Complex Numbers

187 questions · Mathematics · JEE Main

Complex Numbers

187 questions · Mathematics · JEE Main

Let $z$ be a complex number such that $∣ z ∣ = 1$ . If $\frac{2 + k ^{2} z}{k + z ˉ} = k z, k \in R$ , then the maximum distance of $k + i k^{2}$ from the circle $∣ z - (1 + 2 i) ∣ = 1$ is :2025 · 2 Apr · Shift 1 · Q34 · MCQ
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 :2025 · 3 Apr · Shift 1 · Q42 · MCQ
If $z_{1}, z_{2}, z_{3} \in C$ are the vertices of an equilateral triangle, whose centroid is $z_{0}$ , then $k = 1 \sum 3 (z_{k} - z_{0})^{2}$ is equal to2025 · 3 Apr · Shift 2 · Q29 · MCQ
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 .2025 · 4 Apr · Shift 1 · Q50 · Numerical
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 $α + β$ …2025 · 4 Apr · Shift 2 · Q37 · MCQ
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 .2025 · 4 Apr · Shift 2 · Q49 · Numerical
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 ${z \in C - {- 1} : ∣ z ∣ = 1$ and $\frac{z - 1}{z + 1}$ …2025 · 7 Apr · Shift 1 · Q38 · MCQ
If the locus of z ∈ ℂ, such that Re $(\frac{z - 1}{2 z + i}) + Re (\frac{z - 1}{2 z - i}) = 2$ , is a circle of radius r and center $(a, b)$ , then $\frac{15 ab}{r ^{2}}$ is equal to :2025 · 7 Apr · Shift 2 · Q28 · MCQ
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 $θ \in A \sum θ^{2}$ is equal to2025 · 8 Apr · Shift 2 · Q35 · MCQ
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 $∣ z_{1} \overset{z}{ˉ}_{2} + z_{2} \overset{z}{ˉ}_{3} + z_{3} \overset{z}{ˉ}_{1} ∣^{2} = α + β 2, α, β \in Z$ …2025 · 22 Jan · Shift 1 · Q27 · MCQ
Let the curve $z (1 + i) + \overset{z}{ˉ} (1 - i) = 4, z \in C$ , divide the region $∣ z - 3∣ \leq 1$ into two parts of areas $α$ and $β$ . Then $∣ α - β ∣$ equals :2025 · 22 Jan · Shift 2 · Q42 · MCQ
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 $α^{2}$ …2025 · 23 Jan · Shift 1 · Q38 · MCQ
The number of complex numbers $z$ , satisfying $∣ z ∣ = 1$ and $\frac{z}{z ˉ} + \frac{z ˉ}{z} = 1$ , is :2025 · 23 Jan · Shift 2 · Q27 · MCQ
Let $α, β$ be the roots of the equation $x^{2} - ax - b = 0$ with $Im (α) < Im (β)$ . Let $P_{n} = α^{n} - β^{n}$ . If $P_{3} = - 57 i, P_{4} = - 37 i, P_{5} = 117 i$ …2025 · 23 Jan · Shift 2 · Q46 · Numerical
If $α$ and $β$ are the roots of the equation $2 z^{2} - 3 z - 2 i = 0$ , where $i = - 1$ , then $16 \cdot Re (\frac{α ^{19} + β ^{19} + α ^{11} + β ^{11}}{α ^{15} + β ^{15}}) \cdot lm (\frac{α ^{19} + β ^{19} + α ^{11} + β ^{11}}{α ^{15} + β ^{15}})$ …2025 · 24 Jan · Shift 1 · Q32 · MCQ
Let $O$ be the origin, the point $A$ be $z_{1} = 3 + 22 i$ , the point $B (z_{2})$ be such that $3 ∣ z_{2} ∣ = ∣ z_{1} ∣$ and $ar g (z_{2}) = ar g (z_{1}) + \frac{π}{6}$ . Then2025 · 28 Jan · Shift 1 · Q44 · MCQ
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:2025 · 28 Jan · Shift 2 · Q39 · MCQ
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 :2025 · 29 Jan · Shift 1 · Q37 · MCQ
Let integers $a, b \in [- 3, 3]$ be such that $a + b e q 0$ . Then the number of all possible ordered pairs (a, b), for which $\frac{z - a}{z + b} = 1$ and $z + 1 ω ω^{2} ω z + ω^{2} 1 ω^{2} 1 z + ω = 1, z \in C$ …2025 · 29 Jan · Shift 2 · Q48 · Numerical
Let $S = ∣ z \in C : ∣ z - 1 ∣= 1$ and $(2 - 1) (z + \overset{z}{ˉ}) - i (z - \overset{z}{ˉ}) = 22 ∣$ . Let $z_{1}, z_{2} \in S$ be such that $∣ z_{1} ∣ = z \in s max ∣ z ∣$ and $∣ z_{2} ∣ = z \in S min ∣ z ∣$ …2024 · 1 Feb · Shift 1 · Q36 · MCQ
Let $P = {z \in C : ∣ z + 2 - 3 i ∣ \leq 1}$ and $Q = {z \in C : z (1 + i) + \overset{z}{ˉ} (1 - i) \leq - 8}$ . Let in $P \cap Q$ , $∣ z - 3 + 2 i ∣$ be maximum and minimum at $z_{1}$ and $z_{2}$ …2024 · 1 Feb · Shift 1 · Q57 · Numerical
If $z$ is a complex number such that $∣ z ∣ ⩽ 1$ , then the minimum value of $z + \frac{1}{2} (3 + 4 i)$ is :2024 · 1 Feb · Shift 2 · Q32 · MCQ
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 :2024 · 4 Apr · Shift 1 · Q49 · MCQ
The area (in sq. units) of the region $S = {z \in C : ∣ z - 1∣ \leq 2; (z + \overset{z}{ˉ}) + i (z - \overset{z}{ˉ}) \leq 2, lm (z) \geq 0}$ is2024 · 4 Apr · Shift 2 · Q45 · MCQ
Consider the following two statements : Statement I: For any two non-zero complex numbers $z_{1}, z_{2}, (∣ z_{1} ∣ + ∣ z_{2} ∣) \frac{z _{1}}{∣ z _{1} ∣} + \frac{z _{2}}{∣ z _{2} ∣} \leq 2 (∣ z_{1} ∣ + ∣ z_{2} ∣), and$ …2024 · 5 Apr · Shift 1 · Q35 · MCQ
Let $S_{1} = {z \in C : ∣ z ∣ \leq 5}, S_{2} = {z \in C : Im (\frac{z + 1 - 3 i}{1 - 3 i}) \geq 0}$ and $S_{3} = {z \in C : Re (z) \geq 0}$ . Then the area of the…2024 · 5 Apr · Shift 2 · Q35 · MCQ
If $z_{1}, z_{2}$ are two distinct complex number such that $\frac{z _{1} - 2 z _{2}}{\frac{1}{2} - z _{1} z ˉ _{2}} = 2$ , then2024 · 6 Apr · Shift 2 · Q31 · MCQ
Let $z$ be a complex number such that $∣ z + 2∣ = 1$ and $lm (\frac{z + 1}{z + 2}) = \frac{1}{5}$ . Then the value of $∣ Re (\overline{z + 2}) ∣$ is2024 · 8 Apr · Shift 1 · Q43 · MCQ
If the set $R = {(a, b) : a + 5 b = 42, a, b \in N}$ has $m$ elements and $\sum_{n = 1}^{m} (1 - i^{n!}) = x + i y$ , where $i = - 1$ , then the value of $m + x + y$ is2024 · 8 Apr · Shift 1 · Q48 · MCQ
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 :2024 · 8 Apr · Shift 2 · Q45 · MCQ
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 .2024 · 9 Apr · Shift 1 · Q57 · Numerical
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 to2024 · 9 Apr · Shift 2 · Q32 · MCQ
If $S = {z \in C : ∣ z - i ∣ = ∣ z + i ∣ = ∣ z - 1∣}$ , then, $n (S)$ is :2024 · 27 Jan · Shift 1 · Q32 · MCQ
If $α$ satisfies the equation $x^{2} + x + 1 = 0$ and $(1 + α)^{7} = A + B α + C α^{2}, A, B, C ⩾ 0$ , then $5 (3 A - 2 B - C)$ is equal to .2024 · 27 Jan · Shift 1 · Q59 · Numerical
Let the complex numbers $α$ and $\frac{1}{α ˉ}$ 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 $100∣ α ∣^{2}$ is .2024 · 27 Jan · Shift 2 · Q59 · Numerical
If $z = \frac{1}{2} - 2 i$ is such that $∣ z + 1∣ = α z + β (1 + i), i = - 1$ and $α, β \in R$ , then $α + β$ is equal to2024 · 29 Jan · Shift 1 · Q45 · MCQ
Let $α, β$ be the roots of the equation $x^{2} - x + 2 = 0$ with $Im (α) > Im (β)$ . Then $α^{6} + α^{4} + β^{4} - 5 α^{2}$ is equal to .2024 · 29 Jan · Shift 1 · Q58 · Numerical
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 to2024 · 29 Jan · Shift 2 · Q47 · MCQ
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 $\frac{α ^{99}}{β} + α^{98} = 3^{n} (a + ib), i = - 1$ …2024 · 29 Jan · Shift 2 · Q55 · Numerical
If $z = x + i y, x y e q 0$ , satisfies the equation $z^{2} + i \overset{z}{ˉ} = 0$ , then $z^{2}$ is equal to :2024 · 30 Jan · Shift 1 · Q46 · MCQ
If $z$ is a complex number, then the number of common roots of the equations $z^{1985} + z^{100} + 1 = 0$ and $z^{3} + 2 z^{2} + 2 z + 1 = 0$ , is equal to2024 · 30 Jan · Shift 2 · Q35 · MCQ
If $α$ denotes the number of solutions of $∣1 - i ∣^{x} = 2^{x}$ and $β = (\frac{∣ z ∣}{a r g ( z )})$ , where $$z=\frac{\pi}{4}(1+i)^4\left[\frac{1-\sqrt{\pi} i}{\sqrt{\pi}+i}+\frac{\sqrt{\pi}-i}{1+\sqrt{\pi} i}\right],…2024 · 31 Jan · Shift 1 · Q51 · Numerical
Let $z_{1}$ and $z_{2}$ be two complex numbers 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 -2024 · 31 Jan · Shift 2 · Q45 · MCQ
If the center and radius of the circle $\frac{z - 2}{z - 3} = 2$ are respectively $(α, β)$ and $γ$ , then $3 (α + β + γ)$ is equal to :2023 · 1 Feb · Shift 1 · Q28 · MCQ
Let $a, b$ be two real numbers such that $ab < 0$ . IF the complex number $\frac{1 + ai}{b + i}$ is of unit modulus and $a + ib$ lies on the circle $∣ z - 1∣ = ∣2 z ∣$ , then a possible value of $\frac{1 + [ a ]}{4 b}$ , where $[t]$ is greatest integer…2023 · 1 Feb · Shift 2 · Q33 · MCQ
Let $a e q b$ be two non-zero real numbers. Then the number of elements in the set $X = {z \in C : Re (a z^{2} + b z) = a$ and $Re (b z^{2} + a z) = b}$ is equal…2023 · 6 Apr · Shift 2 · Q27 · MCQ
For $α, β, z \in C$ and $λ > 1$ , if $λ - 1$ is the radius of the circle $∣ z - α ∣^{2} + ∣ z - β ∣^{2} = 2 λ$ , then $∣ α - β ∣$ is equal to .2023 · 6 Apr · Shift 2 · Q41 · Numerical
If for $z = α + i β, ∣ z + 2∣ = z + 4 (1 + i)$ , then $α + β$ and $α β$ are the roots of the equation :2023 · 8 Apr · Shift 1 · Q37 · MCQ
Let $A = {θ \in (0, 2 π) : \frac{1 + 2 i s i n θ}{1 - i s i n θ}$ is purely imaginary $}$ . Then the sum of the elements in $A$ is :2023 · 8 Apr · Shift 2 · Q22 · MCQ
Let the complex number $z = x + i y$ be such that $\frac{2 z - 3 i}{2 z + i}$ is purely imaginary. If $x + y^{2} = 0$ , then $y^{4} + y^{2} - y$ is equal to :2023 · 10 Apr · Shift 1 · Q38 · MCQ
Let $S = {z = x + i y : \frac{2 z - 3 i}{4 z + 2 i} is a real number}$ . Then which of the following is NOT correct?2023 · 10 Apr · Shift 2 · Q21 · MCQ
Let $w_{1}$ be the point obtained by the rotation of $z_{1} = 5 + 4 i$ about the origin through a right angle in the anticlockwise direction, and $w_{2}$ be the point obtained by the rotation of $z_{2} = 3 + 5 i$ about the origin through a right…2023 · 11 Apr · Shift 1 · Q28 · MCQ
For $a \in C$ , let $A = {z \in C : Re (a + \overset{z}{ˉ}) > Im (\overset{a}{ˉ} + z)}$ and $B = {z \in C : Re (a + \overset{z}{ˉ}) (S 1) : I f$ \operatorname{Re}(a),…2023 · 11 Apr · Shift 2 · Q26 · MCQ
Let $S = {z \in C - {i, 2 i} : \frac{z ^{2} + 8 i z - 15}{z ^{2} - 3 i z - 2} \in R}$ . If $α - \frac{13}{11} i \in S, α \in R - {0}$ , then $242 α^{2}$ is equal to …2023 · 11 Apr · Shift 2 · Q41 · Numerical
Let $C$ be the circle in the complex plane with centre $z_{0} = \frac{1}{2} (1 + 3 i)$ and radius $r = 1$ . Let $z_{1} = 1 + i$ and the complex number $z_{2}$ be outside the circle $C$ such that $∣ z_{1} - z_{0} ∣ ∣ z_{2} - z_{0} ∣ = 1$ …2023 · 12 Apr · Shift 1 · Q28 · MCQ
Let $w = z \overset{z}{ˉ} + k_{1} z + k_{2} i z + λ (1 + i), k_{1}, k_{2} \in R$ . Let $Re (w) = 0$ be the circle $C$ of radius 1 in the first quadrant touching the line $y = 1$ and the $y$ -axis. If the curve $Im (w) = 0$ …2023 · 13 Apr · Shift 1 · Q39 · Numerical
Let $S = {z \in C : \overset{z}{ˉ} = i (z^{2} + Re (\overset{z}{ˉ}))}$ . Then $\sum_{z \in S} ∣ z ∣^{2}$ is equal to :2023 · 13 Apr · Shift 2 · Q24 · MCQ
If the set ${Re (\frac{z - z ˉ + z z ˉ}{2 - 3 z + 5 z ˉ}) : z \in C, Re (z) = 3}$ is equal to the interval $(α, β]$ , then $24 (β - α)$ is equal to :2023 · 15 Apr · Shift 1 · Q37 · MCQ
Let $p, q \in R$ and $(1 - 3 i)^{200} = 2^{199} (p + i q), i = - 1$ then $p + q + q^{2}$ and $p - q + q^{2}$ are roots of the equation.2023 · 24 Jan · Shift 1 · Q24 · MCQ
The value of $(\frac{1 + s i n \frac{2 π}{9} + i c o s \frac{2 π}{9}}{1 + s i n \frac{2 π}{9} - i c o s \frac{2 π}{9}})^{3}$ is2023 · 24 Jan · Shift 2 · Q34 · MCQ
Let $z_{1} = 2 + 3i$ and $z_{2} = 3 + 4i$ . The set $S = {z \in C : ∣ z - z_{1} ∣^{2} - ∣ z - z_{2} ∣^{2} = ∣ z_{1} - z_{2} ∣^{2}}$ represents a2023 · 25 Jan · Shift 1 · Q35 · MCQ
Let $z$ be a complex number such that $\frac{z - 2 i}{z + i} = 2, z e - i$ . Then $z$ lies on the circle of radius 2 and centre :2023 · 25 Jan · Shift 2 · Q31 · MCQ
For two non-zero complex numbers $z_{1}$ and $z_{2}$ , if $Re (z_{1} z_{2}) = 0$ and $Re (z_{1} + z_{2}) = 0$ , then which of the following are possible? A. $Im (z_{1}) > 0$ …2023 · 29 Jan · Shift 1 · Q31 · MCQ
Let $α = 8 - 14 i, A = {z \in c : \frac{α z - α z}{z ^{2} - ( z ) ^{2} - 112 i} = 1}$ and $B = {z \in c : ∣ z + 3 i ∣ = 4}$ . Then $z \in A \cap B \sum (Re o l imi t s z - Im o l imi t s z)$ …2023 · 29 Jan · Shift 2 · Q41 · Numerical
Let $z = 1 + i$ and $z_{1} = \frac{1 + i z ˉ}{z ˉ ( 1 - z ) + \frac{1}{z}}$ . Then $\frac{12}{π} ar g (z_{1})$ is equal to .2023 · 30 Jan · Shift 1 · Q35 · Numerical
For all $z \in C$ on the curve $C_{1} : ∣ z ∣ = 4$ , let the locus of the point $z + \frac{1}{z}$ be the curve $C_{2}$ . Then :2023 · 31 Jan · Shift 1 · Q37 · MCQ
The complex number $z = \frac{i - 1}{c o s \frac{π}{3} + i s i n \frac{π}{3}}$ is equal to :2023 · 31 Jan · Shift 2 · Q30 · MCQ
Let $A = {z \in C : 1 \leq ∣ z - (1 + i) ∣ \leq 2}$ and $B = {z \in A : ∣ z - (1 - i) ∣ = 1}$ . Then, B :2022 · 24 Jun · Shift 1 · Q22 · MCQ
Let S = {z $\in$ C : |z $-$ 3| $\leq$ 1 and z(4 + 3i) + $\overline{z}$ (4 $-$ 3i) $\leq$ 24}. If $α$ + i $β$ is the point in S which is closest to 4i, then 25( $α$ + $β$ ) is equal to .2022 · 24 Jun · Shift 2 · Q37 · Numerical
For $n \in N$ , let $S_{n} = {z \in C : ∣ z - 3 + 2 i ∣ = \frac{n}{4}}$ and $T_{n} = {z \in C : ∣ z - 2 + 3 i ∣ = \frac{1}{n}}$ . Then the number…2022 · 25 Jul · Shift 1 · Q23 · MCQ
For $z \in C$ if the minimum value of $(∣ z - 32 ∣ + ∣ z - p 2 i ∣)$ is $52$ , then a value Question: of $p$ is .2022 · 25 Jul · Shift 2 · Q23 · MCQ
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 $z (e z_{1})$ is a point on C such that the line through z and z1 is perpendicular to the line through z2 and z3, then $a r g (z)$ …2022 · 25 Jun · Shift 1 · Q38 · MCQ
Let z1 and z2 be two complex numbers such that $\overline{z}_{1} = i \overline{z}_{2}$ and $ar g (\frac{z _{1}}{z _{2}}) = π$ . Then :2022 · 25 Jun · Shift 2 · Q25 · MCQ
Let O be the origin and A be the point $z_{1} = 1 + 2 i$ . If B is the point $z_{2}$ , $Re o l imi t s (z_{2}) < 0$ , such that OAB is a right angled isosceles triangle with OB as hypotenuse, then which of the following is NOT…2022 · 26 Jul · Shift 1 · Q26 · MCQ
If $z = x + i y$ satisfies $∣ z ∣ - 2 = 0$ and $∣ z - i ∣ - ∣ z + 5 i ∣ = 0$ , then :2022 · 26 Jul · Shift 2 · Q21 · MCQ
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 :2022 · 26 Jun · Shift 1 · Q22 · MCQ
If $z^{2} + z + 1 = 0$ , $z \in C$ , then $n = 1 \sum 15 (z^{n} + (- 1)^{n} \frac{1}{z ^{n}})^{2}$ is equal to .2022 · 26 Jun · Shift 2 · Q41 · Numerical
Let the minimum value $v_{0}$ of $v = ∣ z ∣^{2} + ∣ z - 3 ∣^{2} + ∣ z - 6 i ∣^{2}, z \in C$ is attained at $z = z_{0}$ . Then $2 z_{0}^{2} - \overset{z}{ˉ}_{0}^{3} + 3^{2} + v_{0}^{2}$ is equal to :2022 · 27 Jul · Shift 1 · Q26 · MCQ
Let $S = {z \in C : z^{2} + \overset{z}{ˉ} = 0}$ . Then $z \in S \sum (Re (z) + Im (z))$ is equal to .2022 · 27 Jul · Shift 1 · Q46 · Numerical
Let S be the set of all $(α, β), π < α, β < 2 π$ , for which the complex number $\frac{1 - i s i n α}{1 + 2 i s i n α}$ is purely imaginary and $\frac{1 + i c o s β}{1 - 2 i c o s β}$ is purely real. Let $Z_{α β} = sin 2 α + i cos 2 β, (α, β) \in S$ …2022 · 27 Jul · Shift 2 · Q23 · MCQ
The area of the polygon, whose vertices are the non-real roots of the equation $\overline{z} = i z^{2}$ is :2022 · 27 Jun · Shift 1 · Q20 · MCQ
The number of points of intersection of $∣ z - (4 + 3 i) ∣ = 2$ and $∣ z ∣ + ∣ z - 4∣ = 6$ , z $\in$ C, is :2022 · 27 Jun · Shift 2 · Q23 · MCQ
Let $S_{1} = {z_{1} \in C : ∣ z_{1} - 3 ∣ = \frac{1}{2}}$ and $S_{2} = {z_{2} \in C : ∣ z_{2} - ∣ z_{2} + 1∣∣ = ∣ z_{2} + ∣ z_{2} - 1∣∣}$ . Then, for $z_{1} \in S_{1}$ and $z_{2} \in S_{2}$ …2022 · 28 Jul · Shift 1 · Q34 · MCQ
Let $z = a + ib, b e q 0$ be complex numbers satisfying $z^{2} = \overset{z}{ˉ} \cdot 2^{1 - z}$ . Then the least value of $n \in N$ , such that $z^{n} = (z + 1)^{n}$ , is equal to .2022 · 28 Jul · Shift 2 · Q37 · Numerical
The number of elements in the set {z = a + ib $\in$ C : a, b $\in$ Z and 1 < | z $-$ 3 + 2i | < 4} is .2022 · 28 Jun · Shift 1 · Q42 · Numerical
Sum of squares of modulus of all the complex numbers z satisfying $\overline{z} = i z^{2} + z^{2} - z$ is equal to .2022 · 28 Jun · Shift 2 · Q48 · Numerical
If $z = 2 + 3 i$ , then $z^{5} + (\overset{z}{ˉ})^{5}$ is equal to :2022 · 29 Jul · Shift 1 · Q26 · MCQ
If $z e q 0$ be a complex number such that $z - \frac{1}{z} = 2$ , then the maximum value of $∣ z ∣$ is :2022 · 29 Jul · Shift 2 · Q23 · MCQ
Let $S = {z = x + i y : ∣ z - 1 + i ∣ \geq ∣ z ∣, ∣ z ∣ < 2, ∣ z + i ∣ = ∣ z - 1∣}$ . Then the set of all values of $x$ , for which $w = 2 x + i y \in S$ for some $y \in R$ , is :2022 · 29 Jul · Shift 2 · Q34 · MCQ
Let $α$ and $β$ be the roots of the equation x2 + (2i $-$ 1) = 0. Then, the value of | $α$ 8 + $β$ 8| is equal to :2022 · 29 Jun · Shift 1 · Q26 · MCQ
Let $S = {z \in C : ∣ z - 2∣ \leq 1, z (1 + i) + \overline{z} (1 - i) \leq 2}$ . Let $∣ z - 4 i ∣$ attains minimum and maximum values, respectively, at z1 $\in$ S and z2 $\in$ S. If $5 (∣ z_{1} ∣^{2} + ∣ z_{2} ∣^{2}) = α + β 5$ ,…2022 · 29 Jun · Shift 1 · Q36 · Numerical
Let arg(z) represent the principal argument of the complex number z. Then, |z| = 3 and arg(z $-$ 1) $-$ arg(z + 1) = $\frac{π}{4}$ intersect :2022 · 29 Jun · Shift 2 · Q25 · MCQ
The real part of the complex number $\frac{( 1 + 2 i ) ^{8} . ( 1 - 2 i ) ^{2}}{( 3 + 2 i ) . ( 4 - 6 i )}$ is equal to :2022 · 30 Jun · Shift 1 · Q24 · MCQ
If for the complex numbers z satisfying | z $-$ 2 $-$ 2i | $\leq$ 1, the maximum value of | 3iz + 6 | is attained at a + ib, then a + b is equal to .2021 · 1 Sep · Shift 2 · Q40 · Numerical
Let a complex number z, |z| $e$ 1, satisfy $lo g_{\frac{1}{2}} (\frac{∣ z ∣ + 11}{( ∣ z ∣ - 1 ) ^{2}}) \leq 2$ . Then, the largest value of |z| is equal to .2021 · 16 Mar · Shift 1 · Q27 · MCQ
Let z and $ω$ be two complex numbers such that $ω = z \overline{z} - 2 z + 2, \frac{z + i}{z - 3 i} = 1$ and Re( $ω$ ) has minimum value. Then, the minimum value of n $\in$ N for which $ω$ n is real, is…2021 · 16 Mar · Shift 1 · Q42 · Numerical
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 :2021 · 16 Mar · Shift 2 · Q34 · MCQ
The area of the triangle with vertices A(z), B(iz) and C(z + iz) is :2021 · 17 Mar · Shift 1 · Q36 · MCQ
Let S1, S2 and S3 be three sets defined as S1 = {z $\in$ C : |z $-$ 1| $\leq 2$ } S2 = {z $\in$ C : Re((1 $-$ i)z) $\geq$ 1} S3 = {z $\in$ C : Im(z) $\leq$ 1} Then the set S1 $\cap$ S2 $\cap$ S3 :2021 · 17 Mar · Shift 2 · Q24 · MCQ
If the equation $a ∣ z ∣^{2} + \overline{\overline{α} z + α \overline{z}} + d = 0$ represents a circle where a, d are real constants then which of the following condition is correct?2021 · 18 Mar · Shift 1 · Q32 · MCQ
Let z1, z2 be the roots of the equation z2 + az + 12 = 0 and z1, z2 form an equilateral triangle with origin. Then, the value of |a| is :2021 · 18 Mar · Shift 1 · Q43 · Numerical
Let a complex number be w = 1 $- 3$ i. Let another complex number z be such that |zw| = 1 and arg(z) $-$ arg(w) = $\frac{π}{2}$ . Then the area of the triangle with vertices origin, z and w is equal to :2021 · 18 Mar · Shift 2 · Q28 · MCQ
If z and $ω$ are two complex numbers such that $∣ z ω ∣ = 1$ and $ar g (z) - ar g (ω) = \frac{3 π}{2}$ , then $ar g (\frac{1 - 2 z ω}{1 + 3 z ω})$ is :…2021 · 20 Jul · Shift 1 · Q27 · MCQ
Let n denote the number of solutions of the equation z2 + 3 $\overline{z}$ = 0, where z is a complex number. Then the value of $k = 0 \sum \infty \frac{1}{n ^{k}}$ is equal to :2021 · 22 Jul · Shift 2 · Q34 · MCQ
If the least and the largest real values of a, for which the equation z + $α$ |z – 1| + 2i = 0 (z $\in$ C and i = $- 1$ ) has a solution, are p and q respectively; then 4(p2 + q2) is equal to .2021 · 24 Feb · Shift 1 · Q38 · Numerical
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…2021 · 24 Feb · Shift 2 · Q38 · Numerical
Let the lines (2 $-$ i)z = (2 + i) $\overline{z}$ and (2 $+$ i)z + (i $-$ 2) $\overline{z} -$ 4i = 0, (here i2 = $-$ 1) be normal to a circle C. If the line iz + $\overline{z}$ + 1 + i = 0 is tangent to this circle C, then its radius is :2021 · 25 Feb · Shift 1 · Q31 · MCQ
If $α$ , $β \in$ R are such that 1 $-$ 2i (here i2 = $-$ 1) is a root of z2 + $α$ z + $β$ = 0, then ( $α - β$ ) is equal to :2021 · 25 Feb · Shift 2 · Q33 · MCQ
Let $S = {n \in N (01 i 0)^{n} (a c b d) = (a c b d) \forall a, b, c, d \in R}$ …2021 · 25 Jul · Shift 1 · Q45 · Numerical
The equation of a circle is Re(z2) + 2(Im(z))2 + 2Re(z) = 0, where z = x + iy. A line which passes through the center of the given circle and the vertex of the parabola, x2 $-$ 6x $-$ y + 13 = 0, has y-intercept equal to …2021 · 25 Jul · Shift 2 · Q42 · Numerical
The equation $ar g (\frac{z - 1}{z + 1}) = \frac{π}{4}$ represents a circle with :2021 · 26 Aug · Shift 1 · Q29 · MCQ
Let $z = \frac{1 - i 3}{2}$ , $i = - 1$ . Then the value of $21 + (z + \frac{1}{z})^{3} + (z^{2} + \frac{1}{z ^{2}})^{3} + (z^{3} + \frac{1}{z ^{3}})^{3} + .... + (z^{21} + \frac{1}{z ^{21}})^{3}$ …2021 · 26 Aug · Shift 1 · Q34 · Numerical
If $(3 + i)^{100} = 2^{99} (p + i q)$ , then p and q are roots of the equation :2021 · 26 Aug · Shift 2 · Q34 · MCQ
The least positive integer n such that $\frac{( 2 i ) ^{n}}{( 1 - i ) ^{n - 2}}, i = - 1$ is a positive integer, is .2021 · 26 Aug · Shift 2 · Q45 · Numerical
Let z be those complex numbers which satisfy | z + 5 | $\leq$ 4 and z(1 + i) + $\overline{z}$ (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 …2021 · 26 Feb · Shift 2 · Q43 · Numerical
If $S = {z \in C : \frac{z - i}{z + 2 i} \in R}$ , then :2021 · 27 Aug · Shift 1 · Q26 · MCQ
Let z1 and z2 be two complex numbers such that $ar g (z_{1} - z_{2}) = \frac{π}{4}$ and z1, z2 satisfy the equation | z $-$ 3 | = Re(z). Then the imaginary part of z1 + z2 is equal to .2021 · 27 Aug · Shift 2 · Q37 · Numerical
Let C be the set of all complex numbers. Let $S_{1} = {z \in C ∣∣ z - 3 - 2 i ∣^{2} = 8}$ $S_{2} = {z \in C ∣ Re o l imi t s (z) \geq 5}$ and $S_{3} = {z \in C ∣∣ z - \overline{z} ∣ \geq 8}$ . Then the number of elements in $S_{1} \cap S_{2} \cap S_{3}$ …2021 · 27 Jul · Shift 1 · Q26 · MCQ
Let C be the set of all complex numbers. Let S1 = {z $\in$ C : |z $-$ 2| $\leq$ 1} and S2 = {z $\in$ C : z(1 + i) + $\overline{z}$ (1 $-$ i) $\geq$ 4}. Then, the maximum value of $z - \frac{5}{2}^{2}$ for z $\in$ S1 $\cap$ S2…2021 · 27 Jul · Shift 2 · Q26 · MCQ
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 sin23 $θ$ + cos2 $θ$ is equal to .2021 · 27 Jul · Shift 2 · Q38 · Numerical
A point z moves in the complex plane such that $ar g (\frac{z - 2}{z + 2}) = \frac{π}{4}$ , then the minimum value of $z - 92 - 2 i^{2}$ is equal to .2021 · 31 Aug · Shift 1 · Q38 · Numerical
If z is a complex number such that $\frac{z - i}{z - 1}$ is purely imaginary, then the minimum value of | z $-$ (3 + 3i) | is :2021 · 31 Aug · Shift 2 · Q31 · MCQ
The value of $(\frac{1 + s i n \frac{2 π}{9} + i c o s \frac{2 π}{9}}{1 + s i n \frac{2 π}{9} - i c o s \frac{2 π}{9}})^{3}$ is :2020 · 2 Sep · Shift 1 · Q32 · MCQ
The imaginary part of $(3 + 2 - 54)^{\frac{1}{2}} - (3 - 2 - 54)^{\frac{1}{2}}$ can be :2020 · 2 Sep · Shift 2 · Q41 · MCQ
If $(\frac{1 + i}{1 - i})^{\frac{m}{2}} = (\frac{1 + i}{1 - i})^{\frac{n}{3}} = 1$ , (m, n $\in$ N) then the greatest common divisor of the least values of m and n is .2020 · 3 Sep · Shift 1 · Q37 · Numerical
If z1 , z2 are complex numbers such that Re(z1) = |z1 – 1|, Re(z2) = |z2 – 1| , and arg(z1 - z2) = $\frac{π}{6}$ , then Im(z1 + z2 ) is equal to :2020 · 3 Sep · Shift 2 · Q37 · MCQ
Let $u = \frac{2 z + i}{z - k i}$ , z = x + iy and k > 0. If the curve represented by Re(u) + Im(u) = 1 intersects the y-axis at the points P and Q where PQ = 5, then the value of k is :2020 · 4 Sep · Shift 1 · Q32 · MCQ
If a and b are real numbers such that $(2 + α)^{4} = a + b α$ where $α = \frac{- 1 + i 3}{2}$ then a + b is equal to :2020 · 4 Sep · Shift 2 · Q31 · MCQ
If the four complex numbers $z, \overline{z}, \overline{z} - 2 Re o l imi t s (\overline{z})$ and $z - 2 R e (z)$ represent the vertices of a square of side 4 units in the Argand plane, then $∣ z ∣$ is equal to :2020 · 5 Sep · Shift 1 · Q34 · MCQ
The value of $(\frac{- 1 + i 3}{1 - i})^{30}$ is :2020 · 5 Sep · Shift 2 · Q22 · MCQ
The region represented by {z = x + iy $\in$ C : |z| – Re(z) $\leq$ 1} is also given by the inequality : {z = x + iy $\in$ C : |z| – Re(z) $\leq$ 1}2020 · 6 Sep · Shift 1 · Q22 · MCQ
Let z = x + iy be a non-zero complex number such that $z^{2} = i ∣ z ∣^{2}$ , where i = $- 1$ , then z lies on the :2020 · 6 Sep · Shift 2 · Q21 · MCQ
If $Re o l imi t s (\frac{z - 1}{2 z + i}) = 1$ , where z = x + iy, then the point (x, y) lies on a :2020 · 7 Jan · Shift 1 · Q25 · MCQ
If $\frac{3 + i s i n θ}{4 - i c o s θ}$ , $θ \in$ [0, 2 $θ$ ], is a real number, then an argument of sin $θ$ + icos $θ$ is :2020 · 7 Jan · Shift 2 · Q32 · MCQ
If the equation, x2 + bx + 45 = 0 (b $\in$ R) has conjugate complex roots and they satisfy |z +1| = 2 $10$ , then :2020 · 8 Jan · Shift 1 · Q35 · MCQ
Let z be complex number such that $\frac{z - i}{z + 2 i} = 1$ and |z| = $\frac{5}{2}$ . Then the value of |z + 3i| is :2020 · 9 Jan · Shift 1 · Q26 · MCQ
If z be a complex number satisfying |Re(z)| + |Im(z)| = 4, then |z| cannot be :2020 · 9 Jan · Shift 2 · Q25 · MCQ
If $α$ and $β$ be the roots of the equation x2 – 2x + 2 = 0, then the least value of n for which $(\frac{α}{β})^{n} = 1$ is :2019 · 8 Apr · Shift 1 · Q37 · MCQ
If $z = \frac{3}{2} + \frac{i}{2} (i = - 1)$ , then (1 + iz + z5 + iz8)9 is equal to :2019 · 8 Apr · Shift 2 · Q27 · MCQ
All the points in the set $S = {\frac{α + i}{α - i} : α \in R} (i = - 1)$ lie on a :2019 · 9 Apr · Shift 1 · Q29 · MCQ
Let z $\in$ C be such that |z| < 1. If $ω = \frac{5 + 3 z}{5 ( 1 - z )}$ z, then :2019 · 9 Apr · Shift 2 · Q24 · MCQ
Let A = ${θ \in (- \frac{π}{2}, π) : \frac{3 + 2 i s i n θ}{1 - 2 i s i n θ} i s p u r e l y ima g ina r y}$ . Then the sum of the elements in A is :2019 · 9 Jan · Shift 1 · Q31 · MCQ
Let $α$ and $β$ be two roots of the equation x2 + 2x + 2 = 0 , then $α^{15}$ + $β^{15}$ is equal to :2019 · 9 Jan · Shift 1 · Q41 · MCQ
Let z0 be a root of the quadratic equation, x2 + x + 1 = 0, If z = 3 + 6iz $_{0}^{81} -$ 3iz $_{0}^{93}$ , then arg z is equal to :2019 · 9 Jan · Shift 2 · Q38 · MCQ
If a > 0 and z = $\frac{( 1 + i ) ^{2}}{a - i}$ , has magnitude $\frac{2}{5}$ , then $\overline{z}$ is equal to :2019 · 10 Apr · Shift 1 · Q27 · MCQ
If z and w are two complex numbers such that |zw| = 1 and arg(z) – arg(w) = $\frac{π}{2}$ , then :2019 · 10 Apr · Shift 2 · Q39 · MCQ
Let z1 and z2 be any two non-zero complex numbers such that $3 ∣ z_{1} ∣ = 4 ∣ z_{2} ∣ .$ If $z = \frac{3 z _{1}}{2 z _{2}} + \frac{2 z _{2}}{3 z _{1}}$ then :2019 · 10 Jan · Shift 1 · Q37 · MCQ
Let $z = (\frac{3}{2} + \frac{i}{2})^{5} + (\frac{3}{2} - \frac{i}{2})^{5} .$ If R(z) and 1(z) respectively denote the real and imaginary parts of z, then :2019 · 10 Jan · Shift 2 · Q28 · MCQ
Let $(- 2 - \frac{1}{3} i)^{3} = \frac{x + i y}{27} (i = - 1),$ where x and y are real numbers, then y $-$ x equals :2019 · 11 Jan · Shift 1 · Q38 · MCQ
Let z be a complex number such that |z| + z = 3 + i (where i = $- 1$ ). Then |z| is equal to :2019 · 11 Jan · Shift 2 · Q34 · MCQ
The equation |z – i| = |z – 1|, i = $- 1$ , represents :2019 · 12 Apr · Shift 1 · Q25 · MCQ
Let z $\in$ C with Im(z) = 10 and it satisfies $\frac{2 z - n}{2 z + n}$ = 2i - 1 for some natural number n. Then :2019 · 12 Apr · Shift 2 · Q32 · MCQ
If $\frac{z - α}{z + α} (α \in R)$ is a purely imaginary number and | z | = 2, then a value of $α$ is :2019 · 12 Jan · Shift 1 · Q26 · MCQ
Let z1 and z2 be two complex numbers satisfying | z1 | = 9 and | z2 – 3 – 4i | = 4. Then the minimum value of | z1 – z2 | is :2019 · 12 Jan · Shift 2 · Q37 · MCQ
The set of all $α \in$ R, for which w = $\frac{1 + ( 1 - 8 α ) z}{1 - z}$ is purely imaginary number, for all z $\in$ C satisfying |z| = 1 and Re z $e$ 1, is :2018 · 15 Apr · Shift 1 · Q30 · MCQ
If |z $-$ 3 + 2i| $\leq$ 4 then the difference between the greatest value and the least value of |z| is :2018 · 15 Apr · Shift 2 · Q25 · MCQ
The least positive integer n for which $(\frac{1 + i 3}{1 - i 3})^{n} = 1,$ is :2018 · 16 Apr · Shift 1 · Q38 · MCQ
If $α, β \in C$ are the distinct roots of the equation x2 - x + 1 = 0, then $α^{101} + β^{107}$ is equal to :2018 · Shift 0 · Q29 · MCQ
Let z $\in$ C, the set of complex numbers. Then the equation, 2|z + 3i| $-$ |z $-$ i| = 0 represents :2017 · 8 Apr · Shift 1 · Q33 · MCQ
The equation Im $(\frac{i z - 2}{z - i})$ + 1 = 0, z $\in$ C, z $e$ i represents a part of a circle having radius equal to :2017 · 9 Apr · Shift 1 · Q28 · MCQ
Let $ω$ be a complex number such that 2 $ω$ + 1 = z where z = $- 3$ . If $111 1 - ω^{2} - 1 ω^{2} 1 ω^{2} ω^{7} = 3 k$ …2017 · Shift 0 · Q30 · MCQ
The point represented by 2 + i in the Argand plane moves 1 unit eastwards, then 2 units northwards and finally from there $22$ units in the south-westwardsdirection. Then its new position in the Argand plane is at the point…2016 · 9 Apr · Shift 1 · Q28 · MCQ
A value of $θ$ for which $\frac{2 + 3 i s i n θ}{1 - 2 i s i n θ}$ is purely imaginary, is :2016 · Shift 0 · Q22 · MCQ
A complex number z is said to be unimodular if $∣ z ∣ = 1$ . Suppose $z_{1}$ and $z_{2}$ are complex numbers such that $\frac{z _{1} - 2 z _{2}}{2 - z _{1} z _{2}}$ is unimodular and $z_{2}$ is not unimodular.…2015 · Shift 0 · Q41 · MCQ
If z is a complex number such that $∣ z ∣ \geq 2$ , then the minimum value of $z + \frac{1}{2}$ :2014 · Shift 0 · Q42 · MCQ
If z is a complex number of unit modulus and argument $θ$ , then arg $(\frac{1 + z}{1 + z})$ equals :2013 · Shift 0 · Q44 · MCQ
If $z e 1$ and $\frac{z ^{2}}{z - 1}$ is real, then the point represented by the complex number z lies :2012 · Shift 0 · Q43 · MCQ
Let $α, β$ be real and z be a complex number. If $z^{2} + α z + β = 0$ has two distinct roots on the line Re z = 1, then it is necessary that :2011 · Shift 0 · Q33 · MCQ
If $ω (e 1)$ is a cube root of unity, and $(1 + ω)^{7} = A + B ω$ . Then $(A, B)$ equals :2011 · Shift 0 · Q48 · MCQ
The number of complex numbers z such that $∣ z - 1 ∣ = ∣ z + 1 ∣ = ∣ z - i ∣$ equals :2010 · Shift 0 · Q29 · MCQ
If $z - \frac{4}{z} = 2,$ then the maximum value of $∣ z ∣$ is equal to :2009 · Shift 0 · Q30 · MCQ
The conjugate of a complex number is $\frac{1}{i - 1}$ then that complex number is :2008 · Shift 0 · Q54 · MCQ
If $∣ z + 4 ∣ \leq 3$ , then the maximum value of $∣ z + 1 ∣$ is :2007 · Shift 0 · Q61 · MCQ
If $z^{2} + z + 1 = 0$ , where z is complex number, then value of $(z + \frac{1}{z})^{2} + (z^{2} + \frac{1}{z ^{2}})^{2} + (z^{3} + \frac{1}{z ^{3}})^{2} + .......... + (z^{6} + \frac{1}{z ^{6}})^{2}$ …2006 · Shift 0 · Q67 · MCQ
The value of $k = 1 \sum 10 (sin \frac{2 k π}{11} + i cos \frac{2 k π}{11})$ is :2006 · Shift 0 · Q68 · MCQ
If the cube roots of unity are 1, $ω, ω^{2}$ then the roots of the equation $(x - 1)^{3}$ + 8 = 0, are :2005 · Shift 0 · Q70 · MCQ
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 :2005 · Shift 0 · Q94 · MCQ
If $ω = \frac{z}{z - \frac{1}{3} i}$ and $∣ ω ∣ = 1$ , then $z$ lies on :2005 · Shift 0 · Q95 · MCQ
Let z and w be complex numbers such that $\overline{z} + i \overline{w} = 0$ and arg zw = $π$ . Then arg z equals :2004 · Shift 0 · Q97 · MCQ
If $z = x - i y$ and $z^{\frac{1}{3}} = p + i q$ , then $\frac{( \frac{x}{p} + \frac{y}{q} )}{( p ^{2} + q ^{2} )}$ is equal to :2004 · Shift 0 · Q98 · MCQ
If $z^{2} - 1 = ∣ z ∣^{2} + 1$ , then z lies on :2004 · Shift 0 · Q99 · MCQ
If $z$ and $ω$ are two non-zero complex numbers such that $∣ z ω ∣ = 1$ and $A r g (z) - A r g (ω) = \frac{π}{2},$ then $\overline{z} ω$ is equal to2003 · Shift 0 · Q118 · MCQ
Let $Z_{1}$ and $Z_{2}$ be two roots of the equation $Z^{2} + a Z + b = 0$ , Z being complex. Further , assume that the origin, $Z_{1}$ and $Z_{2}$ form an equilateral triangle. Then :2003 · Shift 0 · Q98 · MCQ
If $(\frac{1 + i}{1 - i})^{x} = 1$ then :2003 · Shift 0 · Q99 · MCQ
z and w are two nonzero complex numbers such that $∣ z ∣ = ∣ w ∣$ and Arg z + Arg w = $π$ then z equals2002 · Shift 0 · Q97 · MCQ
If $∣ z - 4 ∣ < ∣ z - 2 ∣$ , its solution is given by :2002 · Shift 0 · Q98 · MCQ
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 :2002 · Shift 0 · Q99 · MCQ