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.

Let $S = {z \in C : ∣ z - 2∣ \leq 1, z (1 + i) + \overset{z}{ˉ} (1 - i) \leq 2}$ . Let $∣ z - 4 i ∣$ attains minimum and maximum values, respectively, at $z_{1} \in S$ and $z_{2} \in S$ . If …
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Let $A = {z \in C : \frac{z + 1}{z - 1} < 1}$
and $B = {z \in C : ar g (\frac{z - 1}{z + 1}) = \frac{2 π}{3}}$ .
Then $A \cap B$ is :
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The area of the polygon, whose vertices are the non-real roots of the equation $\overset{z}{ˉ} = i z^{2}$ is :
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If $z = 2 + 3 i$ , then $z^{5} + (\overline{z})^{5}$ is equal to :
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Let $A = {z \in C : 1 \leq ∣ z - (1 + i) ∣ \leq 2}$ and $B = {z \in A : ∣ z - (1 - i) ∣ = 1}$ . Then, $B :$
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Let $S = {z \in C : ∣ z - 3∣ \leq 1$ and $z (4 + 3 i) + \overset{z}{ˉ} (4 - 3 i) \leq 24}$ .
If $α + i β$ is the point in S which is closest to $4 i$ , then $25 (α + β)$ is equal to ______.
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Let $z_{1}$ and $z_{2}$ be two complex numbers such that $\overline{z}_{1} = i \overline{z}_{2}$ and $ar g (\frac{z _{1}}{z _{2}}) = π$ . Then
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Let a circle C in complex plane pass through the points $z_{1} = 3 + 4 i, z_{2} = 4 + 3 i$ and $z_{3} = 5 i$ . If $z (\neq = z_{1})$ is a point on C such that the line through z and $z_{1}$ is perpendic …
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The equation $ar g (\frac{z - 1}{z + 1}) = \frac{π}{4}$ represents a circle with:
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Let $z = \frac{1 - i 3}{2}, i = - 1$ . Then the value of $21 + (z + \frac{1}{z})^{3} + (z^{2} + \frac{1}{z ^{2}})^{3} + (z^{3} + \frac{1}{z ^{3}})^{3} + \dots\dots + (z^{21} + \frac{1}{z ^{21}})^{3}$ is ____.
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If the least and the largest real values of $α$ , for which the equation $z + α ∣ z - 1∣ + 2 i = 0 (z \in C$ and $i = - 1$ ) has a solution, are $p$ and $q$ respectively, then $4 (p^{2} + q^{2})$ is equal to
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If $(3 + i)^{100} = 2^{99} (p + i q)$ , then $p$ and $q$ are roots of the equation:
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The least positive integer n such that $\frac{( 2 i ) ^{n}}{( 1 - i ) ^{n - 2}}$ , $i = - 1$ , is a positive integer, is ____.
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If $z$ and $ω$ are two complex numbers such that $∣ z ω ∣ = 1 and ar g (z) - ar g (w) = \frac{3 π}{2}$ , then $ar g (\frac{1 - 2 z ˉ ω}{1 + 3 z ˉ ω})$ is :
(Here $ar g (z)$ denotes the principal argument of complex num …
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Let $i = - 1$ . If $\frac{( - 1 + i 3 ) ^{21}}{( 1 - i ) ^{24}} + \frac{( 1 + i 3 ) ^{21}}{( 1 + i ) ^{24}} = k$ , and $n = [∣ k ∣]$ be the greatest integral part of $∣ k ∣$ . Then …
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If the real part of the complex number $(1 - cos θ + 2 i sin θ)^{- 1}$ is $\frac{1}{5}$ for $θ \in (0, π)$ , then the value of the integral $\int_{0}^{θ} sin x d x$ is equal to :
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Let $z_{1}$ and $z_{2}$ be two complex numbers such that $ar g (z_{1} - z_{2}) = \frac{π}{4}$ and $z_{1}, z_{2}$ satisfy the equation $∣ z - 3∣ = Re (z)$ . Then the imaginary part of $z_{1} + z_{2}$ is equal to
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Let the lines $(2 - i) z = (2 + i) \overset{z}{ˉ}$ and $(2 + i) z + (i - 2) \overset{z}{ˉ} - 4 i = 0$ , (here $i^{2} = - 1$ ) be normal to a circle $C$ . If the line $i z + \overset{z}{ˉ} + 1 + i = 0$ is tangent to this circle $C$ , then its radius is :
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Let $n$ denote the number of solutions of the equation $z^{2} + 3 \overset{z}{ˉ} = 0$ , where $z$ is a complex number. Then the value of $\sum_{k = 0}^{\infty} \frac{1}{n ^{k}}$ is equal to :
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If $S = {z \in C : \frac{z - i}{z + 2 i} \in R}$ , then
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Let z be those complex numbers which satisfy
$∣ z + 5∣ \leq 4$ and $z (1 + i) + \overset{z}{ˉ} (1 - i) \geq - 10, i = - 1$ .
If the maximum value of $∣ z + 1 ∣^{2}$ is $α + β 2$ , then the value of $(α + β)$ is ____
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If $z$ is a complex number such that $\frac{z - i}{z - 1}$ is purely imaginary, then the minimum value of $∣ z - (3 + 3 i) ∣$ is:
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If $a_{r} = cos \frac{2 r π}{9} + i sin \frac{2 r π}{9}, r = 1, 2, 3 \dots, i = - 1$ , then the determinant $a_{1} a_{4} a_{7} a_{2} a_{5} a_{8} a_{3} a_{6} a_{9}$ is equal to:
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A point $z$ moves in the complex plane such that arg $(\frac{z - 2}{z + 2}) = \frac{π}{4}$ , then the minimum value of $∣ z - 92 - 2 i ∣^{2}$ is equal to ____.
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Let C be the set of all complex numbers. Let
$S_{1} = {z \in C ∥∣ z - 3 - 2 i ∣^{2} = 8}$ ,
$S_{2} = {z \in C ∣ Re (z) \geq 5}$ and
$S_{3} = {z \in C ∣ ∣ z - \overset{z}{ˉ} ∣ \geq 8}$ .
Then the number of element in …
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The area of the triangle vertices $A (z), B (iz)$ and $C (z + iz)$ is :
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Let c be the set of all complex numbers. Let
$S_{1} = {z \in C : ∣ z - 2∣ \leq 1}$ and
$S_{2} = {z \in C : z (1 + i) + \overset{z}{ˉ} (1 - i) \geq 4}$ .
Then, the maximum value of $z - \frac{5}{2}^{2}$ for $z \in S_{1} \cap S_{2}$ is equal to :
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If the real part of the complex number $z = \frac{3 + 2 i c o s θ}{1 - 3 i c o s θ}, θ \in (0, \frac{π}{2})$ is zero, then the value of $sin^{2} 3 θ + cos^{2} θ$ is equal to ____
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If for the complex numbers z satisfying $∣ z - 2 - 2 i ∣ \leq 1$ , the maximum value of $∣3 i z + 6∣$ is attained at $a + ib$ , then $a + b$ is equal to ______.
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Let a complex number $z, ∣ z ∣ \neq = 1$ , satisfy $lo g_{\frac{1}{2}} (\frac{∣ z ∣ + 11}{( ∣ z ∣ - 1 ) ^{2}}) \leq 2$ . Then, the largest value of $∣ z ∣$ is equal to.........
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Let $z$ and $ω$ be two complex numbers such that $ω = z \overset{z}{ˉ} - 2 z + 2, \frac{z + i}{z - 3 i} = 1$ and $Re (ω)$ has minimum value. Then, the minimum value of $n \in N$ for which $ω^{n}$ is real, is equal to
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The least value of $∣ z ∣$ where $z$ is complex number which satisfies the inequality $exp (\frac{( ∣ z ∣ + 3 ) ( ∣ z ∣ - 1 )}{∣∣ z ∣ + 1∣} lo g_{e} 2) \geq lo g_{2} ∣5 7 + 9 i ∣, i = - 1$ , is equal to :
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Let a complex number be $w = 1 - 3 i$ . Let another complex number z be such that $∣ zw ∣ = 1$ and $ar g (z) - ar g (w) = \frac{π}{2}$ . Then the area of the triangle with vertices origin, $z$ and $w$ is equal to :
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If $f (x)$ and $g (x)$ are two polynomials such that the polynomial $P (x) = f (x^{3}) + xg (x^{3})$ is divisible by $x^{2} + x + 1$ , then $P (1)$ is equal to ______
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Let $S_{1}, S_{2}$ and $S_{3}$ be three sets defined as
$S_{1} = {z \in C : ∣ z - 1∣ \leq 2}$
$S_{2} = {z \in C : Re ((1 - i) z) \geq 1}$
$S_{3} = {z \in C : Im (z) \leq 1}$
Then the set $S_{1} \cap S_{2} \cap S_{3}$
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If the equation $a ∣ z ∣^{2} + \overline{\overset{α}{ˉ} z + α \overline{z}} + d = 0$ represents a circle where $a, d$ are real constants, then which of the following condition is correct?
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Let $z_{1}, z_{2}$ be the roots of the equation $z^{2} + a z + 12 = 0$ and $z_{1}, z_{2}$ form an equilateral triangle with origin. Then, The value of $∣ a ∣$ is ____
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If $Re (\frac{z - 1}{2 z + i}) = 1$ , where $z = x + i y$ , then the point $(x, y)$ lies on a:
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Let $α$ be a root of the equation $x^{2} + x + 1 = 0$ and the matrix $A = \frac{1}{3} 111 1 α α^{2} 1 α^{2} α^{4}$ , then the matrix $A^{31}$ is equal to:
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If $\frac{3 + i s i n θ}{4 - i c o s θ}$ , $θ \in [0, 2 π]$ , is a real number, then an argument of $sin θ + i cos θ$ is:
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If the equation, $x^{2} + b x + 45 = 0 (b \in R)$ has conjugate complex roots and they satisfy $∣ z + 1∣ = 210$ , then:
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Let $α = \frac{- 1 + i 3}{2}$ . If $a = (1 + α) \sum_{k = 0}^{100} α^{2 k}$ and $b = \sum_{k = 0}^{100} α^{3 k}$ , then $a$ and $b$ are the roots of the quadratic equation:
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If z be a complex number satisfying $∣ Re (z) ∣ + ∣ Im (z) ∣ = 4$ , then $∣ z ∣$ cannot be:
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Let $z$ be a complex number such that $\frac{z - i}{z + 2 i} = 1$ and $∣ z ∣ = \frac{5}{2}$ . Then the value of $∣ z + 3 i ∣$ is:
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The value of $(\frac{- 1 + i 3}{1 - i})^{30}$ is:
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If $(\frac{1 + i}{1 - i})^{m /2} = (\frac{1 + i}{i - 1})^{n /3} = 1, (m, n \in N)$

Then the greatest common divisor of the least values of $m$ and $n$ is ____.
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If $z_{1}, z_{2}$ are complex numbers such that $Re (z_{1}) = ∣ z_{1} - 1∣, Re (z_{2}) = ∣ z_{2} - 1∣$ and $ar g (z_{1} - z_{2}) = \frac{π}{6}$ , then $Im (z_{1} + z_{2})$ is equal to:
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Let $z = x + i y$ be a non-zero complex number such that $z^{2} = i ∣ z ∣^{2}$ , where $i = - 1$ , then $z$ lies on the:
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If the four complex numbers, $z, \overset{z}{ˉ}, \overset{z}{ˉ} - 2 Re (\overset{z}{ˉ})$ and $z - 2 Re (z)$ represent the vertices of a square of side 4 units in the Argand plane, then $∣ z ∣$ is equal to:
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The region represented by ${z = x + i y \in C : ∣ z ∣ - Re (z) \leq 1}$ is also given by the inequality:
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