta,z\in\mathbb{C}$ and $\lambda>1$, if $\sqrt{\lambda-1}$ is the radius of the circle $|z-\alpha|^{2}+|z-\beta|^{2}=2\lambda$, then $|\alpha-\beta|$ is …
  44. If $\alpha$ denotes the number of solutions of $|1-i|^{x}=2^{x}$ and $\beta=\left(\frac{|z|}{\arg(z)}\right)$, where …
  45. The sum of the square of the modulus of the elements in the set $\{z=a+ib:a,b\in\mathbf{Z},z\in\mathbf{C},|z-1|\le1,|z-5|\le|z-5i|\}$ is $\_\_\_\_\_\_$ .
  46. Let $S=\{z\in\mathbb{C}:z^2+\bar z=0\}$. Then $\sum_{z\in S}(\mathrm{Re}(z)+\mathrm{Im}(z))$ is equal to ________.
  47. Let $S=\left\{z\in\mathbb{C}-\{i,2i\}:\frac{z^{2}+8iz-15}{z^{2}-3iz-2}\in\mathbb{R}\right\}$. If $\alpha-\frac{13}{11}i\in S$, $\alpha\in\mathbb{R}-\{0\}$, then …
  48. Let $u=\frac{2z+i}{z-ki}$, $z=x+iy$ and $k>0$. If the curve represented by $\mathrm{Re}(u)+\mathrm{Im}(u)=1$ intersects the $y$-axis at the points $P$ and $Q$ where …
  49. Let $z$ be a complex number such that $\left|\frac{z-i}{z+2i}\right|=1$ and $|z|=\frac52$. Then the value of $|z+3i|$ is :
  50. Let $\alpha=\frac{-1+i\sqrt3}{2}$. If $a=(1+\alpha)\sum_{k=0}^{100}\alpha^{2k}$ and $b=\sum_{k=0}^{100}\alpha^{3k}$, then $a$ and $b$ are the roots of the quadratic …

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