Chapter 3

If \(1, \omega, \omega^{2}, \ldots, \omega^{n-1}\) are the \(n, n\)th roots of unity, then \((1-\omega)(1-\omega)^{2} \ldots\left(1-\omega^{n-1}\right)\) is equal to (A) 0 (B) 1 (C) \(n\) (D) \(n^{2}\)

The roots of the equation \(z^{4}+4 z^{2}+16=0\) are (A) \(\pm 1+i \sqrt{3}\) (B) \(\pm 1-i \sqrt{3}\) (C) \(\pm 2+2 \sqrt{3} i\) (D) \(\pm 2-2 \sqrt{3} i\)

The roots of the equation \((2+z)^{6}+(2-z)^{6}=0\) are (A) \(\pm 2 i \tan \pi / 12\) (B) \(\pm 2 i \tan 5 \pi^{\prime} 12\) (C) \(\pm 2 i\) (D) \(\pm 2\)

The roots of the equation \(z^{4}-z^{3}+z^{2}-z+1=0\) are \(\cos \left(\frac{p \pi}{5}\right)+i \sin \left(\frac{p \pi}{5}\right)\) where \(p=\) (A) \(1,3,5,7,9\) (B) \(1,3,7,9\) (C) \(3,5,7,9\) (D) None of these

\hline Column-I \begin{tabular}{l} Column-II \\ \hline (A) \(i-1\) \end{tabular} (I) If \(z_{r}=\cos \left(\frac{\pi}{3^{r}}\right)+i \sin \left(\frac{\pi}{3^{r}}\right) r\) If \(z_{r}\) \(=1,2,3, \ldots\), then \(z_{1} z_{2} z_{3} \ldots \infty=\) (II) If \(i z^{3}+z^{2}-z+i=0\), then \(|z|=\) (B) 1 (III) If \(\frac{z-2}{z+2}(z \neq-2)\) is purely (C) 2 imaginary, then \(|z|=\) (IV) The value of the sum (D) \(i-1\) \(\sum_{n=1}^{13}\left(i^{n}+i^{n+1}\right)\) where \(i=\sqrt{-1}\), equals

Assertion: If the area of the triangle on the arganc plane formed by the complex numbers \(-z, i z, z-i z\) is 600 square units, then \(|z|=20\) Reason: Area of the triangle on the argand plane formed by the complex numbers \(-z, i z, z-i z\) is \(\frac{3}{2}\left|z^{2}\right|\)

Assertion: If \(|z-1|+|z+3| \leq 8\), then the range of values of \(|z-4|\) is \([1,9]\) Reason: \(|z-1|+|z+3| \leq 8 \Rightarrow z\) lies inside or on the ellipse whose foci are \((1,0)\) and \((-3,0)\) and vertices are \((-5,0)\) and \((3,0)\).

Assertion: The locus of the points representing the complex numbers satisfying \(|z|-2=0,|z-i|-\) \(|z+5 i|=0\) is the single point \((0,-2)\) Reason: If \(z\) is a variable point and \(z_{1}, z_{2}\) are two fixed points in the argand plane, then \(\left|z-z_{1}\right|=\left|z-z_{2}\right| \Rightarrow\) locus of \(z\) is the perpendicular bisector of the line segment joining \(z_{1}\) and \(z_{2}\).

\mathrm{\\{} A s s e r t i o n : ~ I f ~ \(z_{0}=\frac{1}{2}(1+i)\), then \(P_{n}(z)=\left(1+z_{0}\right)\left(1+z_{0}^{2}\right)\left(1+z_{0}^{22}\right) \ldots\left(1+z_{0}^{2 n}\right)=\) \((1+i)\left(1-\frac{1}{2^{2^{n}}}\right)\), where \(n>1\) is a positive integer. Reason: \(P_{n}(z)=\frac{1-z_{0}^{2^{n+1}}}{1-z_{0}}\)

If \(\omega\) is an imaginary cube root of unity, then (1 \(\left.-\omega^{2}\right)^{7}\) equals: \(\quad[20\) (A) \(128 \omega\) (B) \(-128 \omega\) (C) \(128 \omega^{2}\) (D) \(-128 \omega^{2}\) Tot 7

If \(\omega\) is an imaginary cube root of unity, then (1 \(\left.-\omega^{2}\right)^{7}\) equals: \(\quad[20\) (A) \(128 \omega\) (B) \(-128 \omega\) (C) \(128 \omega^{2}\) (D) \(-128 \omega^{2}\) Tot 7

If \(z\) and \(\omega\) are two non-zero complex numbers such that \(|z \omega|=1\), and \(\operatorname{Arg}(z)-\operatorname{Arg}(\omega)=\frac{\pi}{2}\), then \(\bar{Z} \omega\) is equal to \([\mathbf{2 0 0 3}]\) (A) 1 (B) \(-1\) (C) \(i\) (D) \(-i\)

If \(\left(\frac{1+i}{1-i}\right)^{x}=1\), then (A) \(x=4 n\), where \(n\) is any positive integer (B) \(x=2 n\), where \(n\) is any positive integer (C) \(x=4 n+1\), where \(n\) is any positive integer (D) \(x=2 n+1\), where \(n\) is any positive integer

Let \(z, w\) be complex numbers such that \(\bar{z}+i \bar{w}=0\) and \(\arg z w=\pi\). Then \(\arg z\) equals [2004] (A) \(\frac{\pi}{4}\) (B) \(\frac{5 \pi}{4}\) (C) \(\frac{3 \pi}{4}\) (D) \(\frac{\pi}{2}\)

If \(z=x-i y\) and \(z^{\frac{1}{3}}=p+i q\), then \(\frac{\left(\frac{x}{p}+\frac{y}{q}\right)}{\left(p^{2}+q^{2}\right)}\) is equal to (A) 1 (B) \(-2\) (C) 2 (D) \(-1\)

If the cubes roots of unity are \(1, \omega, \omega^{2}\) then the roots of the equation \((x-1)^{3}+8=0\), are (A) \(-1,-1+2 \omega,-1-\omega^{2}\) (B) \(-1,-1,-1\) (C) \(-1,1-2 \omega, 1-2 \omega^{2}\) (D) \(-1,1+2 \omega, 1+2 \omega^{2}\)

If \(z_{1}\) and \(\mathrm{z}_{2}\) are two non-zero complex numbers such that \(\left|z_{1}+z_{2}\right|=\left|z_{1}\right|+\left|z_{2}\right|\) then \(\arg \mathrm{z}_{1}-\mathrm{argz}_{2}\) is equal to \([2005]\) (A) \(\frac{\pi}{2}\) (B) \(-\pi\) (C) 0 (D) \(-\frac{\pi}{2}\)

If \(w=\frac{z}{z-\frac{1}{3} i}\) and \(|w|=1\), then \(z\) lies on [2005] (A) an ellipse (B) a circle (C) a straight line (D) a parabola

The value of \(\sum_{k=1}^{10}\left(\sin \frac{2 k \pi}{11}+i \cos \frac{2 k \pi}{11}\right)\) is (A) \(\vec{i}\) (B) 1 (C) \(-1\) (D) \(-i\)

If \(z^{2}+z+1=0\), where \(z\) is a complex number, then the value of \(\left(z+\frac{1}{z}\right)^{2}+\left(z^{2}+\frac{1}{z^{2}}\right)^{2}+\left(z^{3}+\frac{1}{z^{3}}\right)^{2}+\ldots+\left(z^{6}+\frac{1}{z^{6}}\right)^{2}\) is \([\mathbf{2 0 0 6}]\) (A) 18 (B) 54 (C) 6 (D) 12

If \(|z+4| \leq 3\), then the maximum value of \(|z+1|\) is \([2007]\) (A) 4 (B) 10 (C) 6 (D) 0

The conjugate of a complex number is \(\frac{1}{i-1}\) Then the complex number is (A) \(\frac{-1}{i-1}\) (B) \(\frac{1}{i+1}\) (C) \(\frac{-1}{i+1}\) (D) \(\frac{1}{i+1}\)

.If \(\left|Z-\frac{4}{Z}\right|=2\), then the maximum value of \(|Z|\) is equal to (A) \(\sqrt{3}+1\) (B) \(\sqrt{5}+1\) (C) 2 (D) \(2+\sqrt{2}\)

The number of complex numbers \(z\) such that \(|z-1|\) \(=|z+1|=|z-i|\) equals \(\quad[2010]\) (A) 1 (B) 2 (C) \(\infty\) (D) 0

Let \(\alpha, \beta\) be real numbers and \(z\) a complex number. If \(z^{2}+\alpha z+\beta=0\) has two distinct roots on the line \(\operatorname{Re}(z)=1\), then it is necessary that (A) \(\beta \in(-1,0)\) (B) \(|\beta|=1\) (C) \(\beta \in(1, \infty)\) (D) \(\beta \in(0,1)\)

Let \(\alpha, \beta\) be real numbers and \(z\) a complex number. If \(z^{2}+\alpha z+\beta=0\) has two distinct roots on the line \(\operatorname{Re}(z)=1\), then it is necessary that (A) \(\beta \in(-1,0)\) (B) \(|\beta|=1\) (C) \(\beta \in(1, \infty)\) (D) \(\beta \in(0,1)\)

If \(z \neq 1\) and \(\frac{z^{2}}{z-1}\) is real, then the point which is represented by the complex number \(z\) lies (A) either on the real axis or on a circle passing through the origin (B) on a circle with centre at the origin (C) either on the real axis or on a circle not passing through the origin (D) on the imaginary axis

If \(z \neq 1\) and \(\frac{z^{2}}{z-1}\) is real, then the point which is represented by the complex number \(z\) lies (A) either on the real axis or on a circle passing through the origin (B) on a circle with centre at the origin (C) either on the real axis or on a circle not passing through the origin (D) on the imaginary axis

If \(z\) is a complex number such that \(|z| \geq 2\), then the minimum value of \(\left|z+\frac{1}{2}\right|\) [2014] (A) is equal to \(\frac{5}{2}\) (B) lies in the interval \((1,2)\) (C) is strictly greater than \(\frac{5}{2}\) (D) is strictly greater than \(\frac{3}{2}\) but less than \(\frac{5}{2}\)

A complex number \(z\) is said to be unimodular if \(|z|=1\). Suppose \(z_{1}\) and \(z_{2}\) are complex numbers such that \(\frac{z_{1}-2 z_{2}}{2-z_{1} \bar{z}_{2}}\) is unimodular and \(z_{2}\) is not unimodular. Then the point \(z_{1}\) lies on a [2015] (A) straight line parallel to \(y\)-axis. (B) circle of radius 2 . (C) circle of radius \(\sqrt{2}\). (D) straight line parallel to \(x\)-axis.

A value of \(\theta\) for which \(\frac{2+3 i \sin \theta}{1-2 i \sin \theta}\) is purely imagi- nary, is [2016] (a) \(\sin ^{-1}\left(\frac{1}{\sqrt{3}}\right)\) (B) \(\frac{\pi}{3}\) (c) \(\frac{\pi}{6}\) (D) \(\sin ^{-1}\left(\frac{\sqrt{3}}{4}\right)\)