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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.

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Questions 101-150 of 309

  1. If the least and the largest real values of $\alpha$, for which the equation $z+\alpha|z-1|+2i=0$ ($z\in\mathbb{C}$ and $i=\sqrt{-1}$) has a solution, are $p$ and $q$ …
  2. If $f(x)$ and $g(x)$ are two polynomials such that the polynomial $P(x)=f\left(x^{3}\right)+xg\left(x^{3}\right)$ is divisible by $x^{2}+x+1$, then $P(1)$ is equal to …
  3. Let $A=\{z\in\mathbb{C}:|z-2-i|=3\}$, $B=\{z\in\mathbb{C}:\operatorname{Re}(z-iz)=2\}$ and $S=A\cap B$. Then $\sum_{z\in S}|z|^{2}$ is equal to $\_\_\_\_\_\_$ .
  4. The value of $\left(\frac{1+\sin\frac{2\pi}9+i\cos\frac{2\pi}9}{1+\sin\frac{2\pi}9-i\cos\frac{2\pi}9}\right)^3$ is :
  5. The number of elements in the set $\{z=a+ib\in\mathbf{C}:a,b\in\mathbf{Z}$ and $1<|z-3+2i|<4\}$ is $\_\_\_\_\_\_$ .
  6. The sum of $162^{\text{th}}$ power of the roots of the equation $x^{3}-2x^{2}+2x-1=0$ is $\_\_\_\_\_\_$ .
  7. If $\left(\frac{1+i}{1-i}\right)^{m/2}=\left(\frac{1+i}{i-1}\right)^{n/3}=1$, ($m,n\in\mathbf{N}$) then the greatest common divisor of the least values of $m$ and $n$ is …
  8. Let $i=\sqrt{-1}$. If $\frac{(-1+i\sqrt{3})^{21}}{(1-i)^{24}}+\frac{(1+i\sqrt{3})^{21}}{(1+i)^{24}}=k$, and $n=[|k|]$ be the greatest integral part of $|k|$. Then …
  9. Let $S=\{z\in\mathbf{C}:|z-2|\le1,\ z(1+i)+\bar{z}(1-i)\le2\}$. Let $|z-4i|$ attains minimum and maximum values, respectively, at $z_1\in S$ and $z_2\in S$. If …
  10. Let $z$ be a complex number such that $|z+2|=|z-2|$ and $\arg\left(\frac{z+3}{z-i}\right)=\frac{\pi}{4}$. Then $|z|^{2}$ is equal to :
  11. If $z=\left(\dfrac{\sqrt3}{2}+\dfrac{i}{2}\right)^{5}+\left(\dfrac{\sqrt3}{2}-\dfrac{i}{2}\right)^{5}$, then
  12. If $z$ and $\omega$ are two complex numbers such that $z\omega=1$ and $\arg z-\arg\omega=\frac\pi2$, then :
  13. Let $x$ and $y$ be real numbers such that $50\left(\frac{2x}{1+3i}-\frac{y}{1-2i}\right)=31+17i$, $i=\sqrt{-1}$. Then the value of $10(x-3y)$ is :
  14. The complex numbers $z=x+iy$ which satisfy the equation $\left|\dfrac{z-5i}{z+5i}\right|=1$ lie on
  15. Let $z$ be a complex number satisfying $|z|^3+2z^2+4\bar z-8=0$, where $\bar z$ denotes the complex conjugate of $z$. Let the imaginary part of $z$ be nonzero. Match …
  16. If the cube roots of unity are $1,\omega,\omega^{2}$, then the roots of the equation $(x-1)^{3}+8=0$ are
  17. The complex number $z=\frac{i-1}{\cos\frac\pi3+i\sin\frac\pi3}$ is equal to :
  18. For a non-zero complex number $z$, let $\arg(z)$ denote the principal argument of $z$, with $-\pi<\arg(z)\le\pi$. Let $\omega$ be the cube root of unity for which …
  19. Let $z$ be a complex number with non-zero imaginary part. If $$\frac{2+3z+4z^2}{2-3z+4z^2}$$ is a real number, then the value of $|z|^2$ is __________.
  20. Let $\omega\neq1$ be a cube root of unity. Then the minimum of the set $$\{|a+b\omega+c\omega^2|^2:\ a,b,c\ \text{distinct non-zero integers}\}$$ equals ____
  21. Let $A_1, A_2, A_3, \ldots, A_8$ be the vertices of a regular octagon that lie on a circle of radius $2$. Let $P$ be a point on the circle and let $PA_i$ denote the …
  22. The value of $\left(\frac{1+\sin\frac{2\pi}{9}+i\cos\frac{2\pi}{9}}{1+\sin\frac{2\pi}{9}-i\cos\frac{2\pi}{9}}\right)^{3}$ is
  23. Let $S$ be the set of all complex numbers $z$ satisfying $|z-2+i|\geq\sqrt5$. If the complex number $z_0$ is such that $\frac{1}{|z_0-1|}$ is the maximum of the set …
  24. Let $\mathbb{R}$ denote the set of all real numbers. Let $z_1=1+2i$ and $z_2=3i$ be two complex numbers, where $i=\sqrt{-1}$. Let …
  25. State TRUE or FALSE: For complex numbers $z_1=x_1+iy_1$ and $z_2=x_2+iy_2$, we write $z_1\cap z_2$ if $x_1\le x_2$ and $y_1\le y_2$. Then for all complex numbers $z$ …
  26. Find the real values of $x$ and $y$ for which the following equation is satisfied: $$\frac{(1+i)x-2i}{3+i}+\frac{(2-3i)y+i}{3-i}=i$$
  27. Let $\bar{z}$ denote the complex conjugate of a complex number $z$ and let $i=\sqrt{-1}$. In the set of complex numbers, the number of distinct roots of the equation …
  28. Let $a,b$ be two real numbers such that $ab<0$. If the complex number $\frac{1+ai}{b+i}$ is of unit modulus and $a+ib$ lies on the circle $|z-1|=|2z|$, then a possible …
  29. Let $|z_1-8-2i|\le1$ and $|z_2-2+6i|\le2$, $z_1,z_2\in\mathbf C$. Then the minimum value of $|z_1-z_2|$ is :
  30. Let $z\in\mathbb C$ with $\operatorname{Im}(z)=10$ and it satisfies $\frac{2z-n}{2z+n}=2i-1$ for some natural number $n$. Then
  31. Let $p,q\in\mathbb R$ and $\left(1-\sqrt3i\right)^{200}=2^{199}(p+iq)$, $i=\sqrt{-1}$. Then, $p+q+q^{2}$ and $p-q+q^{2}$ are roots of the equation.
  32. For all $z\in\mathbb C$ on the curve $C_1: |z|=4$, let the locus of the point $z+\frac1z$ be the curve $C_2$. Then
  33. If $\alpha$ and $\beta$ are the roots of the equation $2z^{2}-3z-2i=0$, where $i=\sqrt{-1}$, then …
  34. If $z$ is a complex number of unit modulus and argument $\theta$, then $\arg\left(\frac{1+z}{1+\bar z}\right)$ equals :
  35. Let $A=\left\{\frac{1967+1686\,i\sin\theta}{7-3i\cos\theta}: \theta\in\mathbb{R}\right\}$. If $A$ contains exactly one positive integer $n$, then the value of $n$ is …
  36. Express $\dfrac{1}{1-\cos\theta+2i\sin\theta}$ in the form $x+iy$.
  37. The smallest positive integer $n$ for which $\left(\dfrac{1+i}{1-i}\right)^{n}=1$ is
  38. Let $\bar{z}$ denote the complex conjugate of a complex number $z$. If $z$ is a non-zero complex number for which both real and imaginary parts of …
  39. Let $z_1$ and $z_2$ be two complex numbers satisfying $|z_1|=9$ and $|z_2-3-4i|=4$. Then the minimum value of $|z_1-z_2|$ is :
  40. The inequality $|z-4|<|z-2|$ represents the region given by
  41. The least positive integer $n$ for which $\left(\frac{1+i\sqrt3}{1-i\sqrt3}\right)^{n}=1$ is
  42. If $z=x+iy$ and $\omega=(1-iz)/(z-i)$, then $|\omega|=1$ implies that, in the complex plane,
  43. Let $A=\left\{\theta\in\left(-\frac\pi2,\pi\right):\frac{3+2i\sin\theta}{1-2i\sin\theta}\text{ is purely imaginary}\right\}$. Then the sum of the elements in $A$ is:
  44. Let $z$ be a complex number such that $|z|+z=3+i$ (where $i=\sqrt{-1}$). Then $|z|$ is equal to :
  45. Let $z_1$ and $z_2$ be any two non-zero complex numbers such that $3|z_1|=4|z_2|$. If $z=\frac{3z_1}{2z_2}+\frac{2z_2}{3z_1}$ then the maximum value of $|z|$ is
  46. The largest value of $r$, for which the region represented by the set $\{\omega\in C\,|\,|\omega-4-i|\le r\}$ is contained in the region represented by the set …
  47. Let $\left(-2-\frac13i\right)^{3}=\frac{x+iy}{27}$ $(i=\sqrt{-1})$, where $x$ and $y$ are real numbers, then $y-x$ equals
  48. If $\operatorname{Re}\left(\frac{z-1}{2z+i}\right)=1$, where $z=x+iy$, then the point $(x,y)$ lies on a
  49. Let $z_1=2+3i$ and $z_2=3+4i$. The set $S=\left\{z\in C:|z-z_1|^{2}-|z-z_2|^{2}=|z_1-z_2|^{2}\right\}$ represents a
  50. Let $z\in C$, the set of complex numbers. Then the equation $2|z+3i|-|z-i|=0$ represents :

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