Chapter 3

If the complex numbers \(z_{1}, z_{2}, z_{3}\) are the vertices \(A\), \(B, C\) respectively of an isosceles right angled triangle with right angle at \(C\), then \(\left(z_{1}-z_{2}\right)^{2}=k\left(z_{1}-z_{3}\right)\left(z_{3}-z_{2}\right)\), where \(k=\) (A) 1 (B) 2 (C) 4 (D) None of these

If \(2 \sqrt{2 x^{4}}=(\sqrt{3}-1)+i(\sqrt{3}+1)\), then \(x=\cos \frac{1}{4}(2 n \pi+k)+i \sin \frac{1}{4}(2 n \pi+k) ;\) \(n=0,1,2,3\), where \(k=\) (A) \(\frac{\pi}{12}\) (B) \(\frac{5 \pi}{12}\) (C) \(\frac{7 \pi}{12}\) (D) None of these

\(\sum_{p=1}^{32}(3 p+2)\left[\sum_{q=1}^{10}\left(\sin \frac{2 q \pi}{11}-i \cos \frac{2 q \pi}{11}\right)\right]^{p}=\) (A) \(8(1-i)\) (B) \(16(1-i)\) (C) \(48(1-i)\) (D) None of these

The three vertices of a triangle are represented by the complex numbers \(0, z_{1}\) and \(z_{2}\). If the triangle is equilateral, then (A) \(z_{1}^{2}+z_{2}^{2}+z_{1} z_{2}=0\) (B) \(z_{1}^{2}+z_{2}^{2}=z_{1} z_{2}\) (C) \(z_{2}^{2}-z_{1}^{2}=z_{1} z_{2}\) (D) \(z_{1}^{2}-z_{2}^{2}=z_{1} z_{2}\)

If \(|z-25 i| \leq 15\), then |maximum amp \((z)-\) minimum \(\operatorname{amp}(z) \mid\) is equal to (A) \(\sin ^{-1}\left(\frac{3}{5}\right)-\cos ^{-1}\left(\frac{3}{5}\right)\) (B) \(\frac{\pi}{2}+\cos ^{-1}\left(\frac{3}{5}\right)\) (C) \(\pi-2 \cos ^{-1}\left(\frac{3}{5}\right)\) (D) \(\cos ^{-1}\left(\frac{3}{5}\right)\)

If \(z_{1}\) and \(z_{2}\) are any two complex numbers, then \(\left|z_{1}+\sqrt{z_{1}^{2}-z_{2}^{2}}\right|+\left|z_{1}-\sqrt{z_{1}^{2}-z_{2}^{2}}\right|\) is equal to (A) \(\left|z_{1}+z_{2}\right|\) (B) \(\left|z_{1}\right|\) (C) \(\left|z_{2}\right|\) (D) None of these

If \(z=x+i y\) lies in IIIrd quadrant, then \(-\) also lies in the IIIrd quadrant if \(z\) (A) \(y>x>0\) (B) \(yy>0\)

In the Argand diagram, if \(O, P\) and \(Q\) represent respectively the origin and the complex numbers \(z\) and \(z+i z\), then the \(\angle O P Q\) is (A) \(\frac{\pi}{4}\) (B) \(\frac{\pi}{3}\) (C) \(\frac{\pi}{2}\) (D) \(\frac{2 \pi}{3}\)

If \(z\) satisfies \(|z+1|<|z-2|\), and \(\omega=3 z+2+i\), then (A) \(|\omega+1|<|\omega-8|\) (B) \(|\omega+1|<|\omega-7|\) (C) \(\omega+\bar{\omega}>7\) (D) \(|\omega+5|<|\omega-4|\)

If \(z\) satisfies \(|z+1|<|z-2|\), and \(\omega=3 z+2+i\), then (A) \(|\omega+1|<|\omega-8|\) (B) \(|\omega+1|<|\omega-7|\) (C) \(\omega+\bar{\omega}>7\) (D) \(|\omega+5|<|\omega-4|\)

If \(P(x)\) and \(Q(x)\) are two polynomials such that \(f(x)=\) \(P\left(x^{3}\right)+x Q\left(x^{3}\right)\) is divisible by \(x^{2}+x+1\), then (A) \(P(x)\) is divisible by \((x-1)\) but \(Q(x)\) is not divisible by \(x-1\) (B) \(Q(x)\) is divisible by \((x-1)\) but \(P(x)\) is not divisible by \(x-1\) (C) Both \(P(x)\) and \(Q(x)\) are divisible by \(x-1\) (D) \(f(x)\) is divisible by \(x-1\)

\(e^{2 m i \cot ^{-1} p} \cdot\left(\frac{p i+1}{p i-1}\right)^{m}=\) (A) 0 (B) 1 (C) \(-1\) (D) None of these

If \(z_{1}, z_{2}, z_{3}\) are non-zero, non-collinear complex numbers such that \(\frac{2}{z_{1}}=\frac{1}{z_{2}}+\frac{1}{z_{3}}\), then the points \(z_{1}, z_{2}, z_{3}\) lie (A) in the interior of a circle (B) on a circle passing through origin (C) in the exterior of a circle (D) None of these

If \(|z-25 i| \leq 15\), then the least positive value of arg \(z\) is \(\begin{array}{ll}\text { (A) } \pi-\tan ^{-1} \frac{4}{3} & \text { (B) } \tan ^{-1} \frac{4}{3}\end{array}\) (C) \(-\pi+\tan ^{-1} \frac{4}{3}\) (D) None of these

If \(|z-4+3 i| \leq 2\), then the least and the greatest values of \(|z|\) are (A) 3,7 (B) 4,7 (C) 3,9 (D) None of these

If \(z_{1}, z_{2}\) are two complex numbers and \(c>0\) such that \(\left|z_{1}+z_{2}\right|^{2} \leq(1+c)\left|z_{1}\right|^{2}+k\left|z_{2}\right|^{2}\), then \(k=\) (A) \(1-c\) (B) \(c-1\) (C) \(1+c^{-1}\) (D) \(1-c^{-1}\)

If \(|z-4+3 i| \leq 1\) and \(m\) and \(n\) are the least and greatest values of \(|z|\) and \(k\) is the least value of \(\frac{x^{4}+x^{2}+4}{x}\) on the interval \((0, \infty)\), then \(k\) is equal to (A) \(m\) (B) \(n\) (C) \(m+n\) (D) None of these

If \(n>1\), then the roots of \(z^{n}=(z+1)^{n}\) lie on a (A) circle (B) straight line (C) parabola (D) None of these

Let \(z\) be a complex number satisfying \(z^{2}+z+1=0\). If \(n\) is not a multiple of 3, then the value of \(z^{n}+z^{2 n}=\) (A) 2 (B) \(-2\) (C) 0 (D) \(-1\)

If \(z_{1}\) and \(z_{2}\) both satisfy the relation \(z+\bar{z}=2|z-1|\) and \(\arg \left(z_{1}-z_{2}\right)=\frac{\pi}{4}\), then the imaginary part of \(\left(z_{1}+z_{2}\right)\) is (A) 0 (B) 1 (C) 2 (D) None of these

If \(z_{1}+z_{2}+z_{3}=A, z_{1}+z_{2} \omega+z_{3} \omega^{2}=B\) and \(z_{1}+z_{2}\) \(\omega^{2}+z_{3} \omega=C\), where \(1, \omega, \omega^{2}\) are the three cube roots of unity, then \(|A|^{2}+|B|^{2}+|C|^{2}=\) (A) \(3\left(\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}+\left|z_{3}\right|^{2}\right)\) (B) \(2\left(\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}+\left|z_{3}\right|^{2}\right)\) (C) \(\left(\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}+\left|z_{3}\right|^{2}\right)\) (D) None of these

If \(z_{1}+z_{2}+z_{3}=A, z_{1}+z_{2} \omega+z_{3} \omega^{2}=B\) and \(z_{1}+z_{2}\) \(\omega^{2}+z_{3} \omega=C\), where \(1, \omega, \omega^{2}\) are the three cube roots of unity, then \(|A|^{2}+|B|^{2}+|C|^{2}=\) (A) \(3\left(\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}+\left|z_{3}\right|^{2}\right)\) (B) \(2\left(\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}+\left|z_{3}\right|^{2}\right)\) (C) \(\left(\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}+\left|z_{3}\right|^{2}\right)\) (D) None of these

If \(\alpha, \beta\) are the roots of \(z+\frac{1}{z}=2(\cos \theta+\sin \theta)\) Then, (A) \(|\alpha-i|>|\beta-i|\) (B) \(|\alpha-1|<|\beta-i|\) (C) \(|\alpha-i|=|i-\beta|\) (D) \(|\alpha-i|=|\beta-i|\)

If \(\alpha\) is the \(n\)th root of unity, then \(1+2 \alpha+3 \alpha^{2}+\ldots\) to \(n\) terms is equal to (A) \(-\frac{n}{(1-\alpha)^{2}}\) (B) \(-\frac{n}{1-\alpha}\) (C) \(-\frac{2 n}{1-\alpha}\) (D) \(-\frac{2 n}{(1-\alpha)^{2}}\)

Let \(O, A, B\) be three collinear points such that \(O A \cdot O B=1 .\) If \(O\) and \(B\) represent the complex numbers \(o\) and \(z\), then \(A\) represents (A) \(\frac{1}{\bar{z}}\) (B) \(\frac{1}{z}\) (C) \(\bar{z}\) (D) \(z^{2}\)

Let \(O, A, B\) be three collinear points such that \(O A \cdot O B=1 .\) If \(O\) and \(B\) represent the complex numbers \(o\) and \(z\), then \(A\) represents (A) \(\frac{1}{\bar{z}}\) (B) \(\frac{1}{z}\) (C) \(\bar{z}\) (D) \(z^{2}\)

The locus represented by the complex equation \(|z-2-i|=|z| \sin \left(\frac{\pi}{4}-\arg z\right)\) is the part of (A) a pair of straight lines (B) a circle (C) a parabola (D) a rectangular hyperbola

If \(z_{1}, z_{2}, z_{3}\) are three points lying on the circle \(|z|=2\) then the minimum value of \(\left|z_{1}+z_{2}\right|^{2}+\left|z_{2}+z_{3}\right|^{2}+\) \(\left|z_{3}+z_{1}\right|^{2}\) is equal to (A) 6 (B) 12 (C) 15 (D) 24

If \(z_{1}, z_{2}, z_{3}\) are three points lying on the circle \(|z|=2\) then the minimum value of \(\left|z_{1}+z_{2}\right|^{2}+\left|z_{2}+z_{3}\right|^{2}+\) \(\left|z_{3}+z_{1}\right|^{2}\) is equal to (A) 6 (B) 12 (C) 15 (D) 24

\(\sqrt{i}-\sqrt{-i}\) is equal to (A) \(i \sqrt{2}\) \(\begin{array}{lll}\text { (B) } \frac{1}{i \sqrt{2}} & \text { (C) } 0 & \text { (D) }-i \sqrt{2}\end{array}\) (C)

If \(z_{1}, z_{2}, z_{3}, z_{4}\) are the four complex numbers represented by the vertices of a quadrilateral taken in order such that \(z_{1}-z_{4}=z_{2}-z_{3}\) and amp \(\frac{z_{4}-z_{1}}{z_{2}-z_{1}}=\frac{\pi}{2}\) then the quadrilateral is a (A) square (B) rhombus (C) rectangle (D) a cyclic quadrilateral

The sum $$ \sum_{m=1}^{4 n+1}\left[\sum_{k=1^{-}}^{m+1}\left\\{\sin \left(\frac{2 \pi k}{m}\right)-i \cos \left(\frac{2 \pi k}{m}\right)\right\\}\right]^{m} \text { is } $$ (A) independent of \(n\) (B) purely imaginary (C) purely real (D) a root of \(x^{4 n+1}+1=0\)

\(z_{1}=a+i b\) and \(z_{2}=c+i d\) are complex numbers such that \(\left|z_{1}\right|=\left|z_{2}\right|=1\) and \(\operatorname{Re}\left(z_{1} \bar{z}_{2}\right)=0 .\) If \(w_{1}=a+i c\) and \(w_{2}=b+i d(a, b, c, d \in R)\), then (A) \(\left|w_{1}\right|=1\) (B) \(\left|w_{2}\right|=1\) (C) \(\operatorname{Re}\left(w_{1} \bar{w}_{2}\right)=0\) (D) \(\operatorname{Re}\left(w_{1} \bar{w}_{2}\right)=1\)

If \(z_{1}^{2}+2 z_{2}^{2}+z_{3}^{2}=2 z_{2}\left(z_{1}+z_{3}\right)\), where \(z_{1}, z_{2}, z_{3}\) are the vertices of a triangle, then the triangle is (A) isosceles (B) right angled (C) equilateral (D) obtuse angled

If \(\left|z_{1}-z_{2}\right|=\left|z_{1}\right|+\left|z_{2}\right|\), then (A) \(\arg \left(\frac{z_{1}}{z_{2}}\right)=\frac{\pi}{2}\) (B) \(\arg \left(\frac{z_{1}}{z_{2}}\right)=(2 n+1) \pi, n \in I\) (C) \(z_{1} \bar{z}_{2}+\bar{z}_{1} z_{2} \leq 0\) (D) \(z_{1}=l z_{2}, l \in R\)

If \(z_{1}=a+i b\) and \(z_{2}=c+i d\) are two complex numbers such that \(\left|z_{1}\right|=\left|z_{2}\right|=1\) and \(\operatorname{Re}\left(z_{1}, \bar{z}_{2}\right)=0\) then for the pair of complex numbers \(\omega_{1}=a+i c\) and \(\omega_{2}=b+i d\) (A) \(\operatorname{Re}\left(\omega_{1} \bar{\omega}_{2}\right)=0\) (B) \(\operatorname{Re}\left(\omega_{1} \bar{\omega}_{2}\right)=1\) (C) \(\left|\omega_{1}\right|=1\) (D) None of these

If \(z_{1}, z_{2}, z_{3}\) are the vertices of an equilateral triangle in the complex plane and \(z_{0}\) is the centroid, then (A) \(\frac{1}{z_{1}-z_{2}}+\frac{1}{z_{2}-z_{3}}+\frac{1}{z_{3}-z_{1}}=0\) (B) \(\left(z_{1}-z_{2}\right)^{2}+\left(z_{2}-z_{3}\right)^{2}+\left(z_{3}-z_{1}\right)^{2}=0\) (C) \(z_{1}^{2}+z_{2}^{2}+z_{3}^{2}=3 z_{0}^{2}\) (D) \(z_{1}^{2}+z_{2}^{2}+z_{3}^{2}=z_{1} z_{2}+z_{2} z_{3}+z_{3} z_{1}\)

If \(z_{1}, z_{2}, z_{3}\) are the vertices of an equilateral triangle in the complex plane and \(z_{0}\) is the centroid, then (A) \(\frac{1}{z_{1}-z_{2}}+\frac{1}{z_{2}-z_{3}}+\frac{1}{z_{3}-z_{1}}=0\) (B) \(\left(z_{1}-z_{2}\right)^{2}+\left(z_{2}-z_{3}\right)^{2}+\left(z_{3}-z_{1}\right)^{2}=0\) (C) \(z_{1}^{2}+z_{2}^{2}+z_{3}^{2}=3 z_{0}^{2}\) (D) \(z_{1}^{2}+z_{2}^{2}+z_{3}^{2}=z_{1} z_{2}+z_{2} z_{3}+z_{3} z_{1}\)

If \(z_{1}, z_{2}, z_{3}\) are the vertices of an isosceles triangle and right angled at \(z_{2}\), then (A) \(z_{1}^{2}+z_{3}^{2}+2 z_{2}^{2}=2\left(z_{1}+z_{3}\right) z_{2}\) (B) \(z_{1}^{2}+z_{3}^{2}=2 z_{2}\left(z_{1}+z_{3}-z_{2}\right)\) (C) \(\left(z_{1}-z_{2}\right)^{2}+\left(z_{2}-z_{3}\right)^{2}=0\) (D) \(\frac{z_{1}-z_{2}}{z_{2}-z_{3}}\) is imaginary

\(A, B, C\) are the points representing the complex numbers \(z_{1}, z_{2}, z_{3}\), respectively on the complex plane and the circumcentre of the triangle \(A B C\) lies at the origin. If the altitude \(A D\) of the triangle \(A B C\) meets circumcircle again at \(P\), then \(P\) represents the \(\begin{array}{ll}\text { complex number } \\ \text { (A) }-\bar{z}_{1} z_{2} z_{3} & \text { (B) }-\frac{\bar{z}_{1} z_{2}}{\bar{z}_{3}}\end{array}\) (C) \(-\frac{\bar{z}_{1} z_{3}}{\bar{z}_{2}}\) (D) \(-\frac{z_{2} z_{3}}{z_{1}}\)

\(A, B, C\) are the points representing the complex numbers \(z_{1}, z_{2}, z_{3}\), respectively on the complex plane and the circumcentre of the triangle \(A B C\) lies at the origin. If the altitude \(A D\) of the triangle \(A B C\) meets circumcircle again at \(P\), then \(P\) represents the \(\begin{array}{ll}\text { complex number } \\ \text { (A) }-\bar{z}_{1} z_{2} z_{3} & \text { (B) }-\frac{\bar{z}_{1} z_{2}}{\bar{z}_{3}}\end{array}\) (C) \(-\frac{\bar{z}_{1} z_{3}}{\bar{z}_{2}}\) (D) \(-\frac{z_{2} z_{3}}{z_{1}}\)

If \(f(x)\) and \(g(x)\) are two polynomials such that the polynomial \(h(x)=x f\left(x^{3}\right)+x^{2} g\left(x^{6}\right)\) is divisible by \(x^{2}+x+1\), then (A) \(f(1)=g(1)\) (B) \(f(1)=-g(1)\) (C) \(h(1)=0\) (D) \(h(-1)=0\)

If \(\alpha\) is the fifth root of unity, then (A) \(\left|1+\alpha+\alpha^{2}+\alpha^{3}+\alpha^{4}\right|=0\) (B) \(\left|1+\alpha+\alpha^{2}+\alpha^{3}\right|=1\) (C) \(\left|1+\alpha+\alpha^{2}\right|=2 \cos \frac{\pi}{5}\) (D) \(|1+\alpha|=2 \cos \frac{\pi}{10}\)

The value of \(\sum_{r=1}^{16}\left(\sin \frac{2 r \pi}{17}+i \cos \frac{2 r \pi}{17}\right)\) is (A) 1 (B) \(-1\) (C) \(\vec{i}\) (D) \(-i\)

One of the values of \((a+i b)^{m / n}+(a-i b)^{m / n}\) is (A) \(2\left(a^{2}+b^{2}\right)^{m / n} \cos \left(\frac{m}{n} \tan ^{-1} \frac{b}{a}\right)\) (B) \(2\left(a^{2}+b^{2}\right)^{m / 2 n} \cos \left(\frac{m}{n} \tan ^{-1} \frac{b}{a}\right)\)

The values of \((16)^{1 / 4}\) are (A) \(\pm 2, \pm 2 i\) (B) \(\pm 4, \pm 4 i\) (C) \(\pm 1, \pm i\) (D) None of these

The roots of the equation \(z^{4}+1=0\) are (A) \((\pm 1 \pm i)\) (B) \((\pm 2 \pm 2 i)\) (C) \(\frac{1}{\sqrt{2}}(\pm 1 \pm i)\) (D) None of these

The roots of the equation \(z^{4}+1=0\) are (A) \((\pm 1 \pm i)\) (B) \((\pm 2 \pm 2 i)\) (C) \(\frac{1}{\sqrt{2}}(\pm 1 \pm i)\) (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 \(1, a_{1}, a_{2}, \ldots, a_{n-1}\) 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.