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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 251-300 of 309

  1. Let $A=\{z\in\mathbf{C}:|z-2|\le4\}$ and $B=\{z\in\mathbf{C}:|z-2|+|z+2|=5\}$. Then the $\max\{|z_1-z_2|:z_1\in A$ and $z_2\in B\}$ is :
  2. A man walks a distance of 3 units from the origin towards the north-east ($N\,45^\circ E$) direction. From there, he walks a distance of 4 units towards the north-west …
  3. If $a,b,c$ and $u,v,w$ are complex numbers representing the vertices of two triangles such that $c=(1-r)a+rb$ and $w=(1-r)u+rv$, where $w=(1-r)u+rv$, is a complex …
  4. If $z$ is a complex number such that $|z|\ge2$, then the minimum value of $\left|z+\frac12\right|$ :
  5. If $|z|=1$ and $z\neq\pm1$, then all the values of $\frac{z}{1-z^2}$ lie on
  6. A particle $P$ starts from the point $z_0 = 1+2i$, where $i=\sqrt{-1}$. It moves first horizontally away from origin by 5 units and then vertically away from origin by 3 …
  7. If $z$ is any complex number satisfying $|z-3-2i|\le2$, then the minimum value of $|2z-6+5i|$ is
  8. Let $z=\frac{-1+\sqrt{3}i}{2}$, where $i=\sqrt{-1}$, and $r,s\in\{1,2,3\}$. Let $P=\begin{bmatrix}(-z)^r&z^{2s}\\z^{2s}&z^r\end{bmatrix}$ and $I$ be the identity matrix …
  9. Let $\theta_1,\theta_2,\ldots,\theta_{10}$ be positive valued angles (in radian) such that $\theta_1+\theta_2+\cdots+\theta_{10}=2\pi$. Define the complex numbers …
  10. Let $z=x+iy$ be a complex number where $x$ and $y$ are integers. Then the area of the rectangle whose vertices are the roots of the equation $$z\bar z^3+\bar z z^3=350$$ …
  11. Let $z=\cos\theta+i\sin\theta$. Then the value of $$\sum_{m=1}^{15}\mathrm{Im}\left(z^{2m-1}\right)$$ at $\theta=2^\circ$ is
  12. Let $z$ be a complex number such that the imaginary part of $z$ is nonzero and $a=z^2+z+1$ is real. Then $a$ cannot take the value
  13. Let complex numbers $\alpha$ and $\frac{1}{\alpha}$ lie on circles $(x-x_0)^2+(y-y_0)^2=r^2$ and $(x-x_0)^2+(y-y_0)^2=4r^2$, respectively. If $z_0=x_0+iy_0$ satisfies …
  14. Let $w=\frac{\sqrt3+i}{2}$ and $P=\{w^n : n=1,2,3,\ldots\}$. Further $H_1=\left\{z\in\mathbb{C}:\operatorname{Re}z>\frac12\right\}$ and …
  15. For any integer $k$, let $\alpha_k=\cos\left(\frac{k\pi}{7}\right)+i\sin\left(\frac{k\pi}{7}\right)$, where $i=\sqrt{-1}$. The value of the expression …
  16. Let $S$ be the set of all complex numbers $z$ satisfying $|z^2+z+1|=1$. Then which of the following statements is/are TRUE?
  17. For a complex number $z$, let $\mathrm{Re}(z)$ denote the real part of $z$. Let $S$ be the set of all complex numbers $z$ satisfying $z^4-|z|^4=4iz^2$, where …
  18. The number of complex numbers $z$, satisfying $|z|=1$ and $\left|\frac{z}{\bar z}+\frac{\bar z}{z}\right|=1$, is :
  19. Let $\alpha,\beta$ be the roots of the equation $x^2-ax-b=0$ with $\text{Im}(\alpha)<\text{Im}(\beta)$. Let $P_n=\alpha^n-\beta^n$. If $P_3=-5\sqrt7\,i$ …
  20. The number of elements in the set $A\cap B\cap C$ is
  21. Area of $S=$
  22. $\min_{z\in S}|1-3i-z|=$
  23. Let $z$ be any point in $A\cap B\cap C$. Then, $|z+1-i|^2+|z-5-i|^2$ lies between
  24. Let $z$ be any point in $A\cap B\cap C$ and let $w$ be any point satisfying $|w-2-i|<3$. Then, $|z|-|w|+3$ lies between
  25. Let $\omega$ be a solution of $x^3-1=0$ with $\mathrm{Im}(\omega)>0$. If $a=2$ with $b$ and $c$ satisfying (E), then the value of …
  26. If $\alpha$ and $\beta$ are the roots of the equation $x^{2}-x+1=0$, then $\alpha^{2009}+\beta^{2009}=$
  27. If $|z+4|\le3$, then the maximum value of $|z+1|$ is :
  28. If $\omega(\neq1)$ is a cube root of unity, and $(1+\omega)^{7}=A+B\omega$. Then $(A,B)$ equals :
  29. Match the statements in Column-I with those in Column-II. [Note: Here $z$ takes values in the complex plane and $\operatorname{Im}z$ and $\operatorname{Re}z$ denote …
  30. If $x=a+b$, $y=a\gamma+b\beta$ and $z=a\beta+b\gamma$, where $\gamma$ and $\beta$ are the complex cube roots of unity, show that $xyz=a^{3}+b^{3}$.
  31. If $x+iy=\sqrt{\dfrac{a+ib}{c+id}}$, prove that $\left(x^{2}+y^{2}\right)^{2}=\dfrac{a^{2}+b^{2}}{c^{2}+d^{2}}$.
  32. Let the complex numbers $z_1$, $z_2$ and $z_3$ be the vertices of an equilateral triangle. Let $z_0$ be the circumcentre of the triangle. Then prove that …
  33. Prove that the complex numbers $z_1$, $z_2$ and the origin form an equilateral triangle only if $z_1^{2}+z_2^{2}-z_1z_2=0$.
  34. If $1,a_1,a_2,\ldots,a_{n-1}$ are the $n$ roots of unity, then show that $(1-a_1)(1-a_2)(1-a_3)\cdots(1-a_{n-1})=n$.
  35. Show that the area of the triangle on the Argand diagram formed by the complex numbers $z$, $iz$ and $z+iz$ is $\dfrac12|z|^{2}$.
  36. Let $z_1=10+6i$ and $z_2=4+6i$. If $Z$ is any complex number such that the argument of $\dfrac{(z-z_1)}{(z-z_2)}$ is $\dfrac{\pi}{4}$, then prove that $|z-7-9i|=3\sqrt2$.
  37. If $|Z|\le1$, $|W|\le1$, show that $|Z-W|^{2}\le(|Z|-|W|)^{2}+(\operatorname{Arg}Z-\operatorname{Arg}W)^{2}$.
  38. If $iz^{3}+z^{2}-z+i=0$, then show that $|z|=1$.
  39. Let $z_1$ and $z_2$ be roots of the equation $z^{2}+pz+q=0$, where the coefficients $p$ and $q$ may be complex numbers. Let $A$ and $B$ represent $z_1$ and $z_2$ in the …
  40. Let a complex number $\alpha$, $\alpha\neq1$, be a root of the equation $z^{p+q}-z^{p}-z^{q}+1=0$, where $p,q$ are distinct primes. Show that either …
  41. If $z_1$ and $z_2$ are two complex numbers such that $|z_1|<1<|z_2|$ then prove that $\left|\dfrac{1-z_1\overline{z_2}}{z_1-z_2}\right|<1$.
  42. If $z\ne1$ and $\frac{z^2}{z-1}$ is real, then the point represented by the complex number $z$ lies :
  43. For complex numbers $z$ and $w$, prove that $|z|^{2}w-|w|^{2}z=z-w$ if and only if $z=w$ or $z\bar w=1$.
  44. Prove that there exists no complex number $z$ such that $|z|<\dfrac13$ and $\displaystyle\sum_{r=1}^{n}a_rz^r=1$ where $|a_r|<2$.
  45. If $z=\dfrac{\sqrt{3}-i}{2}$ and $\left(z^{95}+i^{95}\right)^{94}=z^{n}$, then the minimum positive integral value of '$n$' is $\_\_\_\_\_\_\_\_$ .
  46. Let a complex number $z$, $|z|\neq1$, satisfy $\log_{\frac{1}{\sqrt{2}}}\left(\dfrac{|z|+11}{(|z|-1)^{2}}\right)\le2$. Then, the largest value of $|z|$ is equal to …
  47. Let $z$ and $w$ be two complex numbers such that $w=z\bar{z}-2z+2$, $\left|\dfrac{z+i}{z-3i}\right|=1$ and $\operatorname{Re}(w)$ has minimum value. Then, the minimum …
  48. Let $\alpha$ and $\beta$ be the sum and the product of all the non-zero solutions of the equation $(\bar{z})^{2}+|z|=0$, $z\in\mathbb{C}$. Then …
  49. The area (in sq. units) of the region $S=\left\{z\in\mathbb{C}:|z-1|\le2;\ (z+\bar{z})+i(z-\bar{z})\le2,\ \operatorname{Im}(z)\ge0\right\}$ is
  50. Consider the following two statements : Statement I : For any two non-zero complex numbers $z_1,z_2$ …

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