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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 201-250 of 309

  1. For all complex numbers $z$ of the form $1+i\alpha$, $\alpha\in R$, if $z^{2}=x+iy$, then
  2. Let $z=1+ai$, be a complex number, $a>0$, such that $z^{3}$ is a real number. Then, the sum $1+z+z^{2}+\ldots+z^{11}$ is equal to :
  3. If $\alpha$ and $\beta$ be the roots of the equation $x^{2}-2x+2=0$, then the least value of $n$ for which $\left(\frac{\alpha}{\beta}\right)^{n}=1$ is
  4. If $z_1$ and $z_2$ are two non-zero complex numbers such that $|z_1+z_2|=|z_1|+|z_2|$, then $\arg z_1-\arg z_2$ is equal to :
  5. Let the complex numbers $\alpha$ and $\frac1{\bar\alpha}$ lie on the circles $|z-z_0|^2=4$ and $|z-z_0|^2=16$ respectively, where $z_0=1+i$. Then, the value of …
  6. The locus of the centre of a circle which touches the circle $|z-z_1|=a$ and $|z-z_2|=b$ externally ($z,z_1\ \&\ z_2$ are complex numbers) will be :
  7. Let $z$ satisfy $|z|=1$ and $z=1-\bar z$. Statement 1 : $z$ is a real number. Statement 2 : Principal argument of $z$ is $\frac\pi3$.
  8. Suppose $Z_1,Z_2,Z_3$ are the vertices of an equilateral triangle inscribed in the circle $|Z|=2$. If $Z_1=1+i\sqrt3$ then $Z_2=$ ______, $Z_3=$ ______.
  9. If $\omega$ is an imaginary cube root of unity then the value of $\sin\left\{\left(\omega^{10}+\omega^{23}\right)\pi-\dfrac{\pi}{4}\right\}$ is
  10. If $z_1=a+ib$ and $z_2=c+id$ are complex numbers such that $|z_1|=|z_2|=1$ and $\mathrm{Re}(z_1\overline{z_2})=0$, then the pair of complex numbers $w_1=a+ic$ and …
  11. If the expression $$\frac{\left[\sin\left(\frac{x}{2}\right)+\cos\left(\frac{x}{2}\right)+i\tan(x)\right]}{\left[1+2i\sin\left(\frac{x}{2}\right)\right]}$$ is real, then …
  12. For any two complex numbers $z_1,z_2$ and any real numbers $a$ and $b$, $|az_1-bz_2|^{2}+|bz_1+az_2|^{2}=$ ______.
  13. Let $z_1$ and $z_2$ be complex numbers such that $z_1\neq z_2$ and $|z_1|=|z_2|$. If $z_1$ has positive real part and $z_2$ has negative imaginary part, then …
  14. $z\neq0$ is a complex number. Column I: (A) $\operatorname{Re}z=0$; (B) $\operatorname{Arg}z=\dfrac{\pi}{4}$. Column II: (p) $\operatorname{Re}z^{2}=0$; (q) …
  15. State TRUE or FALSE: If three complex numbers are in A.P. then they lie on a circle in the complex plane.
  16. State TRUE or FALSE: The cube roots of unity when represented on the Argand diagram form the vertices of an equilateral triangle.
  17. If $a,b,c$ are the numbers between $0$ and $1$ such that the points $z_1=a+i$, $z_2=1+bi$ and $z_3=0$ form an equilateral triangle, then $a=$ ______ and $b=$ ______.
  18. State TRUE or FALSE: If the complex numbers $Z_1,Z_2$ and $Z_3$ represent the vertices of an equilateral triangle such that $|Z_1|=|Z_2|=|Z_3|$, then $Z_1+Z_2+Z_3=0$.
  19. If $z_1$ and $z_2$ are two nonzero complex numbers such that $|z_1+z_2|=|z_1|+|z_2|$, then $\operatorname{Arg}z_1-\operatorname{Arg}z_2$ is equal to
  20. $ABCD$ is a rhombus. Its diagonals $AC$ and $BD$ intersect at the point $M$ and satisfy $BD=2AC$. If the points $D$ and $M$ represent the complex numbers $1+i$ and $2-i$ …
  21. The value of $\displaystyle\sum_{k=1}^{6}\left(\sin\dfrac{2\pi k}{7}-i\cos\dfrac{2\pi k}{7}\right)$ is
  22. Let $z$ and $\omega$ be two non zero complex numbers such that $|z|=|\omega|$ and $\operatorname{Arg}z+\operatorname{Arg}\omega=\pi$, then $z$ equals
  23. If $\omega(\neq1)$ is a cube root of unity and $(1+\omega)^{7}=A+B\omega$ then $A$ and $B$ are respectively
  24. The value of the sum $\displaystyle\sum_{n=1}^{13}\left(i^{n}+i^{n+1}\right)$, where $i=\sqrt{-1}$, equals
  25. The value of the expression $1\cdot(2-\omega)(2-\omega^{2})+2\cdot(3-\omega)(3-\omega^{2})+\cdots+(n-1)\cdot(n-\omega)(n-\omega^{2})$, where $\omega$ is an imaginary …
  26. If $i=\sqrt{-1}$, then $4+5\left(-\dfrac12+\dfrac{i\sqrt3}{2}\right)^{334}+3\left(-\dfrac12+\dfrac{i\sqrt3}{2}\right)^{365}$ is equal to
  27. Find all non-zero complex numbers $Z$ satisfying $\overline{Z}=iZ^{2}$.
  28. Let $z_1$ and $z_2$ be $n$th roots of unity which subtend a right angle at the origin. Then $n$ must be of the form
  29. Let $z$ and $\omega$ be two complex numbers such that $|z|\le1$, $|\omega|\le1$ and $|z+i\omega|=|z-i\bar\omega|=2$ then $z$ equals
  30. For positive integers $n_1,n_2$ the value of the expression $(1+i)^{n_1}+\left(1+i^{3}\right)^{n_1}+\left(1+i^{5}\right)^{n_2}+\left(1+i^{7}\right)^{n_2}$, where …
  31. The complex numbers $z_1$, $z_2$ and $z_3$ satisfying $\dfrac{z_1-z_3}{z_2-z_3}=\dfrac{1-i\sqrt3}{2}$ are the vertices of a triangle which is
  32. If $z_1,z_2$ and $z_3$ are complex numbers such that $|z_1|=|z_2|=|z_3|=\left|\dfrac{1}{z_1}+\dfrac{1}{z_2}+\dfrac{1}{z_3}\right|=1$, then $|z_1+z_2+z_3|$ is
  33. If $\arg(z)<0$, then $\arg(-z)-\arg(z)=$
  34. If $\omega$ is an imaginary cube root of unity, then $(1+\omega-\omega^{2})^{7}$ equals
  35. If one of the vertices of the square circumscribing the circle $|z-1|=\sqrt2$ is $2+\sqrt3\,i$, find the other vertices of the square.
  36. If $|z|=1$ and $\omega=\dfrac{z-1}{z+1}$ (where $z\neq-1$), then $\mathrm{Re}(\omega)$ is
  37. Find the centre and radius of the circle given by $\left|\dfrac{z-\alpha}{z-\beta}\right|=k,\ k\neq1$, where $z=x+iy$, $\alpha=\alpha_1+i\alpha_2$ …
  38. $a,b,c$ are integers, not all simultaneously equal, and $\omega$ is a cube root of unity ($\omega\ne 1$), then the minimum value of $|a+b\omega+c\omega^{2}|$ is
  39. For all complex numbers $z_1,z_2$ satisfying $|z_1|=12$ and $|z_2-3-4i|=5$, the minimum value of $|z_1-z_2|$ is
  40. If $\omega\ (\neq1)$ be a cube root of unity and $\left(1+\omega^{2}\right)^{n}=\left(1+\omega^{4}\right)^{n}$, then the least positive value of $n$ is
  41. If $w=\alpha+i\beta$, where $\beta\neq0$ and $z\neq1$, satisfies the condition that $\left(\dfrac{w-\bar wz}{1-z}\right)$ is purely real, then the set of values of $z$ is
  42. Let $z=(1+i)(1+2i)(1+3i)\ldots(1+ni)$, where $i=\sqrt{-1}$. If $|z|^2=44200$, then $n$ is equal to $\_\_\_\_\_\_$ .
  43. Let $z$ be a complex number such that $|z-6|=5$ and $|z+2-6i|=5$. Then the value of $z^3+3z^2-15z+141$ is equal to :
  44. Let $S=\left\{z\in\mathbf{C}:\left|\frac{z-6i}{z-2i}\right|=1\text{ and }\left|\frac{z-8+2i}{z+2i}\right|=\frac35\right\}$. Then $\sum_{z\in S}|z|^2$ is equal to :
  45. Let the set of all values of $k\in\mathbb{R}$ such that the equation $z(\bar z+2+i)+k(2+3i)=0$, $z\in\mathbb{C}$, has at least one solution, be the interval …
  46. Let $z$ be the complex number satisfying $|z-5|\le3$ and having maximum positive principal argument. Then $34\left|\frac{5z-12}{5iz+16}\right|^2$ is equal to :
  47. Let $S=\{z\in\mathbf{C}:4z^2+\bar z=0\}$. Then $\sum_{z\in S}|z|^2$ is equal to :
  48. Let $S=\{z:3\le|2z-3(1+i)|\le7\}$ be a set of complex numbers. Then $\min_{z\in S}\left|z+\frac12(5+3i)\right|$ is equal to :
  49. If $x^2+x+1=0$, then the value of $\left(x+\frac1x\right)^4+\left(x^2+\frac1{x^2}\right)^4+\left(x^3+\frac1{x^3}\right)^4+\ldots+\left(x^{25}+\frac1{x^{25}}\right)^4$ is …
  50. If $z=\frac{\sqrt3}{2}+\frac i2$, $i=\sqrt{-1}$, then $\left(z^{201}-i\right)^8$ is equal to :

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