Complex Numbers Questions (654)

Difference between the square of the least and the square of the greatest values of $|z|$, where $z = e^{i2\theta} \sin\theta + \cos\theta - \cos(\phi + R)$ is ____.

If the roots $Z_1, Z_2, Z_3$ of the equation $Z^3 - Z^2 + mZ - 1 = 0$ lie on $|Z| = 1$ and $|(Z_1 + 3)(Z_2 + 3)(Z_3 + 3)| = 10$, then $\lambda = $ ____.

Let $\alpha = \cos\frac{2\pi}{5} + i\sin\frac{2\pi}{5}$ and let $A_k = x + y\alpha^k + p\alpha^{2k} + w\alpha^{3k} + f\alpha^{4k}$ where $x, y, p, w, f$ are points on the circle $|z| = 1$, then $\frac{|A_0|^2 + |A_1|^2 + |A_2|^2 + |A_3|^2 + |A_4|^2}{5}$ is equal to ___________.

Number of functions from set $S \to C$, where $C$ has 8 distinct elements is:

The area of the region bounded by curves. (i) $|z - z_1| = |z - z_3|$ (ii) $|\text{Re}(z) - \text{Re}(z_3)| = |\text{Re}(z) - \text{Re}(z_1)|$ (iii) $|z - z_2| = |z - z_1| = |z_1 - z_2|$ (where $z_1 = 1 + i, z_2 = 2 + i, z_3 = -3 + 3i$) is $\frac{p}{q}$, (p, q are co-prime) then find $p + q$.

Given $|3z_1 - 2z_2 - 4|^2 = |3z_1 - 1|^2 + |2z_2 + 3|^2$, $z_2 = -\frac{3}{2}$. If cube roots of $w = \frac{3z_1 - 1}{2z_2 + 3}$ are $w_1, w_2, w_3$ where $(\arg w_1 < \arg w_2 < \arg w_3)$, then the value of $\frac{w_2}{w_1 w_3}$ is ____.

Let $z = (\cos 12° + i \sin 12° + \cos 48° + i \sin 48°)^9$, then $\text{Im}(z)$ is equal to ___________.

Let $A, B, C$ be the set of complex numbers defined as $A = \{z: |z + 2| - |z - 2|| = 2\}$, $B = \left\{z: \arg\left(\frac{z - 1}{z}\right) = \frac{\pi}{2}\right\}$ and $C = \{z: \arg(z - 1) = \pi\}$, then $n(A \cap B \cap C) = $ ___________.

If z and w are two non-zero complex numbers such that |zw| = 1, and arg(z) - arg(w) = π/2, then zw is equal to:

If x = a + b, y = a\alpha + b\beta and z = a\beta + b\alpha, where \alpha and \beta are complex cube roots of unity, then xyz is equal to

52. Let \(x_1, x_2, \ldots, x_{10}\) be the roots of the polynomial equation \(x^{10} + x^9 + \cdots + x + 1 = 0\). Then the value of \(\displaystyle\sum_{n=1}^{10} \left(\frac{1}{1 - x_n}\right)\):

If $|z| = 1$ and $z^{2n} + 1 + z^{-n} = 0$ then $\frac{z^{2n}}{2^{2n} + 1} - \frac{\left(\bar{z}\right)^n}{\left(\bar{z}\right)^{2n} + 1}$ is equal to ____.

Let z1 and z2 be two roots of the equation z2 + az + b = 0, z being complex number. Further, assume that the origin, z1 and z2 form an equilateral triangle, then:

Locus of $z$, if $\arg(z - (1+i)) = \begin{cases} \frac{3\pi}{4}, & \text{when } |z| \leq |z-2| \\ -\frac{\pi}{4}, & \text{when } |z| > |z-2| \end{cases}$ is:

The roots of the equation \((x-1)^3 + 8 = 0\) are:

The value of $i\log\left(x-i\right)+i^2\pi+i^3\log\left(x+i\right)+i^4\left(2\tan^{-1}x\right)$, (where, $x>0$ and $i=\sqrt{-1}$), is :

The equation $\left(1+a\right)x^2+2a^2x+a^2+b^2-1=0$ has roots of opposite sign, if $a+b$ lies, $(a>-1)$ :

If z is a complex number in the argand plane, the equation |z - 2| + |z + 2| = 8 represents

Let $AB$ and $CD$ be parallel chords of the circle $|z| = r$. If $z_1, z_2, z_3$ and $z_4$ represent $A, B, C$ and $D$, respectively, and $z_1 z_2 = kz_3 z_4$ then $\frac{z_3 z_4}{4}$ equals ____.

The points \((a_1, b_1), (a_2, b_2), \ldots, (a_n, b_n), (a_{n+1}, b_{n+1})\) are given with \((a_{n+1} + i b_{n+1}) = z_{n+1}\), where \(z_{n+1} = (\sqrt{3}\, a_n - b_n) + i(\sqrt{3}\, b_n + a_n)\). If \(z_n = a_n + i b_n\), then \(z_{n+1} = z_n(\sqrt{3} + i) = 2z_n\left(\cos\dfrac{\pi}{6} + i\sin\dfrac{\pi}{6}\right)\). Which option correctly represents the recursive relation?

Sum of all the solutions of $z = |z| + z^2$ is ____.

Let \(\theta = \dfrac{2\pi}{n}\) and \(P_1, P_2, \ldots, P_n\) be vertices of a regular polygon inscribed in a unit circle. Find the value of \(n\) such that \((P_1P_2)(P_1P_3)\cdots(P_1P_{17}) = 17\), i.e., the product of all chord lengths from \(P_1\) equals 17. What is \(n\)?

If $8z^3 + 12z^2 - 18z + 27i = 0$, then $2|z|$ is equal to ___________.

If $\arg(z) < 0$, then $\frac{10}{\pi} \arg\left(\frac{z - \bar{z}}{2}\right)$ is equal to ___________.

Sum of all the solutions of $z^2 + |z| = |z(z)|^2$ is ____.

Let $z_1$ and $z_2$ be two complex numbers such that $z_1+z_2=5$ and $z_1^3+z_2^3=20+15i$. Then $|z_1^4+z_2^4|$ equals

If $z$ is a complex number and the minimum value of $|z| + |z - 1| + |2z - 3|$ is $\lambda$ and if $y = 2[x] + 3 = 3[x - \lambda]$ then find the value of $\frac{1}{5}([x + y])$. (where $[.]$ denotes the greatest integer function).

Let \(z\) and \(\omega\) be two complex numbers such that \(|z| \leq 1\), \(|\omega| \leq 1\) and \(|z + i\omega| = |z - i\bar{\omega}| = 2\). Then \(z\) equals

If $A(z_1), B(z_2), C(z_3)$ are the vertices of triangle such that $z_1 = \frac{z_2 - iz_1}{1 - i}$, $|z_1| = 3$, $|z_2| = 4$ and $|z_2 + iz_1| = |z_1| + |z_2|$, then area of $\triangle ABC$ is ____.

The complex number $z=\dfrac{i-1}{\cos\dfrac{\pi}{3}+i\sin\dfrac{\pi}{3}}$ is equal to:

If \alpha \neq 1 is any nth root of unity, then S = 1^2 + 3\alpha + 5\alpha^2 + \cdots up to n terms, is equal to

If \(z + z^{-1} = 1\), then find the value of \(z^{100} + z^{-100}\).

Let $z = x + iy$ and $\arg\left(e^z\right) = \arg\left(e^{i(z+y)}\right)$. If $y = f(x)$ is a function, then $f(3)$ is equal to ____.

The value of $4\alpha\left(\beta^4 - \alpha^4\right)$, if $\alpha + i\beta, \beta \neq 0$ is a root of $z^5 = -1$, is ____.

If $|1 + 2|z^2| = |z^2 + 1| + |2|z| + 1|^2$, then the value of $\frac{|z(z+1)|}{2}$ is ____.

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

Given that the complex number \(\bar{z} = \dfrac{1}{i-1}\), then \(z\) equals:

The value of \(\displaystyle\sum_{k=1}^{10}\left(\sin\dfrac{2k\pi}{11} + i\cos\dfrac{2k\pi}{11}\right)\) is

One vertex of the triangle of maximum area that can be inscribed in the curve \(|z - (1+i)| = \sqrt{10}\) is \(-2 + 2i\), then the remaining vertices are

\(z_1\) and \(z_2\) lie on a circle with center at the origin. The point of intersection \(z_3\) of the tangents at \(z_1\) and \(z_2\) is given by

Equation of tangent drawn to circle \(|z| = r\) at the point \(A(z_0)\) is

Let \(s_1 = z_1 + z_2 + z_3\), \(s_2 = z_1z_2 + z_2z_3 + z_3z_1\), and \(s_3 = z_1z_2z_3\). Consider the cubic equation \(z^3 - s_1z^2 + s_2z - s_3 = 0\) having three roots \(z_1, z_2, z_3\). Given that \(z_1^2 + z_2^2 + z_3^2 = 0\), find the value of \(s_1^2 - 2s_2\).

Find the modulus, argument, and the principal argument of the complex number \[\dfrac{i-1}{i\left(1 - \cos\dfrac{2\pi}{5}\right) + \sin\dfrac{2\pi}{5}}\]

For Problems 20–22: Consider the equation of line \(a\bar{z} + \bar{a}z + b = 0\), where \(b\) is a real parameter and \(a\) is fixed non-zero complex number.The intercept of line on real axis is given by

The set of all \(\alpha \in R\), for which \(w = \dfrac{1+(1-8\alpha)z}{1-z}\) is a purely imaginary number, for all \(z \in C\) satisfying \(|z|=1\) and \(\text{Re } z \neq 1\), is

If z is any complex number satisfying \(|z - 3 - 2i| \leq 2\), then the minimum value of \(|2z - 6 + 5i|\) is _______.(IIT-JEE, 2011)

If \(\omega \neq 1\) is a cube root of unity, and \((1 + \omega)^7 = A + B\omega\). Then \((A, B)\) equals

If \(z^2 + az + b = 0\) has roots \(z_1\) and \(z_2\), and \(O\) (origin), \(z_1\), \(z_2\) form an equilateral triangle, then which of the following is true?

Find the principal argument of (d) \((1+i\sqrt{3})^2\)

Since \(\omega^3 = 1\) and \(\omega^2 + \omega + 1 = 0\), find the value of \((1+\omega)(1+\omega^2)(1+\omega^3)\cdots(1+\omega^{1987})\), given that \(1987 = 662 \times 3 + 1\).