Chapter 3

Let \(A_{0} A_{1} A_{2} A_{3} A_{4} A_{5}\) be a regular hexagon inscribed in a circle of unit radius. Then the product of the lengths of the line segments \(A_{0} A_{1}, A_{0} A_{2}\) and \(A_{0} A_{4}\) is (A) \(\frac{3}{4}\) (B) \(3 \sqrt{3}\) (C) 3 (D) \(\frac{3 \sqrt{3}}{2}\)

If \(z_{1}\) and \(z_{2}\) are the two complex roots of equal magnitude and their arguments differ by \(\frac{\pi}{2}\), of the quadratic equation \(a x^{2}+b x+c=0(a \neq 0)\) then \(a\) (in terms of \(b\) and \(c\) ) is (A) \(\frac{b^{2}}{2 c}\) (B) \(\frac{b^{2}}{c}\) (C) \(\frac{b}{2 c}\) (D) None of these

Common roots of the equations \(z^{3}+2 z^{2}+2 z+1=0\) and \(z^{1985}+z^{100}+1=0\) are (A) \(\omega, \omega^{2}\) (B) \(1, \omega, \omega^{2}\) (C) \(-1, \omega, \omega^{2}\) (D) \(-\omega,-\omega^{2}\)

\(\sin ^{-1}\left[\frac{1}{i}(z-1)\right]\), where \(z\) is non-real, can be the angle of a triangle if (A) \(\operatorname{Re}(z)=1, \operatorname{Im}(z)=2\) (B) \(\operatorname{Re}(z)=1,-1 \leq \operatorname{Im}(z) \leq 1\) (C) \(\operatorname{Re}(z)+\operatorname{Im}(z)=0\) (D) \(\operatorname{Re}(z)=\operatorname{Im}(z)\)

If \(x^{2}-x+1=0\) then the value of \(\sum_{n=1}^{5}\left(x^{n}+\frac{1}{x^{n}}\right)^{2}\) is (A) 8 (B) 10 (C) 12 (D) None of these

The triangle formed by the points \(1, \frac{1+i}{\sqrt{2}}\) and \(i\) vertices in the Argand diagram is (A) scalene (B) equilateral (C) isosceles (D) right-angled

The triangle formed by the points \(1, \frac{1+i}{\sqrt{2}}\) and \(i\) vertices in the Argand diagram is (A) scalene (B) equilateral (C) isosceles (D) right-angled

If \(|z-i| \leq 2\) and \(z_{0}=5+3 i\), the maximum value of \(\mid i z\) \(+z_{0} \mid\) is (A) 7 (B) 9 (C) 13 (D) None of these

The solutions of the equation \(z(\overline{z-2 i})=2(2+i)\) are (A) \(3+i, 3-i\) (B) \(1+3 i, 1-3 i\) (C) \(1+3 i, 1-i\) (D) \(1-3 i, 1+i\)

Let \(\alpha, \beta\) be real and \(z\) be 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) \(\beta \in(1, \infty)\) (B) \(\beta \in(0,1)\) (C) \(\beta \in(-1,0)\) (D) \(|\beta|=1\)

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

If \(z \neq 1\) and \(\frac{z^{2}}{z-1}\) is real, then the point 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.

Two circles in complex plane are \(C_{1}:|z-i|=2\) \(C_{2}:|z-1-2 i|=4\). Then (A) \(C_{1}\) and \(C_{2}\) touch each other. (B) \(C_{1}\) and \(C_{2}\) intersect at two distinct points. (C) \(C_{1}\) lies within \(C_{2}\). (D) \(C_{2}\) lies within \(C_{1}\) -

Two circles in complex plane are \(C_{1}:|z-i|=2\) \(C_{2}:|z-1-2 i|=4\). Then (A) \(C_{1}\) and \(C_{2}\) touch each other. (B) \(C_{1}\) and \(C_{2}\) intersect at two distinct points. (C) \(C_{1}\) lies within \(C_{2}\). (D) \(C_{2}\) lies within \(C_{1}\) -

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) \(2+\sqrt{2}\)

If \(\sqrt{1-C^{2}}=n c-1\) and \(z=e^{i \theta}\), then \(\frac{c}{2 n}(1+n z)\left(1+\frac{n}{z}\right)=\) (A) \(1+c \cos \theta\) (B) \(1-c \cos \theta\) (C) \(1+2 c \cos \theta\) (D) \(1-2 c \cos \theta\)

Let ' \(a\) ' be a complex number such that \(|a|<1\) and \(z_{1}\), \(z_{2}, \ldots, z_{n}\) be the vetices of a polygon such that \(z_{k}=1+\) \(a+a^{2}+\ldots+a^{k}\), then the vertices of the polygon lie within the circle (A) \(|z|=\frac{1}{|1-a|}\) (B) \(|z-a|=\frac{1}{|1-a|}\) (C) \(\left|z-\frac{1}{1-a}\right|=\frac{1}{|1-a|}\) (D) None of these

If \(z^{4}=(z-1)^{4}\), then the roots are represented in the argand plane by the points that are (A) collinear (B) concyclic (C) vertices of a parallelogram (D) None of these

The maximum value of \(|z|\) when \(z\) satisfies the condition \(\left|z+\frac{2}{z}\right|=2\) is (A) \(\sqrt{3}-1\) (B) \(\sqrt{3}+1\) (C) \(\sqrt{3}\) (D) \(\sqrt{2}+\sqrt{3}\)

If \(|z+\bar{z}|+|z-\bar{z}|=8\), then \(z\) lies on (A) a circle (B) a straight line (C) a square (D) None of these

The complex number which satisfies the equation \(z+\sqrt{2}|z+1|+i=0\) is (A) \(2-i\) (B) \(-2-i\) (C) \(2+i\) (D) \(-2+i\)

\(\tan \left[i \log \frac{a-i b}{a+i b}\right]\) is equal to (A) \(\frac{2 a b}{a^{2}+b^{2}}\) (B) \(\frac{a^{2}-b^{2}}{2 a b}\) (C) \(\frac{2 a b}{a^{2}-b^{2}}\) (D) \(a b\)

If \(z=x+i y, x, y\) real, then \(|x|+|y| \leq k|z|\), where \(k\) is equal to (A) 1 (B) \(\sqrt{2}\) (C) \(\sqrt{3}\) (D) None of these

If \(z=a+i b\) where \(a>0, b>0\), then (A) \(|z| \geq \frac{1}{\sqrt{2}}(a-b)\) (B) \(|z| \geq \frac{1}{\sqrt{2}}(a+b)\) (C) \(|z|<\frac{1}{\sqrt{2}}(a+b)\) (D) None of these

If \(\left(1+x+x^{2}\right)^{n}=a_{0}+a_{1} x+a_{2} x^{2}+\ldots+a_{2 n} x^{2 n}\), then \(a_{0}+a_{3}+a_{6}+\ldots=\) (A) \(3^{n}\) (B) \(3^{n-1}\) (C) \(3^{n-2}\) (D) None of these

The closest distance of the origin from a curve given as \(a \bar{z}+\bar{a} z+a \bar{a}=0\) ( \(a\) is a complex number) is (A) 1 (B) \(\frac{|a|}{2}\) (B) \(\frac{\operatorname{Re} a}{|a|}\) (D) \(\frac{\operatorname{Im} a}{|a|}\)

The integral solution of the equation \((1-i)^{n}=2^{n}\) is (A) \(n=0\) (B) \(n=1\) (C) \(n=-1\) (D) None of these

The locus of the complex number \(z\) in an argand plane satisfying the equation $$ \operatorname{Arg}(z+i)-\operatorname{Arg}(z-i)=\frac{\pi}{2} \text { is } $$ (A) boundary of a circle (B) interior of a circle (C) exterior of a circle (D) None of these

If for the complex numbers \(z_{1}\) and \(z_{2},\left|z_{1}+z_{2}\right|=\left|z_{1}-z_{2}\right|\), then \(\operatorname{amp} z_{1} \sim \operatorname{amp} z_{2}=\) (A) \(\pi\) (B) \(\frac{\pi}{2}\) (C) \(\frac{\pi}{4}\) (D) None of these

The locus of the complex number \(z\) in an argand plane satisfying the inequality \(\log _{1 / 2}\left(\frac{|z-1|+4}{3|z-1|-2}\right)>1\left(\right.\) where \(\left.|z-1| \neq \frac{2}{3}\right)\) is (A) a circle (B) interior of a circle (C) exterior of a circle (D) None of these

Let \(z_{1}\) and \(z_{2}\) be two complex numbers such that \(\frac{z_{1}}{z_{2}}+\frac{z_{2}}{z_{1}}=1\), then (A) \(z_{1}, z_{2}\) are collinear (B) \(z_{1}, z_{2}\) and the origin from a right angled triangle (C) \(z_{1}, z_{2}\) and the origin form an equilateral triangle (D) None of these

If \(a, b, c, p, q, r\) are three non-zero complex numbers such that \(\frac{p}{a}+\frac{q}{b}+\frac{r}{c}=1+i\) and \(\frac{a}{p}+\frac{b}{q}+\frac{c}{r}=0\), then value of \(\frac{p^{2}}{a^{2}}+\frac{q^{2}}{b^{2}}+\frac{r^{2}}{c^{2}}\) is (A) 0 (B) \(-1\) (C) \(2 i\) (D) \(-2 i\)

If \(z_{1}, z_{2}\) are two complex numbers such that \(\left|\frac{z_{1}-z_{2}}{z_{1}+z_{2}}\right|=1\) and \(t z_{1}=k z_{2}\) where \(k \in R\), then the angle between \(\left(z_{1}-z_{2}\right)\) and \(\left(z_{1}+z_{2}\right)\) is (A) \(\tan ^{-1}\left(\frac{2 k}{k^{2}+1}\right)\) (B) \(\tan ^{-1}\left(\frac{2 k}{1-k^{2}}\right)\) (C) \(-2 \tan ^{-1}(k)\) (D) \(2 \tan ^{-1}(k)\)

\(1+x^{2}=\sqrt{3} x\), then \(\sum_{n=1}^{24}\left(x^{n}-\frac{1}{x^{n}}\right)^{2}\) is equal to (A) 48 (B) \(-48\) (C) \(\pm 48\left(\omega-\omega^{2}\right)\) (D) \(1 \pm 48 \omega\)

\(1+x^{2}=\sqrt{3} x\), then \(\sum_{n=1}^{24}\left(x^{n}-\frac{1}{x^{n}}\right)^{2}\) is equal to (A) 48 (B) \(-48\) (C) \(\pm 48\left(\omega-\omega^{2}\right)\) (D) \(1 \pm 48 \omega\)

If the roots of \((z-1)^{25}=2 \omega^{2}(z+1)^{25}\) (where \(\omega\) is a complex cube root of unity) are plotted in the argand plane, they lie on (A) a straight line (B) a circle (C) an ellipse (D) None of these

If the roots of \((z-1)^{25}=2 \omega^{2}(z+1)^{25}\) (where \(\omega\) is a complex cube root of unity) are plotted in the argand plane, they lie on (A) a straight line (B) a circle (C) an ellipse (D) None of these

\(z_{1}, z_{2}\) are two non-real complex numbers such that \(\frac{z_{1}}{z_{2}}+\frac{z_{2}}{z_{1}}=1 .\) Then, \(z_{1}, z_{2}\) and the origin (A) are collinear (B) form right angled triangle (C) form right angle isosceles triangle (D) form an equilateral triangle

If \(a, b, c\) are real, \(a^{2}+b^{2}+c^{2}=1\) and \(b+i c=(1+a) z\), then \(\frac{1+i z}{1-i z}=\) (A) \(\frac{a-i b}{1+c}\) (B) \(\frac{a+i b}{1+c}\) (C) \(\frac{a+i b}{1-c}\) (D) \(\frac{a-i b}{1-c}\)

If \(a, b, c\) are real, \(a^{2}+b^{2}+c^{2}=1\) and \(b+i c=(1+a) z\), then \(\frac{1+i z}{1-i z}=\) (A) \(\frac{a-i b}{1+c}\) (B) \(\frac{a+i b}{1+c}\) (C) \(\frac{a+i b}{1-c}\) (D) \(\frac{a-i b}{1-c}\)

If \(z_{1}, z_{2}\) are two complex numbers and \(w^{k}, k=0,1, \ldots\), \(n-1\) are the \(n\)th roots of unity, then \(\sum_{k=0}^{n-1}\left|z_{1}+z_{2} w^{k}\right|^{2}\) \((\mathrm{A})n\left(\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}\right)\) (D) can't say

If \(z_{1}, z_{2}\) are two complex numbers and \(w^{k}, k=0,1, \ldots\), \(n-1\) are the \(n\)th roots of unity, then \(\sum_{k=0}^{n-1}\left|z_{1}+z_{2} w^{k}\right|^{2}\) \((\mathrm{A})n\left(\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}\right)\) (D) can't say

The value of the expression \((\omega-1)\left(\omega-\omega^{2}\right)\left(\omega-\omega^{3}\right)\) ... \(\left(\omega-\omega^{n-1}\right)\), where \(w\) is the \(n^{\text {th }}\) root of unity, is (A) \(n \omega^{n-1}\) (B) \(n \omega^{n}\) (C) \((n-1) \omega^{n}\) (D) \((n-1) \omega^{n-1}\)

If \(|z-i|=1\) and \(\arg (z)=\theta, \theta \in\left(0, \frac{\pi}{2}\right)\), then the value of \(\cot \theta-\frac{2}{z}\) is equal to (A) 0 (B) \(i\) (C) \(-i\) (D) 1

If \(i=\sqrt{-1}\), then \(4+5\left(-\frac{1}{2}+\frac{i \sqrt{3}}{2}\right)^{334}+3\left(-\frac{1}{2}+\frac{i \sqrt{3}}{2}\right)^{365}\) is equal to (A) \(1-i \sqrt{3}\) (B) \(-1+i \sqrt{3}\) (C) \(i \sqrt{3}\) (D) \(-\sqrt{3} i\)

Let \(\bar{b} z+b \bar{z}=c, b \neq 0\), be a line in the complex plane, where \(\bar{b}\) is the complex conjugate of \(b .\) If a point \(z_{1}\) is the reflection of a point \(z_{2}\) through the line, then \(\bar{z}_{1} b+z_{2} \bar{b}=\) (A) \(4 c\) (B) \(2 c\) (C) \(c\) (D) None of these

Let \(\bar{b} z+b \bar{z}=c, b \neq 0\), be a line in the complex plane, where \(\bar{b}\) is the complex conjugate of \(b .\) If a point \(z_{1}\) is the reflection of a point \(z_{2}\) through the line, then \(\bar{z}_{1} b+z_{2} \bar{b}=\) (A) \(4 c\) (B) \(2 c\) (C) \(c\) (D) None of these

Let \(z_{1}\) and \(z_{2}\) be roots of the equation \(z^{2}+p z+q=0\), where the coefficients \(p\) and \(q\) may be complex numbers. Let \(A\) and \(B\) represent \(z_{1}\) and \(z_{2}\) in the complex plane. If \(\angle A O B=\alpha \neq 0\) and \(O A=O B\), where \(O\) is the origin, then \(p^{2}=k \cos ^{2} \frac{\alpha}{2}\), where \(k=\) (A) \(q\) (B) \(2 q\) (C) \(4 q\) (D) None of these

If \(|z| \leq 1,|\omega| \leq 1\), then \(|\mathrm{z}-\omega|^{2}\) \((\mathrm{A}) \leq(|z|-|\omega|)^{2}-(\operatorname{Arg} z-\operatorname{Arg} \omega)^{2}\) (B) \(\leq(|z|-|\omega|)^{2}+(\operatorname{Arg} z-\operatorname{Arg} \omega)^{2}\) (C) \(\leq(|z|-|\omega|)^{2}+2(\operatorname{Arg} z-\operatorname{Arg} \omega)^{2}\) (D) None of these

If \(k=\frac{3 n}{2}\), where \(n\) is an even positive integer, then \(\sum_{r=1}^{k}(-3)^{r-1} \cdot{ }^{3 n} C_{2 r-1}=\) (A) 0 (B) 1 (C) \(-1\) (D) None of these