Chapter 3

If \(1, \omega, \omega^{2}, \ldots \omega^{n-1}\) are the \(n, n\)th roots of unity and \(z_{1}\) and \(z_{2}\) are any two complex numbers, then $$ \sum_{k=0}^{n-1}\left|z_{1}+\omega^{k} z_{2}\right|^{2}= $$ (A) \(n\left[\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}\right]\) (B) \((n-1)\left[\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}\right]\) (C) \((n+1)\left[\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}\right]\) (D) None of these

If \(1, a_{1}, a_{2}, \ldots, a_{n-1}\) are the \(n, n\)th roots of unity, then \(\left(1-a_{1}\right)\left(1-a_{2}\right)\left(1-a_{3}\right) \ldots\left(1-a_{n-1}\right)=\) (A) \(n+1\) (B) \(n\) (C) \(n-1\) (D) None of these.

If \(1, \alpha, \alpha^{2}, \ldots, \alpha^{n}{\underline{\phantom{xx}}}^{1}\) are the \(n n\)th roots of unity then \(\sum_{i=1}^{n-1} \frac{1}{2-\alpha^{i}}\) is equal to (A) \(\frac{(n-2) 2^{n-1}+1}{2^{n}-1}\) (B) \((n-2) \times 2^{n}\) (C) \(\frac{(n-2) \cdot 2^{n-1}}{2^{n}-1}\) (D) None of these

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) \(n^{2}\)Passage 3 Solution of Equations Certain types of algebraic equations can be solved with the help of De'Moivre's theorem Equations of the type \(p z^{n}+q=0:\) If \(p z^{n}+q=0\), where \(p\) and \(q\) are complex numbers, and \(p \neq 0\), then $$ z^{n}=-q / p $$ The roots of the given equation are, therefore, the \(n\) values of \((-q / p)^{1 / n}\). For example, consider the equation \(z^{7}+1=0\).\(z^{7}+1=0 \Rightarrow z^{7}=-1=\) cis \((2 p+1) \pi\), where \(p\) is an integer. Therefore, \(z=\operatorname{cis}[(2 p+1) \pi 7], p=0,1, \ldots, 6\) On putting \(p=0,1,2,3,4,5,6\), the roots are seen to be \(\cos (\pi 7) \pm i \sin (\pi / 7), \cos (3 \pi / 7) \pm i \sin \left(3 \pi^{\prime} 7\right), \cos\) \((5 \pi / 7) \pm i \sin (5 \pi / 7),-1 .\) Equations of the type \(p z^{2 n}+q z^{n}+r=0\), where \(p, q\) and \(r\) are complex numbers and \(p \neq 0\). $$ z^{n}=\frac{-q \pm \sqrt{q^{2}-4 p r}}{2 p} $$ Denoting these values of \(z^{n}\) by \(\alpha\) and \(\beta\), we have two equations \(z^{n}=\alpha\) and \(z^{n}=\beta\), each of which can be solved by the method given in the above example. Equations of the type \(a(p z+q)^{n}+b(r z+s)^{n}=0:\) The substitution \(\frac{p z+q}{n+s}=w\) reduces the given equation to the form $$ a w^{n}+b=0 $$ which can be solved by the method given above. If \(w_{k}\) be a root of the equation (i), the corresponding root \(z_{k}\) of the given equation is obtained by solving the equation $$ \frac{p z_{k}+q}{r z_{k}+s}=w_{k} $$

The roots of the equation \(x^{6}+x^{3}+1=0\) are \(\cos \left(\frac{p \pi}{9}\right) \pm i \sin \left(\frac{p \pi}{9}\right)\), where \(p=\) (A) 2 (B) 8 (C) 14 (D) 20

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

Column-I Column-II (1) If \(z_{r}=\cos \left(\frac{\pi}{3^{r}}\right)+i \sin \left(\frac{\pi}{3^{r}}\right) r\) (A) \(i-1\) \(=1,2,3, \ldots\), 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

Column-I Column-II (I) \(\left(\frac{\sqrt{3}+i}{2}\right)^{6}+\left(\frac{i-\sqrt{3}}{2}\right)^{6}\) (A) \(-\frac{7}{2}\) (II) If \(\frac{z-1}{z+1}\) is purely imaginary, (B) 0 then \(|z|=\) (III) \((i+\sqrt{3})^{100}+(i-\sqrt{3})^{100}+2^{100}=\) (' (C) \(-2\) (IV) Let \(z_{k}=(k=0,1,2, \ldots 6)\) be the (D) 1 roots of the equation \((z+1)^{7}\) \(+z^{7}=0\), then \(\sum_{k=0}^{6} \operatorname{Re}\left(z_{k}\right)\) is

Assertion: If \(a=\cos \alpha+i \sin \alpha, b=\cos \beta+i \sin \beta\), \(c=\cos \gamma+i \sin \gamma\) and \(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}=-1\), then \(\cos (\beta-\gamma)\) \(+\cos (\gamma-\alpha)+\cos (\alpha-\beta)=-1\) Reason: \(\left(\cos \alpha_{1}+i \sin \alpha_{1}\right)\left(\cos \alpha_{2}+i \sin \alpha_{2}\right)=\) \(\cos \left(\alpha_{1}+\alpha_{2}\right)+i \sin \left(\alpha_{1}+\alpha_{2}\right)\)

Assertion: The locus of the point \(z\) satisfying the condition arg \(\frac{z-1}{z+1}=\frac{\pi}{3}\) is a parabola Reason: \(\operatorname{Arg} \frac{z_{1}}{z_{2}}=\operatorname{Arg} z_{1}-\operatorname{Arg} z_{2}\)

Assertion: If the area of the triangle on the argand 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 greatest value of the moduli of complex numbers \(z\) satisfying the equation \(\left|z-\frac{4}{z}\right|=2\) is \(\sqrt{5}+1\) Reason: For any two complex numbers \(z_{1}\) and \(z_{2}\), \(\left|z_{1}-z_{2}\right| \geq\left|z_{1}\right|-\left|z_{2}\right|\)

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 \(z_{2}\).

Assertion: If \(z_{0}=\frac{1}{2}(1+i)\), then \(P_{n}(z)=\left(1+z_{0}\right)\left(1+z_{0}^{2}\right)\left(1+z_{0}^{22}\right) \ldots\left(1+z_{0}^{2 n}\right)=\) \((1+i)\left(1-\frac{1}{2^{2^{*}}}\right)\), 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 \((1+\omega\) \(\left.-\omega^{2}\right)^{7}\) equals: (A) \(128 \omega\) (B) \(-128 \omega\) (C) \(128 \omega^{2}\) (D) \(-128 \omega^{2}\)

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 \([2003]\) (A) \(a^{2}=b\) (B) \(a^{2}=2 b\) (C) \(a^{2}=3 b\) (D) \(a^{2}=4 b\)

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 (A) 1 (B) \(-1\) (C) \(i\) (D) \(-i\)

If \(\left(\frac{1+i}{1-i}\right)^{x}=1\), then \(\quad[2003]\) (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 \(\left|z^{2}-1\right|=|z|^{2}+1\), then \(z\) lies on \(\quad\) [2004] (A) the real axis (B) an ellipse (C) a circle (D) the imaginary axis.

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 \(z_{2}\) are two non-zero complex numbers such that \(\left|z_{1}+z_{2}\right|=\left|z_{1}\right|+\left|z_{2}\right|\) then \(\arg z_{1}-\arg z_{2}\) 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) \(i\) (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 (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 \(\mathrm{z}\) such that \(|z-1|\) \(=|z+1|=|z-i|\) equals \([2010]\) (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) \(\beta \in(0,1)\)

If \(\omega(\neq 1)\) is a cube root of unity, and \((1+\omega)^{7}=A+\) \(B \omega\). Then \((A, B)\) equals [2011] (A) \((1,1)\) (B) \((1,0)\) (C) \((-1,1)\) (D) \((0,1)\)

If \(z \neq 1\) and \(\frac{z^{2}}{z-1}\) is real, then the point which is represented by the complex number \(z\) lies [2012] (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 of unit modulus and argument \(\theta\), then \(\left(\frac{1+z}{1+\bar{z}}\right)\) equals \(\quad\) [2013] (A) \(\frac{\pi}{2}-\theta\) (B) \(\theta\) (C) \(\pi-\theta\) (D) \(-\theta\)

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 imaginary, is (a) \(\sin ^{-1}\left(\frac{1}{\sqrt{3}}\right)\) (B) \(\frac{\pi}{3}\) (c) \(\frac{\pi}{6}\) (D) \(\sin ^{-1}\left(\frac{\sqrt{3}}{4}\right)\)