Complex Numbers — JEE Main PYQs

188 previous year questions on Complex Numbers (Mathematics), newest first. Open any question for its complete step-by-step solution.

If $x^{2} + x + 1 = 0$ , then the value of $(x + \frac{1}{x})^{4} + (x^{2} + \frac{1}{x ^{2}})^{4} + (x^{3} + \frac{1}{x ^{3}})^{4} + \dots + (x^{25} + \frac{1}{x ^{25}})^{4}$ is :
JEE Main 2026View solution →
Let z be the complex number satisfying $∣ z - 5∣ \leq 3$ and having maximum positive principal argument.
Then $34 \frac{5 z - 12}{5 i z + 16}^{2}$ is equal to:
JEE Main 2026View solution →
Let $α = \frac{- 1 + i 3}{2}$ and $β = \frac{- 1 - i 3}{2}, i = - 1$ . If $(7 - 7 α + 9 β)^{20} + (9 + 7 α - 7 β)^{20} + (- 7 + 9 α + 7 β)^{20} + (14 + 7 α + 7 β)^{20} = m^{10}$ , then m is
JEE Main 2026View solution →
Let $S = {z \in C : 4 z^{2} + \overline{z} = 0}$ . Then $\sum_{z \in S} ∣ z ∣^{2}$ is equal to :
JEE Main 2026View solution →
Let $S = {z : 3 \leq ∣2 z - 3 (1 + i) ∣ \leq 7}$ be a set of complex numbers. Then $Min_{z \in S} (z + \frac{1}{2} (5 + 3 i))$ is equal to:
JEE Main 2026View solution →
If $z = \frac{3}{2} + \frac{i}{2}, i = - 1$ , then $(z^{201} - i)^{8}$ is equal to
JEE Main 2026View solution →
Let $S = {z \in C : \frac{z - 6 i}{z - 2 i} = 1$ and $\frac{z - 8 + 2 i}{z + 2 i} = \frac{3}{5}}$ .

Then …
JEE Main 2026View solution →
Let $z = (1 + i) (1 + 2 i) (1 + 3 i) \dots (1 + ni)$ , where i $= - 1$ . If $∣ z ∣^{2} = 44200$ , then $n$ is equal to ______.
JEE Main 2026View solution →
Let z be a complex number such that $∣ z - 6∣ = 5$ and $∣ z + 2 - 6 i ∣ = 5$ .
Then the value of $z^{3} + 3 z^{2} - 15 z + 141$ is equal to
JEE Main 2026View solution →
Let
$A = {z \in C : ∣ z - 2∣ \leq 4}$ and
$B = {z \in C : ∣ z - 2∣ + ∣ z + 2∣ = 5}$ .
Then the max ${∣ z_{1} - z_{2} ∣ : z_{1} \in A$ and …
JEE Main 2026View solution →
Let $x$ and $y$ be real number such that $50 (\frac{2 x}{1 + 3 i} - \frac{y}{1 - 2 i}) = 31 + 17 i$ , $i = - 1$ . Then the value of $10 (x - 3 y)$ is :
JEE Main 2026View solution →
Let $S = {z \in C : z^{2} + 4 z + 16 = 0}$ .

Then $\sum_{z \in S} ∣ z + 3 i ∣^{2}$ is equal to :
JEE Main 2026View solution →
Let z be a complex number such that $∣ z + 2∣ = ∣ z - 2∣$ and $ar g (\frac{z + 3}{z - i}) = \frac{π}{4}$ . Then $∣ z ∣^{2}$ is equal to :
JEE Main 2026View solution →
Let $z_{1}, z_{2} \in C$ be the distinct solutions of the equation $z^{2} + 4 z - (1 + 12 i) = 0$ . Then $∣ z_{1} ∣^{2} + ∣ z_{2} ∣^{2}$ is equal to :
JEE Main 2026View solution →
Let $S = {z \in C : z^{2} + 6 i z - 3 = 0}$ . Then $\sum_{z \in S} z^{8}$ is equal to :
JEE Main 2026View solution →
Let the set of all values of $k \in R$ such that the equation $z (\overline{z} + 2 + i) + k (2 + 3 i) = 0, z \in C$ , has at least one solution, be the interval $[α, β]$ . Then $9 (α + β)$ is equal to:
JEE Main 2026View solution →
The number of values of $z \in C$ , satisfying the equations $∣ z - (4 + 8 i) ∣ = 10$ and $∣ z - (3 + 5 i) ∣ + ∣ z - (5 + 11 i) ∣ = 45$ , is :
JEE Main 2026View solution →
Let the circle $C_{1} : ∣ z ∣ = r$ and $C_{2} : ∣ z - 3 - 4 i ∣ = 5, z \in C$ , be such that $C_{2}$ lies within $C_{1}$ . If $z_{1}$ moves on $C_{1}$ , $z_{2}$ moves o …
JEE Main 2026View solution →
Let $z_{1}, z_{2}$ and $z_{3}$ be three complex numbers on the circle $∣ z ∣ = 1$ with $ar g (z_{1}) = \frac{- π}{4}, ar g (z_{2}) = 0$ and $ar g (z_{3}) = \frac{π}{4}$ . If …
JEE Main 2025View solution →
Let the curve $z (1 + i) + \overline{z} (1 - i) = 4, z \in C$ , divide the region $∣ z - 3∣ \leq 1$ into two parts of areas $α$ and $β$ . Then $∣ α - β ∣$ equals :
JEE Main 2025View solution →
Let $\frac{z ˉ - i}{2 z ˉ + i} = \frac{1}{3}, z \in C$ , be the equation of a circle with center at C . If the area of the triangle, whose vertices are at the points $(0, 0), C$ and $(α, 0)$ is 11 square units, then …
JEE Main 2025View solution →
The number of complex numbers z, satisfying $∣ z ∣ = 1$ and $\frac{z}{z ˉ} + \frac{z ˉ}{z} = 1$ , is:
JEE Main 2025View solution →
Let $α, β$ be the roots of the equation $x^{2} - a x - b = 0$ with $Im (α) < Im (β)$ . Let $P_{n} = α^{n} - β^{n}$ . If $P_{3} = - 57 i$ , $P_{4} = - 37 i$ , $P_{5} = 117 i$ and $P_{6} = 457 i$ , then …
JEE Main 2025View solution →
Let integers $a, b \in [- 3, 3]$ be such that $a + b \neq = 0$ . Then the number of all possible ordered pairs $(a, b)$ , for which $\frac{z - a}{z + b} = 1$ and …
JEE Main 2025View solution →
Let $∣ z_{1} - 8 - 2 i ∣ \leq 1$ and $∣ z_{2} - 2 + 6 i ∣ \leq 2$ , $z_{1}, z_{2} \in C$ . Then the minimum value of $∣ z_{1} - z_{2} ∣$ is :
JEE Main 2025View solution →
If $α + i β$ and $γ + i δ$ are the roots of $x^{2} - (3 - 2 i) x - (2 i - 2) = 0, i = - 1$ , then $α γ + β δ$ is equal to :
JEE Main 2025View solution →
If $α$ and $β$ are the roots of the equation $2 z^{2} - 3 z - 2 i = 0$ , where $i = - 1$ , then …
JEE Main 2025View solution →
If $z_{1}, z_{2}, z_{3} \in C$ are the vertices of an equilateral triangle, whose centroid is $z_{0}$ , then $\sum_{k = 1}^{3} (z_{k} - z_{0})^{2}$ is equal to
JEE Main 2025View solution →
Let $z$ be a complex number such that $∣ z ∣ = 1$ . If $\frac{2 + k ^{2} z}{k + z} = k z$ , $k \in R$ , then the maximum distance of $k + i k^{2}$ from the circle $∣ z - (1 + 2 i) ∣ = 1$ is:
JEE Main 2025View solution →
Let $z \in C$ be such that $\frac{z ^{2} + 3 i}{z - 2 + i} = 2 + 3 i$ . Then the sum of all possible values of $z^{2}$ is
JEE Main 2025View solution →
Let the product of $ω_{1} = (8 + i) sin θ + (7 + 4 i) cos θ$ and $ω_{2} = (1 + 8 i) sin θ + (4 + 7 i) cos θ$ be $α + i β$ , $i = - 1$ . Let p and q be the maximum and the minimum values of …
JEE Main 2025View solution →
If $α$ is a root of the equation $x^{2} + x + 1 = 0$ and $\sum_{k = 1}^{n} (α^{k} + \frac{1}{α ^{k}})^{2} = 20$ , then n is equal to
JEE Main 2025View solution →
Let $A = {z \in C : ∣ z - 2 - i ∣ = 3}$ , $B = {z \in C : Re (z - i z) = 2}$ and $S = A \cap B$ . Then $\sum_{z \in S} ∣ z ∣^{2}$ is equal to
JEE Main 2025View solution →
If the locus of $z \in C$ , such that
$Re (\frac{z - 1}{2 z + i}) + Re (\frac{z ˉ - 1}{2 z ˉ - i}) = 2,$

is a circle of radius $r$ and center $(a, b)$ then $\frac{15 ab}{r ^{2}}$ is equal to :
JEE Main 2025View solution →
Among the statements
(S1) : The set ${z \in C - {- i} : ∣ z ∣ = 1$ and $\frac{z - i}{z + i}$ is purely real} contains exactly two elements, and
(S2) : The set …
JEE Main 2025View solution →
Let $α$ be a solution of $x^{2} + x + 1 = 0$ , and for some a and b in $R, [4 a b] 1 - 1 - 2 16 - 1 - 14 132 - 8 = [000]$ . If …
JEE Main 2025View solution →
Let $A = {θ \in [0, 2 π] : 1 + 10 Re (\frac{2 c o s θ + i s i n θ}{c o s θ - 3 i s i n θ}) = 0}$ . Then $\sum_{θ \in A} θ^{2}$ is equal to
JEE Main 2025View solution →
Let O be the origin, the point A be $z_{1}$ , and the point B be $z_{2}$ such that $∣ z_{2} ∣ = 3 ∣ z_{1} ∣$ and $ar g (z_{2}) = ar g (z_{1}) + \frac{π}{6}$ . Then the triangle ABO is:
JEE Main 2025View solution →
Let $α$ and $β$ be the sum and the product of all the non-zero solutions of the equation $(\overset{z}{ˉ})^{2} + ∣ z ∣ = 0$ , $z \in C$ . Then $4 (α^{2} + β^{2})$ is equal to
JEE Main 2024View solution →
Consider the following two statements: Statement I: For any two non-zero complex numbers $z_{1}, z_{2}$ , $(∣ z_{1} ∣ + ∣ z_{2} ∣) \frac{z _{1}}{∣ z _{1} ∣} + \frac{z _{2}}{∣ z _{2} ∣} \leq 2 (∣ z_{1} ∣ + ∣ z_{2} ∣)$ ; Statement II: if $x, y, z$ are three distinct complex numbers and $a, b, c$ are three …
JEE Main 2024View solution →
If $z_{1}, z_{2}$ are two distinct complex numbers such that $\frac{z _{1} - 2 z _{2}}{\frac{1}{2} - z _{1} z ˉ _{2}} = 2$ , then
JEE Main 2024View solution →
Let z be a complex number such that $∣ z + 2∣ = 1$ and $Im (\frac{z + 1}{z + 2}) = \frac{1}{5}$ . Then the value of $∣ Re (\overline{z + 2}) ∣$ is :
JEE Main 2024View solution →
If the set $R = {(a, b); a + 5 b = 42, a, b \in N}$ has $m$ elements and $n = 1 \sum m (1 - i^{n!}) = x + i y$ , where $i = - 1$ , then the value of $m + x + y$ is :
JEE Main 2024View solution →
The sum of all possible values of $θ \in [- π, 2 π]$ , for which $\frac{1 + i c o s θ}{1 - 2 i c o s θ}$ is purely imaginary, is equal to
JEE Main 2024View solution →
The sum of the square of the modulus of the elements in the set
${z = a + ib : a, b \in Z, z \in C, ∣ z - 1∣ \leq 1, ∣ z - 5∣ \leq ∣ z - 5 i ∣}$ is ______.
JEE Main 2024View solution →
Let z be a complex number such that the real part of $\frac{z - 2 i}{z + 2 i}$ is zero. Then, the maximum value of $∣ z - (6 + 8 i) ∣$ is equal to :
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Let r and $θ$ respectively be the modulus and amplitude of the complex number $z = 2 - i (2 tan \frac{5 π}{8})$ , then $(r, θ)$ is equal to
JEE Main 2024View solution →
Let $α, β$ be the roots of the equation $x^{2} - 6 x + 3 = 0$ such that $Im (α) > Im (β)$ . Let $a, b$ be integers not divisible by 3 and n be a natural number such that …
JEE Main 2024View solution →
Let $z_{1}$ and $z_{2}$ be two complex number such that $z_{1} + z_{2} = 5$ and $z_{1}^{3} + z_{2}^{3} = 20 + 15 i$ . Then $z_{1}^{4} + z_{2}^{4}$ equals-
JEE Main 2024View solution →
If $z = x + iy, xy \neq = 0$ , satisfies the equation $z^{2} + i \overset{z}{ˉ} = 0$ , then $z^{2}$ is equal to :
JEE Main 2024View solution →