Complex Numbers

18 JEE Main questions · Maths

MathsJEE Main 21 Jan 2026, Shift 1·Q17·MCQ·Moderate
Open
Cube roots of unity
If $x^2 + x + 1 = 0$ , then the value of $\left(x + \frac1x\right)^4 + \left(x^2 + \frac{1}{x^2}\right)^4 + \left(x^3 + \frac{1}{x^3}\right)^4 + \ldots + \left(x^{25} + \frac{1}{x^{25}}\right)^4$ is:
1. (1)128
2. (2)162
3. (3)175
4. (4)145
MathsJEE Main 21 Jan 2026, Shift 2·Q19·MCQ·Moderate
Open
Maximum argument in a disc
Let $z$ be the complex number satisfying $\lvert z - 5\rvert \le 3$ and having maximum positive principal argument. Then $34\left\lvert\dfrac{5z - 12}{5iz + 16}\right\rvert^2$ is equal to:
1. (1)16
2. (2)12
3. (3)26
4. (4)20
MathsJEE Main 22 Jan 2026, Shift 1·Q21·Numerical·Moderate
Open
Cube roots of unity
Let $\alpha = \dfrac{-1 + i\sqrt3}{2}$ and $\beta = \dfrac{-1 - i\sqrt3}{2}$ , $i = \sqrt{-1}$ . If $(7 - 7\alpha + 9\beta)^{20} + (9 + 7\alpha - 7\beta)^{20} + (-7 + 9\alpha + 7\beta)^{20} + (14 + 7\alpha + 7\beta)^{20} = m^{10},$ then $m$ is .
MathsJEE Main 22 Jan 2026, Shift 2·Q4·MCQ·Moderate
Open
Solving equations in conjugate
Let $S = \left\{z \in \mathbb{C} : 4z^2 + \bar{z} = 0\right\}$ . Then $\displaystyle\sum_{z\in S}\lvert z\rvert^2$ is equal to:
1. (1) $\dfrac{3}{16}$
2. (2) $\dfrac{7}{64}$
3. (3) $\dfrac{1}{16}$
4. (4) $\dfrac{5}{64}$
MathsJEE Main 23 Jan 2026, Shift 1·Q17·MCQ·Moderate
Open
Locus and minimum modulus
Let $S = \{z : 3 \le \lvert 2z - 3(1 + i)\rvert \le 7\}$ be a set of complex numbers. Then $\displaystyle\min_{z\in S}\left\lvert\left(z + \frac12(5 + 3i)\right)\right\rvert$ is equal to:
1. (1) $\dfrac12$
2. (2) $\dfrac32$
3. (3)2
4. (4) $\dfrac52$
MathsJEE Main 23 Jan 2026, Shift 2·Q17·MCQ·Easy–moderate
Open
Powers in polar form
If $z = \dfrac{\sqrt3}{2} + \dfrac{i}{2}$ , $i = \sqrt{-1}$ , then $(z^{201} - i)^8$ is equal to
1. (1) $-1$
2. (2)0
3. (3)1
4. (4)256
MathsJEE Main 24 Jan 2026, Shift 1·Q9·MCQ·Moderate
Open
Locus of complex numbers
Let $S = \left\{z \in \mathbb{C} : \left\lvert\dfrac{z - 6i}{z - 2i}\right\rvert = 1 \text{ and } \left\lvert\dfrac{z - 8 + 2i}{z + 2i}\right\rvert = \dfrac35\right\}$ . Then $\displaystyle\sum_{z \in S}\lvert z\rvert^2$ is equal to
1. (1)398
2. (2)413
3. (3)423
4. (4)385
We corrected a printing error in the original paper
Source printed the summation index as “ $z \in s$ ” (lower-case $s$ ; the set is $S$ ).
MathsJEE Main 24 Jan 2026, Shift 2·Q23·Numerical·Easy–moderate
Open
Modulus of a product
Let $z = (1 + i)(1 + 2i)(1 + 3i)\ldots(1 + ni)$ , where $i = \sqrt{-1}$ . If $\lvert z\rvert^2 = 44200$ , then $n$ is equal to .
MathsJEE Main 28 Jan 2026, Shift 1·Q5·MCQ·Moderate
Open
Modulus and touching circles
Let $z$ be a complex number such that $\lvert z - 6\rvert = 5$ and $\lvert z + 2 - 6i\rvert = 5$ . Then the value of $z^3 + 3z^2 - 15z + 141$ is equal to
1. (1)42
2. (2)37
3. (3)50
4. (4)61
MathsJEE Main 28 Jan 2026, Shift 2·Q18·MCQ·Moderate
Open
Locus and maximum modulus
Let $A = \{z \in \mathbb{C} : \lvert z - 2\rvert \le 4\} \quad\text{and}\quad B = \{z \in \mathbb{C} : \lvert z - 2\rvert + \lvert z + 2\rvert = 5\}.$ Then the $\max\{\lvert z_1 - z_2\rvert : z_1 \in A \text{ and } z_2 \in B\}$ is
1. (1) $\dfrac{15}{2}$
2. (2)8
3. (3) $\dfrac{17}{2}$
4. (4)9
MathsJEE Main 2 Apr 2026, Shift 1·Q2·MCQ·Easy–moderate
Open
Equality of complex numbers
Let $x$ and $y$ be real numbers such that $50\left(\dfrac{2x}{1+3i} - \dfrac{y}{1-2i}\right) = 31 + 17i,\quad i = \sqrt{-1}.$ Then the value of $10(x - 3y)$ is:
1. (1)20
2. (2)31
3. (3)35
4. (4)75
We corrected a printing error in the original paper
Source printed “Let $x$ and $y$ be real number”.
MathsJEE Main 2 Apr 2026, Shift 2·Q2·MCQ·Moderate
Open
Circles in the complex plane
Let the circles $C_1 : \lvert z\rvert = r$ and $C_2 : \lvert z - 3 - 4i\rvert = 5$ , $z \in \mathbb{C}$ , be such that $C_2$ lies within $C_1$ . If $z_1$ moves on $C_1$ , $z_2$ moves on $C_2$ and $\min\lvert z_1 - z_2\rvert = 2$ , then $\max\lvert z_1 - z_2\rvert$ is equal to:
1. (1)12
2. (2)17
3. (3)22
4. (4)24
We corrected a printing error in the original paper
Source printed “Let the circle $C_1$ … and $C_2$ …” (singular for two circles).
MathsJEE Main 4 Apr 2026, Shift 1·Q3·MCQ·Moderate
Open
Locus and argument of complex numbers
Let $z$ be a complex number such that $\lvert z + 2\rvert = \lvert z - 2\rvert$ and $\arg\left(\dfrac{z + 3}{z - i}\right) = \dfrac{\pi}{4}$ . Then $\lvert z\rvert^2$ is equal to:
1. (1)9
2. (2)4
3. (3)5
4. (4)1
MathsJEE Main 4 Apr 2026, Shift 2·Q2·MCQ·Easy–moderate
Open
Modulus of complex numbers
Let $S = \{z \in \mathbb{C} : z^2 + 4z + 16 = 0\}$ . Then $\displaystyle\sum_{z\in S}\left\lvert z + \sqrt3\,i\right\rvert^2$ is equal to:
1. (1)42
2. (2)23
3. (3)27
4. (4)38
MathsJEE Main 5 Apr 2026, Shift 2·Q2·MCQ·Easy–moderate
Open
Quadratic with complex coefficients
Let $z_1, z_2 \in \mathbb{C}$ be the distinct solutions of the equation $z^2 + 4z - (1 + 12i) = 0$ . Then $\lvert z_1\rvert^2 + \lvert z_2\rvert^2$ is equal to:
1. (1)18
2. (2)22
3. (3)29
4. (4)34
MathsJEE Main 6 Apr 2026, Shift 1·Q4·MCQ·Moderate
Open
Complex equation with real parameter
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 $[\alpha, \beta]$ . Then $9(\alpha + \beta)$ is equal to:
1. (1) $-10$
2. (2) $-8$
3. (3) $10\sqrt{13}$
4. (4) $8\sqrt{13}$
MathsJEE Main 6 Apr 2026, Shift 2·Q3·MCQ·Easy–moderate
Open
Roots and powers of complex numbers
Let $S = \left\{z \in \mathbb{C} : z^2 + \sqrt6\,iz - 3 = 0\right\}$ . Then $\displaystyle\sum_{z\in S} z^8$ is equal to:
1. (1)162
2. (2)184
3. (3)262
4. (4)324
MathsJEE Main 8 Apr 2026, Shift 2·Q2·MCQ·Moderate
Open
Locus of complex numbers
The number of values of $z \in \mathbb{C}$ , satisfying the equations $\lvert z - (4 + 8i)\rvert = \sqrt{10}$ and $\lvert z - (3 + 5i)\rvert + \lvert z - (5 + 11i)\rvert = 4\sqrt5$ , is:
1. (1)0
2. (2)2
3. (3)1
4. (4)4