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Complex Numbers

38 questions · Mathematics · JEE Advanced

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PYQ Library
JEE Advanced
Mathematics
Complex Numbers

Complex Numbers

38 questions · Mathematics · JEE Advanced

Let ℝ denote the set of all real numbers. Let $z_{1} = 1 + 2 i$ and $z_{2} = 3 i$ be two complex numbers, where $i = - 1$ . Let $S = {(x, y) \in R \times R : ∣ x + i y - z_{1} ∣ = 2∣ x + i y - z_{2} ∣} .$ Then which of the…2025 · Shift 1 · Q23 · Multiple correct
For a non-zero complex number $z$ , let $ar g (z)$ denote the principal argument of $z$ , with $- π$ $α = ar g (n = 1 \sum 2025 (- ω)^{n})$ $T h e n t h e v a l u eo f$ \frac{3 \alpha}{\pi} $i s$ \underline{\hspace{2cm}}$.2025 · Shift 2 · Q29 · Numerical
Let $S = {a + b 2 : a, b \in Z}, T_{1} = {(- 1 + 2)^{n} : n \in N}$ , and $T_{2} = {(1 + 2)^{n} : n \in N}$ . Then which of the following statements is (are) TRUE?2024 · Shift 1 · Q22 · Multiple correct
Let $f (x) = x^{4} + a x^{3} + b x^{2} + c$ be a polynomial with real coefficients such that $f (1) = - 9$ . Suppose that $i 3$ is a root of the equation $4 x^{3} + 3 a x^{2} + 2 b x = 0$ , where $i = - 1$ . If $α_{1}, α_{2}, α_{3}$ , and $α_{4}$ …2024 · Shift 1 · Q26 · Numerical
Let $A = {\frac{1967 + 1686 i s i n θ}{7 - 3 i c o s θ} : θ \in R}$ . If $A$ contains exactly one positive integer $n$ , then the value of $n$ is2023 · Shift 1 · Q28 · Numerical
Let $z$ be a complex number satisfying $∣ z ∣^{3} + 2 z^{2} + 4 \overset{z}{ˉ} - 8 = 0$ , where $\overset{z}{ˉ}$ denotes the complex conjugate of $z$ . Let the imaginary part of $z$ be nonzero. Match each entry in List-I to the correct entries in List-II. The correct… Includes table2023 · Shift 1 · Q34 · MCQ
Let $z$ be a complex number with a non-zero imaginary part. If $\frac{2 + 3 z + 4 z ^{2}}{2 - 3 z + 4 z ^{2}}$ is a real number, then the value of $∣ z ∣^{2}$ is .2022 · Shift 1 · Q22 · Numerical
Let $\overset{z}{ˉ}$ denote the complex conjugate of a complex number $z$ and let $i = - 1$ . In the set of complex numbers, the number of distinct roots of the equation $\overset{z}{ˉ} - z^{2} = i (\overset{z}{ˉ} + z^{2})$ is …2022 · Shift 1 · Q23 · Numerical
Let $\overset{z}{ˉ}$ denote the complex conjugate of a complex number $z$ . If $z$ is a non-zero complex number for which both real and imaginary parts of $(\overset{z}{ˉ})^{2} + \frac{1}{z ^{2}}$ are integers, then which of the following is/are possible…2022 · Shift 2 · Q29 · Multiple correct
Let $θ_{1}, θ_{2}, \dots, θ_{10}$ be positive valued angles (in radian) such that $θ_{1} + θ_{2} + \dots + θ_{10} = 2 π$ . Define the complex numbers $z_{1} = e^{i θ_{1}}, z_{k} = z_{k - 1} e^{i θ_{k}}$ for $k=2,3, \ldots,…2021 · Shift 1 · Q23 · MCQ
For any complex number w = c + id, let $ar g (w) \in (- π, π]$ , where $i = - 1$ . Let $α$ and $β$ be real numbers such that for all complex numbers z = x + iy satisfying $ar g (\frac{z + α}{z + β}) = \frac{π}{4}$ …2021 · Shift 1 · Q35 · Multiple correct
Let S be the set of all complex numbers z satisfying |z2 + z + 1| = 1. Then which of the following statements is/are TRUE?2020 · Shift 1 · Q27 · Multiple correct
For a complex number z, let Re(z) denote that real part of z. Let S be the set of all complex numbers z satisfying $z^{4} - ∣ z ∣^{4} = 4 i z^{2}$ , where i = $- 1$ . Then the minimum possible value of |z1 $-$ z2|2, where z1, z2 $\in$ S…2020 · Shift 2 · Q19 · Numerical
Let S be the set of all complex numbers z satisfying $∣ z - 2 + i ∣ \geq 5$ . If the complex number z0 is such that $\frac{1}{∣ z _{0} - 1 ∣}$ is the maximum of the set ${\frac{1}{∣ z _{0} - 1 ∣} : z \in S}$ …2019 · Shift 1 · Q19 · MCQ
Let $ω e 1$ be a cube root of unity. Then the minimum of the set ${a + bω + c ω^{2}^{2} : a, b, c$ distinct non-zero integers} equals ..................2019 · Shift 1 · Q33 · Numerical
For a non-zero complex number z, let arg(z) denote the principal argument with $- π$ < arg(z) $\leq π$ . Then, which of the following statement(s) is (are) FALSE?2018 · Shift 1 · Q19 · Multiple correct
Let s, t, r be non-zero complex numbers and L be the set of solutions $z = x + i y (x, y \in R, i = - 1)$ of the equation $sz + t \overline{z} + r = 0$ where $\overline{z}$ = x $-$ iy. Then, which of the following statement(s) is(are)…2018 · Shift 2 · Q23 · Multiple correct
Let a, b, x and y be real numbers such that a $-$ b = 1 and y $e$ 0. If the complex number z = x + iy satisfies $Im o l imi t s (\frac{a z + b}{z + 1}) = y$ , then which of the following is(are) possible…2017 · Shift 1 · Q21 · Multiple correct
Let $a, b \in R an d a^{2} + b^{2} e 0$ . Suppose $S = {Z \in C : Z = \frac{1}{a + ib t}, + \in R, t e 0}$ , where $i = - 1$ . Ifz = x + iy and z $\in$ S, then (x, y) lies on2016 · Shift 2 · Q19 · Multiple correct
For any integer k, let $a_{k} = cos (\frac{k π}{7}) + i sin (\frac{k π}{7})$ , where $i = - 1$ . The value of the expression $\frac{k = 1 \sum 12 ∣ α _{k + 1} - a _{k} ∣}{k = 1 \sum 3 ∣ α _{4 k - 1} - α _{4 k - 2} ∣}$ …2015 · Shift 2 · Q22 · Numerical
Let $z_{k}$ = $cos (\frac{2 k π}{10}) + i sin (\frac{2 k π}{10}); k = 1, 2...., 9$ List-I P. For each $z_{k}$ = there exits as $z_{j}$ such that $z_{k}$ . $z_{j}$ = 1 Q. There exists a $k \in {1, 2, ...., 9}$ …2014 · Shift 2 · Q27 · MCQ
Let complex numbers $α an d \frac{1}{α}$ lie on circles $(x - x_{0})^{2} + (y - y_{0})^{2} = r^{2}$ and $(x - x_{0})^{2} + (y - y_{0})^{2} = 4 r^{2}$ …2013 · Shift 1 · Q23 · MCQ
Let $ω = \frac{3 + i}{2}$ and $P = {ω^{n} : n = 1, 2, 3, \dots}$ . Further $H_{1} = {z \in C : Re z < \frac{1}{2}}$ and $H_{2} = {z \in C : Re z < \frac{- 1}{2}}$ …2013 · Shift 2 · Q22 · Multiple correct
Let $S = S_{1} \cap S_{2} \cap S_{3}$ , where $S_{1} = {z \in C : ∣ z ∣ < 4}, S_{2} = {z \in C : Im o l imi t s [\frac{z - 1 + 3 i}{1 - 3 i}] > 0}$ and $S_{3} = {z \in C : Re o l imi t s z > 0}$ …2013 · Shift 2 · Q36 · MCQ
Let $S = S_{1} \cap S_{2} \cap S_{3}$ , where $S_{1} = {z \in C : ∣ z ∣ < 4}, S_{2} = {z \in C : Im [\frac{z - 1 + 3 i}{1 - 3 i}] > 0}$ and $S_{3} = {z \in C : Re (z) > 0}$ …2013 · Shift 2 · Q37 · MCQ
Let z be a complex number such that the imaginary part of z is non-zero and $a = z^{2} + z + 1$ is real. Then a cannot take the value2012 · Shift 1 · Q22 · MCQ
If z is any complex number satisfying $∣ z - 3 - 2 i ∣ \leq 2$ , then the minimum value of $∣ 2 z - 6 + 5 i ∣$ is2011 · Shift 1 · Q24 · Numerical
Let $ω = e^{\frac{iπ}{3}}$ , and a, b, c, x, y, z be non-zero complex numbers such that $a + b + c = x a + bω + c ω^{2} = y a + b ω^{2} + c ω = z$ Then the value of $\frac{∣ x ∣ ^{2} + ∣ y ∣ ^{2} + ∣ z ∣ ^{2}}{∣ a ∣ ^{2} + ∣ b ∣ ^{2} + ∣ c ∣ ^{2}}$ …2011 · Shift 2 · Q22 · Numerical
Let $z_{1}$ and $z_{2}$ be two distinct complex number and let z =( 1 - t) $z_{1}$ + t $z_{2}$ for some real number t with 0 < t < 1. IfArg (w) denote the principal argument of a non-zero complex number w, then2010 · Shift 1 · Q32 · Multiple correct
Let $z_{1}$ and $z_{2}$ be two distinct complex numbers let $z = (1 - t) z_{1} + t z_{2}$ for some real number t with $0 I f$ \operatorname{Arg}(w) $d e n o t es t h e p r in c i p a l a r g u m e n t o f an o n z er oco m pl e x n u mb er$ w$, then :2010 · Shift 1 · Q53 · Multiple correct
Match the statements in Column I with those in Column II. [Note : Here z takes value in the complex plane and Im z and Re z denotes, respectively, the imaginary part and the real part of z.] Column I (A) The set of points z satisfying $∣ z - i ∣ z ∥ = ∣ z + i ∣ z ∥$ …2010 · Shift 2 · Q21 · MCQ
Let $z = x + i y$ be a complex number where x and y are integers. Then the area of the rectangle whose vertices are the roots of the equation $\overline{z} z^{3} + z \overline{z}^{3} = 350$ is2009 · Shift 1 · Q21 · MCQ
Let $z = cos θ + i sin θ$ . Then the value of $m = 1 \sum 15 Im o l imi t s (z^{2 m - 1}) a t θ = 2^{\circ}$ is2009 · Shift 1 · Q22 · MCQ
Let A, B, C be three sets of complex numbers as defined below : $A = {z : Im o l imi t s z \geq 1} B = {z : ∣ z - 2 - i ∣ = 3} C = {z : Re o l imi t s (1 - i) z) = 2}$ …2008 · Shift 1 · Q24 · MCQ
Let A, B, C be three sets of complex numbers as defined below : $A = {z : Im o l imi t s z \geq 1} B = {z : ∣ z - 2 - i ∣ = 3} C = {z : Re o l imi t s (1 - i) z) = 2}$ …2008 · Shift 1 · Q25 · MCQ
Let A, B, C be three sets of complex numbers as defined below $A = {z : Im o l imi t s z \geq 1} B = {z : ∣ z - 2 - i ∣ = 3} C = {z : Re o l imi t s (1 - i) z) = 2}$ …2008 · Shift 1 · Q35 · MCQ
A particle P stats from the point $z_{0}$ = 1 +2i, where $i = - 1$ . It moves horizontally away from origin by 5 unit and then vertically away from origin by 3 units to reach a point $z_{1}$ . From $z_{1}$ the particle moves $\sqrt…2008 · Shift 2 · Q24 · MCQ
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 (N 45 $^{\circ}$ W) direction to reach a point P. Then the position of P…2007 · Shift 1 · Q30 · MCQ