Complex Numbers

Let $\mathbb{R}$ denote the set of all real numbers and let $i=\sqrt{-1}$. Consider the matrices
$$ S=\left[\begin{array}{rr} 0 & -1 \\ 1 & 0 \end{array}\right] \quad \text { and } \quad T=\left[\begin{array}{ll} 1 & 1 \\ 0 & 1 \end{array}\...

Match each entry in List-I to the correct entry in List-II and choose the correct option.



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Let$$ \alpha = \left( 1 - 2\cos\left(\frac{\pi}{11}\right) \right) \left( 1 - 2\cos\left(\frac{3\pi}{11}\right) \right) \left( 1 - 2\cos\left(\frac{9\pi}{11}\right) \right) \left( 1 - 2\cos\left(\frac{27\pi}{11}\right) \right) \left( 1 - 2\...

For a non-zero complex number $z$, let $\arg (z)$ denote the principal argument of $z$, with $-\pi<\arg (z) \leq \pi$. Let $\omega$ be the cube root of unity for which $0<\arg (\omega)<\pi$. Let
$$ \alpha=\arg \left(\sum\limits_{n=1}^{2025}...

Let ℝ denote the set of all real numbers. Let $z_1 = 1 + 2i$ and $z_2 = 3i$ be two complex numbers, where $i = \sqrt{-1}$. Let\(S = \{(x, y) \in \mathbb{R} \times \mathbb{R} : |x + iy - z_1| = 2|x + iy - z_2| \}.\)Then which of the followin...

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 $i \sqrt{3}$ is a root of the equation $4 x^3+3 a x^2+2 b x=0$, where $i=\sqrt{-1}$. If $\alpha_1, \alpha_2, \alpha_3$, and $\alpha_4$ are...

Let $S=\{a+b \sqrt{2}: a, b \in \mathbb{Z}\}, T_1=\left\{(-1+\sqrt{2})^n: n \in \mathbb{N}\right\}$, and $T_2=\left\{(1+\sqrt{2})^n: n \in \mathbb{N}\right\}$. Then which of the following statements is (are) TRUE?

Let $z$ be a complex number satisfying $|z|^3+2 z^2+4 \bar{z}-8=0$, where $\bar{z}$ 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.

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Let $A=\left\{\frac{1967+1686 i \sin \theta}{7-3 i \cos \theta}: \theta \in \mathbb{R}\right\}$. If $A$ contains exactly one positive integer $n$, then the value of $n$ is

Let $\bar{z}$ 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

\((\bar{z})^{2}+\frac{1}{z^{2}}\)

are integers, then which of the following is/are possib...

Let \(\bar{z}\) denote the complex conjugate of a complex number \(z\) and let \(i=\sqrt{-1}\). In the set of complex numbers, the number of distinct roots of the equation

\(\bar{z}-z^{2}=i\left(\bar{z}+z^{2}\right)\)

is _________.

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 _________.

For any complex number w = c + id, let \(\arg (w) \in ( - \pi ,\pi ]\), where \(i = \sqrt { - 1}\). Let \(\alpha\) and \(\beta\) be real numbers such that for all complex numbers z = x + iy satisfying $$\arg \left( {{{z + \alpha } \over {z...

Let $\theta_1, \theta_2, \ldots, \theta_{10}$ be positive valued angles (in radian) such that $\theta_1+\theta_2+\cdots+\theta_{10}=2 \pi$. Define the complex numbers $z_1=e^{i \theta_1}, z_k=z_{k-1} e^{i \theta_k}$ for $k=2,3, \ldots, 10$,...

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 \({z^4} - |z{|^4} = 4i{z^2}\), where i = \(\sqrt { - 1}\). Then the minimum possible value of |z1 \(-\) z2|2, where z1, z2$$...

Let S be the set of all complex numbers z satisfying |z2 + z + 1| = 1. Then which of the following statements is/are TRUE?

Let \(\omega \ne 1\) be a cube root of unity. Then the minimum of the set \(\{ {\left| {a + b\omega + c{\omega ^2}} \right|^2}:a,b,c\) distinct non-zero integers} equals ..................

Let S be the set of all complex numbers z satisfying \(\left| {z - 2 + i} \right| \ge \sqrt 5\). If the complex number z0 is such that \({1 \over {\left| {{z_0} - 1} \right|}}\) is the maximum of the set $$\left\{ {{1 \over {\left| {{z_0} ...

Let s, t, r be non-zero complex numbers and L be the set of solutions \(z = x + iy(x,y \in R,\,i = \sqrt { - 1} )\) of the equation \(sz + t\overline z + r = 0\) where \(\overline z\) = x \(-\) iy. Then, which of the following statement(s...

For a non-zero complex number z, let arg(z) denote the principal argument with \(-\) \(\pi\) < arg(z) \(\le\) \(\pi\). Then, which of the following statement(s) is (are) FALSE?

Let a, b, x and y be real numbers such that a \(-\) b = 1 and y \(\ne\) 0. If the complex number z = x + iy satisfies \({\mathop{\rm Im}\nolimits} \left( {{{az + b} \over {z + 1}}} \right) = y\), then which of the following is(are) possi...

Let \(a,\,b \in R\,and\,{a^{2\,}} + {b^2} \ne 0\). Suppose \(S = \left\{ {Z \in C:Z = {1 \over {a + ibt}}, + \in R,t \ne 0} \right\}\), where \(i = \sqrt { - 1}\). Ifz = x + iy and z \(\in\) S, then (x, y) lies on

For any integer k, let \({a_k} = \cos \left( {{{k\pi } \over 7}} \right) + i\,\,\sin \left( {{{k\pi } \over 7}} \right)\), where \(i = \sqrt { - 1} \,\). The value of the expression $${{\sum\limits_{k = 1}^{12} {\left| {{\alpha _{k + 1}} -...

Let \({z_k}\) = \(\cos \left( {{{2k\pi } \over {10}}} \right) + i\,\,\sin \left( {{{2k\pi } \over {10}}} \right);\,k = 1,2....,9\)
List-I
P. For each \({z_k}\) = there exits as \({z_j}\) such that \({z_k}\).\({z_j}\) = 1
Q. T...

Let \(S = {S_1} \cap {S_2} \cap {S_3}\), where \({S_1} = \left\{ {z \in C:\left| z \right| < 4} \right\},{S_2} = \left\{ {z \in C:{\mathop{\rm Im}\nolimits} \left[ {{{z - 1 + \sqrt 3 i} \over {1 - \sqrt 3 i}}} \right] > 0} \right\}\) and $$...

Let \(S = {S_1} \cap {S_2} \cap {S_3}\), where \({S_1} = \left\{ {z \in C:\left| z \right| < 4} \right\},{S_2} = \left\{ {z \in C:{\mathop{\rm Im}\nolimits} \left[ {{{z - 1 + \sqrt 3 i} \over {1 - \sqrt 3 i}}} \right] > 0} \right\}\) and $$...

Let $\omega=\frac{\sqrt{3}+i}{2}$ and $P=\left\{\omega^n: n=1,2,3, \ldots\right\}$. Further
$\mathrm{H}_1=\left\{z \in \mathrm{C}: \operatorname{Re} z<\frac{1}{2}\right\}$ and
$\mathrm{H}_2=\left\{z \in \mathrm{C}: \operatorname{Re} z<\fr...

Let complex numbers \(\alpha \,and\,{1 \over {\overline \alpha }}\,\) lie on circles \({\left( {x - {x_0}} \right)^2} + \,\,{\left( {y - {y_0}} \right)^2} = {r^2}\) and $$\,{\left( {x - {x_0}} \right)^2} + \,\,{\left( {y - {y_0}} \right)^2...

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 value

Let \(\omega = {e^{{{i\pi } \over 3}}}\), and a, b, c, x, y, z be non-zero complex numbers such that
\(a + b + c = x\)
\(a + b\omega + c{\omega ^2} = y\)
\(a + b{\omega ^2} + c\omega = z\)
Then the value of $${{{{\left| x \righ...

If z is any complex number satisfying \(\,\left| {z - 3 - 2i} \right| \le 2\), then the minimum value of \(\left| {2z - 6 + 5i} \right|\) is

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 sat...

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 $0 < t < 1$.
If $\operatorname{Arg}(w)$ denotes the principal argument of a nonzero complex number $w$, then :

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, then

Let \(z = \,\cos \,\theta \, + i\,\sin \,\theta\) . Then the value of \(\sum\limits_{m = 1}^{15} {{\mathop{\rm Im}\nolimits} } ({z^{2m - 1}})\,at\,\theta \, = {2^ \circ }\) is

Let \(z = x + iy\) be a complex number where x and y are integers. Then the area of the rectangle whose vertices are the roots of the equation \(\overline z {z^3} + z{\overline z ^3} = 350\) is

A particle P stats from the point \({z_0}\) = 1 +2i, where \(i = \sqrt { - 1}\). 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...

Let z be any point \(A \cap B \cap C\) and let w be any point satisfying \(\left| {w - 2 - i} \right| < 3\,\). Then, \(\left| z \right| - \left| w \right| + 3\) lies between :

Let z be any point in \(A \cap B \cap C\)
Then, \({\left| {z + 1 - i} \right|^2} + {\left| {z - 5 - i} \right|^2}\) lies between :

The number of elements in the set \(A \cap B \cap C\) is

If \(|z|=1\) and \(z \neq \pm 1\), then all the values of \(\frac{z}{1-z^{2}}\) lie on

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 i...

If \(\left| z \right|\, =1\,and\,z\, \ne \, \pm \,1,\) then all the values of \({z \over {1 - {z^2}}}\) lie on

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 \(\left( {N\,{{45}^ \circ }\,W} \right)\) direction to reach a poi...

If $P$ is a point on $C_1$ and $Q$ in another point on $\mathrm{C}_2$, then $\frac{\mathrm{PA}^2+\mathrm{PB}^2+\mathrm{PC}^2+\mathrm{PD}^2}{\mathrm{QA}^2+\mathrm{QB}^2+\mathrm{QC}^2+\mathrm{QD}^2}$ is equal to :

If \(w=\alpha+\mathrm{i} \beta\), where \(\beta \neq 0\) and \(z \neq 1\), satisfies the condition that \(\left(\frac{w-\bar{w} z}{1-z}\right)\) is purely real, then the set of values of \(z\) is:

\(a,\,b,\,c\) are integers, not all simultaneously equal and \(\omega\) is cube root of unity \(\left( {\omega \ne 1} \right),\) then minimum value of \(\left| {a + b\omega + c{\omega ^2}} \right|\) is

If one of the vertices of the square circumscribing the circle \(|z-1|=\sqrt{2}\) is \((2+\sqrt{3 i})\). Find the other vertices of square.

If one the vertices of the square circumscribing the circle \(\left| {z - 1} \right| = \sqrt 2 \,is\,2 + \sqrt {3\,} \,i\). Find the other vertices of the square.

If \(\omega\) \(\left( { \ne 1} \right)\) be a cube root of unity and \({\left( {1 + {\omega ^2}} \right)^n} = {\left( {1 + {\omega ^4}} \right)^n},\) then the least positive value of n is