IIT JEE Super Course in Mathematics ALGEBRA II

Trishna Knowledge Systems

Chapter 1

Complex Numbers - all with Video Answers

Educators

Chapter Questions

Problem 1

Find the value of $x$ and $y$ in the following
(i) $(2+3 i)(4-7 i)=x+i y$
(ii) $\frac{(2 x+4 i)(1+i)}{(3-2 i)}=2+i$
(iii) $(1-3 i)^{2}=x+i y$
(iv) $x+i 5 y=i x+y+7$
(v) $\frac{1}{2-\sqrt{3} \mathrm{i}}+\frac{1}{2+\sqrt{3} \mathrm{i}}=\mathrm{x}+\mathrm{iy}$

Anas Venkitta

Anas Venkitta

Numerade Educator

Problem 2

If $z_{1}=1+i, z_{2}=3 i, z_{3}=\sqrt{2}-i$, then find
(i) $\frac{z_{1} \bar{z}_{2}}{\bar{z}_{3}}+\frac{z_{2} \bar{z}_{3}}{z_{1}}$
(ii) $\mathrm{z}_{1} \overline{\mathrm{z}}_{2} \overline{\mathrm{z}}_{3}+\overline{\mathrm{z}}_{1} \mathrm{z}_{2} \mathrm{z}_{3}$
(iii) $z_{1}^{2}-2 z_{1}+2$
(iv) $\left|z_{1} z_{2}+z_{1} z_{3}+z_{2} z_{3}\right|$
(v) $\left(z_{1} z_{2} z_{3}\right)\left(\overline{z_{1} z_{2} z_{3}}\right)$

Anas Venkitta

Numerade Educator

Problem 3

Express in the Modulus-amplitude form
(i) $-\mathrm{i}$
(ii) $\sqrt{3}-\mathrm{i}$
(iii) $(1+i)(-1+i \sqrt{3})$
(iv) $\frac{(1-i)(1+\sqrt{3} i)}{(1+i)}$
(v) $\frac{1+7 i}{(2-i)^{2}}$

Anas Venkitta

Numerade Educator

Problem 4

If $z_{1}, z_{2}, z_{y}, \ldots .$ is a sequence of complex numbers defined by $z_{n}=\sum_{k=0}^{n} i^{k} .$ Then prove that $z_{100}+z_{101}+z_{102}+z_{103}=$ $2(1+i)$

Anas Venkitta

Numerade Educator

Problem 5

Find $z$ (if it exists) satisfying the relations $\left|\frac{z+6}{z+4 i}\right|=\frac{5}{3}$ and $\left|\frac{z+2}{z+4}\right|=1$.

Anas Venkitta

Numerade Educator

Problem 6

If $(1+z)^{n}=p_{0}+p_{1} z+p_{2} z^{2}+\ldots+p_{n} z^{n}$, show that
(i) $\mathrm{P}_{0}-\mathrm{P}_{2}+\mathrm{P}_{4}-+\ldots=2^{\mathrm{n} / 2} \cos \frac{\mathrm{n} \pi}{4}$
(ii) $\mathrm{p}_{1}-\mathrm{p}_{3}+\mathrm{P}_{5}-+\ldots=2^{\mathrm{n} / 2} \sin \frac{\mathrm{n} \pi}{4}$
(iii) $\mathrm{p}_{0}+\mathrm{P}_{4}+\mathrm{p}_{8}+\ldots=2^{(\mathrm{n} / 2)-1} \cos \frac{\mathrm{n} \pi}{4}+2^{\mathrm{n}-2}$

Uma Kumari

Numerade Educator

Problem 7

If $\omega$ is a complex cube root of unity, show that
(i) $\left(1-\omega+\omega^{2}\right)\left(1-\omega^{2}+\omega^{4}\right)\left(1-\omega^{4}+\omega^{8}\right) \ldots \ldots \ldots 2 n$ factors $=2^{2 n}$
(ii) $(a+b+c)\left(a+b \omega+c \omega^{2}\right)\left(a+b \omega^{2}+c \omega\right)=a^{3}+b^{3}+c^{3}-3 a b c$
where $a, b, c$ are real numbers
(iii) $\left(\mathrm{k}+\omega+\omega^{2}\right)\left(\mathrm{k}+\omega^{2}+\omega^{4}\right)\left(\mathrm{k}+\omega^{4}+\omega^{8}\right) \ldots 4 \mathrm{n}$ factors $=(\mathrm{k}-1)^{\mathrm{tm}}$
where $\mathrm{k}$ is a real number
(iv) $1+\omega^{n}+\omega^{2 n}=\left\{\begin{array}{l}0 \text { when } n \text { is not a multiple of } 3 \\ 3 \text { when } n \text { is a multiple of } 3\end{array}\right.$

Uma Kumari

Numerade Educator

Problem 8

Solve the equations
(i) $(x-1)^{3}+64=0$
(ii) $(2 z-1)^{4}+(z+2)^{4}=0$.

Uma Kumari

Numerade Educator

Problem 9

Solve the equation $x^{11}-1=0 .$ Deduce the value of
$\sin \frac{\pi}{11} \sin \frac{2 \pi}{11} \sin \frac{3 \pi}{11} \sin \frac{4 \pi}{11} \sin \frac{5 \pi}{11}$

Uma Kumari

Numerade Educator

Problem 10

If $\mathrm{z}_{1}, \mathrm{z}_{2}, \mathrm{z}_{3}$ are non-zero complex numbers such that $\frac{2}{\mathrm{z}_{1}}=\frac{1}{\mathrm{z}_{2}}+\frac{1}{\mathrm{z}_{3}}$ prove that $\mathrm{z}_{1}, \mathrm{z}_{2}, \mathrm{z}_{3}$ lie on a circle passing through the origin.

Uma Kumari

Numerade Educator

Problem 11

If $(x+i y)(3-4 i)=5+12 i$, then $|x+i y|$ is equal to
(a) 65
(b) $5 / 3$
(c) $13 / 5$
(d) 18

Anas Venkitta

Numerade Educator

Problem 12

If $\sqrt{(x+i y)}=\pm(a+i b)$, then $\sqrt{-x-i y}$ is equal to
(a) $\pm(\mathrm{b}+\mathrm{ia})$
(b) $\pm(a-i b)$
(c) $\pm(\mathrm{ai}-\mathrm{b})$
(d) $\pm(-b-i a)$

Anas Venkitta

Numerade Educator

Problem 13

The value of $\left(\sin \frac{\pi}{6}+i \cos \frac{\pi}{6}\right)^{7}$ is
(a) $\left(\cos \frac{\pi}{6}+\mathrm{i} \sin \frac{\pi}{6}\right)$
(b) $\left(\cos \frac{\pi}{6}-\mathrm{i} \sin \frac{\pi}{6}\right)$
(c) $\left(\sin \frac{\pi}{6}+i \cos \frac{\pi}{6}\right)$
(d) $\left(\sin \frac{\pi}{6}-i \cos \frac{\pi}{6}\right)$

Anas Venkitta

Numerade Educator

Problem 14

If $\omega$ is a complex cube root of unity, then $\frac{1+5 \omega+9 \omega^{2}}{\omega^{2}+5+9 \omega}+\frac{2+3 \omega+5 \omega^{2}}{5+2 \omega+3 \omega^{2}}$ is equal to
(a) $-1$
(b) 20
(c) 0
(d) $-2 \omega$

Anas Venkitta

Numerade Educator

Problem 15

The value of $\log (-\mathrm{i})$ is
(a) $-\frac{\pi}{2}$
(b) $\frac{\pi}{2}$
(c) $-\mathrm{i} \frac{\pi}{2}$
(d) $\mathrm{i} \frac{\pi}{2}$

Anas Venkitta

Numerade Educator

Problem 16

Statement 1
For any non-zero complex number arg $z+$ arg $\bar{z}=\pi$. and
Statement 2
$\arg \left(z_{1} z_{2}\right)=\arg z_{1}+\arg z_{2}$

Uma Kumari

Numerade Educator

Problem 17

Statement 1
If $z_{r}=\cos \frac{\pi}{2^{r}}+i \sin \frac{\pi}{2^{r}}$ then $z_{1}, z_{2}, z_{3} \ldots \ldots ., \infty$ is equal to $-1$.
and
Statement 2
The argument of a product of complex numbers is equal to the sum of the arguments of the factors.

Uma Kumari

Numerade Educator

Problem 18

Let $z=\sin 2 \theta+i(1+\cos 2 \theta)$
Statement 1 $|z|=|2 \cos \theta|$
and
Statement 2
Modulus of a complex number $\mathrm{z}=\mathrm{r}(\sin \theta+\mathrm{i} \cos \theta)$ is $|\mathrm{r}|$

Uma Kumari

Numerade Educator

Problem 19

Statement 1 The roots of the equation $x^{4}+x^{2}+1=0$ are $\pm \omega$ and $\pm \frac{1}{\omega}$ where $\omega$ is a complex cube root of unity. and
Statement 2 If the product of the roots of the equation $a x^{4}+b x^{3}+c x^{2}+d x+e=0$ is 1, then the roots of the equation are of the form $\alpha, \frac{1}{\alpha}, \beta, \frac{1}{\beta}$

Uma Kumari

Numerade Educator

Problem 20

Statement 1
Let $z=x+$ iy and $w=u+$ iv where, $x, y, u, v$ are real If $w=3 i+z$ and $z$ moves along a straight line, then, $w$ also will move along a straight line. and
Statement 2
arg $\mathrm{z}=\alpha$ represents a straight line

Uma Kumari

Numerade Educator

Problem 21

Real part of $\log \left(\frac{1-\mathrm{i}}{1+\mathrm{i}}\right)$ is
(a) 1
(b) 0
(c) $\frac{1}{2}$
(d) $\frac{1}{4}$

Anas Venkitta

Numerade Educator

Problem 22

If $\mathrm{i}^{r^{\prime}}(\mathrm{A}, \mathrm{B}$ are real $), \mathrm{A}^{2}+\mathrm{B}^{2}=$
(a) $\mathrm{e}^{\mathrm{nB}}$
(b) $\mathrm{e}^{\pi}$
(c) $\mathrm{e}^{-\pi \overline{8}}$
(d) $\mathrm{e}^{-\mathrm{n} / 2}$

Anas Venkitta

Numerade Educator

Problem 23

Real part of $\mathrm{i}^{\log (1+1)}$ is
(a) $\mathrm{e}^{\pi^{2} / 8} \cos (\log 2)$
(b) $\mathrm{e}^{-\pi / 8}$
(c) $\mathrm{e}^{\mathrm{x} / \mathrm{s}}$
(d) $e^{-\pi^{2} / 8} \cos \left(\frac{\pi}{4} \log 2\right)$

Anas Venkitta

Numerade Educator

Problem 24

$\cos 6 \theta=$
(a) $32 \cos ^{6} \theta+48 \cos ^{\prime} \theta-18 \cos ^{2} \theta-1$
(b) $32 \cos ^{6} \theta-48 \cos ^{4} \theta+18 \cos ^{2} \theta-1$
(c) $32 \cos ^{6} \theta+48 \cos ^{4} \theta+8 \cos ^{2} \theta-1$
(d) None of the above

Uma Kumari

Numerade Educator

Problem 25

$\sin 7 \theta=$
(a) $\sin ^{7} \theta-35 \sin ^{5} \theta \cos ^{2} \theta+21 \sin ^{3} \theta \cos ^{4} \theta-\cos ^{7} \theta$
(b) $\sin ^{6} \theta \cos \theta-35 \sin ^{5} \theta \cos ^{2} \theta+21 \sin ^{4} \theta \cos ^{3} \theta-\cos ^{7} \theta$
(c) $7 \cos ^{7} \theta-35 \cos ^{4} \theta \sin ^{3} \theta+21 \cos ^{2} \theta \sin ^{5} \theta+\sin ^{7} \theta$
(d) $(\sin \theta)\left[7 \cos ^{6} \theta-35 \cos ^{4} \theta \sin ^{2} \theta+21 \cos ^{2} \theta \sin ^{4} \theta-\sin ^{6} \theta\right]$

Uma Kumari

Numerade Educator

Problem 26

$\cos ^{6} \theta=$
(a) $\frac{1}{32}(\cos 6 \theta+8 \cos 4 \theta+12 \cos 2 \theta-5)$
(b) $\frac{1}{32}(\cos 6 \theta+6 \cos 4 \theta+15 \cos 2 \theta+10)$
(c) $\frac{1}{32}(3 \cos 6 \theta+6 \cos 4 \theta-15 \cos 2 \theta+10)$
(d) None of these

Uma Kumari

Numerade Educator

Problem 27

If $\arg \left(z_{1} z_{2}\right)=0$ and $\left|z_{1}\right|=\left|z_{2}\right|=1$ then
(a) $\mathrm{z}_{1}+\mathrm{z}_{2}=0$
(b) $z_{1} z_{2}=1$
(c) $\mathrm{z}_{1}=\overline{\mathrm{z}}_{2}$
(d) $\arg z_{1}=\arg \bar{z}_{2}$

Uma Kumari

Numerade Educator

Problem 28

Let $\frac{2-3 z_{1} z_{2}}{3 z_{1}-2 z_{2}}$ be a point lying inside the circle $|z|=1$ for any two complex numbers $z_{1}$ and $z_{z^{\prime}}$. Then
(a) $\mathrm{z}_{1}$ lies inside $|\mathrm{z}|=\frac{2}{3}$ and $\mathrm{z}_{2}$ inside $|\mathrm{z}|=1$
(b) $\mathrm{z}_{1}$ lies inside $|\mathrm{z}|=\frac{2}{3}$ and $\mathrm{z}_{2}$ outside $|\mathrm{z}|=1$
(c) $\mathrm{z}_{1}$ lies outside $|\mathrm{z}|=\frac{2}{3}$ and $\mathrm{z}_{2}$ inside $|\mathrm{z}|=1$
(d) $\mathrm{z}_{1}$ lies outside $|\mathrm{z}|=\frac{2}{3}$ and $\mathrm{z}_{2}$ outside $|\mathrm{z}|=1$

Uma Kumari

Numerade Educator

Problem 29

If $1, \alpha_{1}, \alpha_{2}, \ldots \ldots, \alpha_{\mathrm{n}-1}$ are the nth roots of unity and if $\omega$ is a non real 5 th root of unity then $\left(\omega-\alpha_{1}\right)\left(\omega-\alpha_{2}\right) \ldots \ldots$ $\left(\omega-\alpha_{n-1}\right)$ is not non-real provided $n$ is of the form
(a) $5 \mathrm{~K}+2$
(b) $5 \mathrm{~K}+1$
(c) $5 \mathrm{~K}$
(d) $5 \mathrm{~K}+3$

Uma Kumari

Numerade Educator

Problem 30

Column I $\quad$ Column II
(a) Minimum value of $|z-2+i|+|z-2 i+1|$ is
(p) 6
(b) Minimum value of $\left|z_{1}-z_{2}\right|$ where $\left|z_{1}+1-i\right|=13$
(q) 5 and $\left|z_{2}-3-4 i\right|=2$ is
(c) Possible value of $\alpha$ so that $\left|z-\alpha^{2}\right|+|z-4 \alpha|=5$
(r) 4 represents an ellipse is
(d) Maximum value of $|z|$ if $\left|z+\frac{4}{z}\right|=3$ is
(s) $3 \sqrt{2}$

Uma Kumari

Numerade Educator

Problem 31

The conjugate of $\frac{2-i}{2+i}$ is
(a) $\frac{3+4 \mathrm{i}}{6}$
(b) $\frac{3+4 \mathrm{i}}{5}$
(c) $\frac{2+3 \mathrm{i}}{5}$
(d) $\frac{3-4 \mathrm{i}}{5}$

Anas Venkitta

Numerade Educator

Problem 32

$\operatorname{Im}(\mathrm{z})$ is equal to
(a) $\frac{1}{2 \mathrm{i}}(\mathrm{z}+\overline{\mathrm{z}})$
(b) $\frac{1}{2}(z-\bar{z})$
(c) $\frac{1}{2}(z+\bar{z})$
(d) $\frac{1}{2 \mathrm{i}}(z-\bar{z})$

Anas Venkitta

Numerade Educator

Problem 33

$(-4+\sqrt{48} \mathrm{i})^{2}+(-4-\sqrt{48} \mathrm{i})^{2}=$
(a) 64
(b) 32
(c) $-64$
(d) $-16$

Ekaveera Kumar

Numerade Educator

Problem 34

$\sqrt{-28} \times \sqrt{-7} \times-3=$
(a) $-42$
(b) 42
(c) $-56$
(d) 56

Ekaveera Kumar

Numerade Educator

Problem 35

If $\mathrm{k}, \mathrm{l}, \mathrm{m}$ and $\mathrm{n}$ are four consecutive integers, then $\mathrm{i}^{\mathrm{k}}+\mathrm{i}^{1}+\mathrm{i}^{\mathrm{m}}+\mathrm{i}^{\mathrm{n}}$ is equal to
(a) 1
(b) 0
(c) 2
(d) 4

Ekaveera Kumar

Numerade Educator

Problem 36

The smallest positive integral value of $\mathrm{n}$ for which $\left(\frac{1-\mathrm{i}}{1+\mathrm{i}}\right)^{n}$ is purely imaginary with positive imaginary part is
(a) 2
(b) 3
(c) 4
(d) 5

Ekaveera Kumar

Numerade Educator

Problem 37

If $x, a, b$ are real numbers and $\frac{1-i x}{1+i x}=a-i b$, then $a^{2}+b^{2}$ is
(a) $2\left(1+x^{2}\right)$
(b) $2\left(1-x^{2}\right)$
(c) 2
(d) 1

Anas Venkitta

Numerade Educator

Problem 38

Argument of $\frac{1+i \sqrt{3}}{1+i}$ is
(a) $\frac{\pi}{12}$
(b) $\frac{\pi}{4}$
(c) $\frac{\pi}{6}$
(d) $\frac{\pi}{3}$

Anas Venkitta

Numerade Educator

Problem 39

If $z$ is any complex number satisfying $|z-1|=1$, then which of the following is correct?
(a) $\arg (z-1)=2 \arg (z)$
(b) $2 \arg (z)=\frac{2}{3} \arg \left(z^{3}-4\right)$
(c) $\arg (z+1)=\arg (z-1)$
(d) $2 \arg (z+1)=\arg (z-1)$

Uma Kumari

Numerade Educator

Problem 40

The value of $2\left[\left(\cos 50^{\circ}+\mathrm{i} \sin 50^{\circ}\right) \times\left(\cos 40^{\circ}+\mathrm{i} \sin 40^{\circ}\right)\right]$ is
(a) $3+2 \mathrm{i}$
(b) $(1-i)$
(c) 5i
(d) 2i

Anas Venkitta

Numerade Educator

Problem 41

Given $x=-2+7 i$, the value of $x^{3}+4 x^{2}+53 x+5$ is
(a) 51
(b) 7
(c) 5
(d) 0

Ekaveera Kumar

Numerade Educator

Problem 42

If $\mathrm{i} z^{3}+\mathrm{z}^{2}-\mathrm{z}+\mathrm{i}=0$ then $|\mathrm{z}|$ is
(a) 1
(b) 2
(c) 0
(d) None of these

Anas Venkitta

Numerade Educator

Problem 43

The complex numbers $z_{1}, z_{2}, z_{3}$ and $z_{4}$ represent the vertices of a parallelogram taken in order if and only if
(a) $\mathrm{z}_{1}+\mathrm{z}_{3}=\mathrm{z}_{2}+\mathrm{z}_{4}$
(b) $\mathrm{z}_{1}+\mathrm{z}_{2}+\mathrm{z}_{3}=\mathrm{z}_{4}$
(c) $\mathrm{z}_{1}+\mathrm{z}_{2}=\mathrm{z}_{3}+\mathrm{z}_{4}$
(d) $\mathrm{z}_{1}-\mathrm{z}_{2}=\mathrm{z}_{3}-\mathrm{z}_{4}$

Uma Kumari

Numerade Educator

Problem 44

The points $5 \pm 2 \mathrm{i},-5 \pm 2 \mathrm{i}$
(a) lie on a circle
(b) are the vertices of a square
(c) are the vertices of a rectangle
(d) lie on an ellipse

Anas Venkitta

Numerade Educator

Problem 45

$\mathrm{i}=\sqrt{-1}$, then $4+5\left[\frac{-1}{2}+\frac{\mathrm{i} \sqrt{3}}{2}\right]^{334}+3\left[\frac{-1}{2}+\frac{\sqrt{3} \mathrm{i}}{2}\right]^{3,35}$ is equal to
(a) $1-\mathrm{i} \sqrt{3}$
(b) $-1+\mathrm{i} \sqrt{3}$
(c) $\mathrm{i} \sqrt{3}$
(d) $-\mathrm{i} \sqrt{3}$

Anas Venkitta

Numerade Educator

Problem 46

If $\omega$ is a complex cube root of unity, then the value of $\sin \left[\left(\omega^{10}+\omega^{26}\right) \pi-\frac{\pi}{4}\right]$ is
(a) $\frac{-\sqrt{3}}{2}$
(b) $\frac{-1}{\sqrt{2}}$
(c) $\frac{1}{\sqrt{3}}$
(d) $\frac{1}{\sqrt{2}}$

Anas Venkitta

Numerade Educator

Problem 47

If $x^{2}-x+1=0$, the value of $\sum_{n=1}^{5}\left(x^{n}+\frac{1}{x^{n}}\right)^{2}$ is
(a) 8
(b) 10
(c) 12
(d) 5

Anas Venkitta

Numerade Educator

Problem 48

If $\mathrm{x}=\mathrm{a}+\mathrm{b}, \mathrm{y}=\mathrm{a} \omega+\mathrm{b} \omega^{2}$ and $z=\mathrm{a} \omega^{2}+\mathrm{b} \omega$, the value of $\mathrm{x}^{3}+\mathrm{y}^{3}+\mathrm{z}^{3}$ is equal to
(a) $a^{3}+b^{3}$
(b) 0
(c) $3\left(a^{3}+b^{3}\right)$
(d) $\left(a^{2}-a b+b^{2}\right)$

Anas Venkitta

Numerade Educator

Problem 49

If $\omega$ is a non-real cube root of unity then $(1+\omega)\left(1+\omega^{2}\right)\left(1+\omega^{4}\right)\left(1+\omega^{8}\right)\left(1+\omega^{10}\right)\left(1+\omega^{32}\right)$ is equal to
(a) 1
(b) $-1$
(c) 32
(d) 64

Anas Venkitta

Numerade Educator

Problem 50

If $\omega$ is a complex cube root of unity, then $\cos \left\{\left[(1-\omega)\left(1-\omega^{2}\right)+(2-\omega)\left(2-\omega^{2}\right)+(3-\omega)\left(3-\omega^{2}\right)+(4-\omega)\left(4-\omega^{2}\right)\right.\right.$
$\left.\left.+(5-\omega)\left(5-\omega^{2}\right)\right] \frac{2 \pi}{75}\right\}$ equals
(a) 0
(b) 1
(c) $-1$
(d) $\frac{1}{2}$

Anas Venkitta

Numerade Educator

Problem 51

If $\mathrm{z}_{1}=5+10 \mathrm{i}, \mathrm{z}_{2}=3+8 \mathrm{i}$ and $\mathrm{z}_{3}=1+2 \mathrm{i}$ are three complex numbers, then they represent the vertices of
(a) an isosceles triangle
(b) a right angled triangle
(c) an equilateral triangle
(d) a scalene triangle

Uma Kumari

Numerade Educator

Problem 52

If $3^{19}[x+i y]=\left[\frac{3}{2}+\frac{\sqrt{3}}{2} i\right]$ and $x=k y$, then $k$ equals
(a) $\frac{1}{\sqrt{3}}$
(b) $\sqrt{3}$
(c) $\frac{-1}{\sqrt{3}}$
(d) $-\sqrt{3}$

Anas Venkitta

Numerade Educator

Problem 53

If $z$ is a non-real root of $z^{7}+1=0$, then $z^{86}+z^{175}+z^{289}$ is equal to
(a) 0
(b) $-1$
(c) 3
(d) 1

Anas Venkitta

Numerade Educator

Problem 54

The value of $(1+i)^{3}+(1-i)^{6}$ is
(a) $10-2 \mathrm{i}$
(b) 2
(c) $\frac{(1-5 i)}{\sqrt{2}}$
(d) $-2+10 \mathrm{i}$

Anas Venkitta

Numerade Educator

Problem 55

If $(\alpha+\mathrm{i} \beta)=\log (\mathrm{x}+\mathrm{iy})$ then $\alpha$ is
(a) $\frac{1}{2} \log \left(x^{2}+y^{2}\right)$
(b) $\sqrt{x^{2}+y^{2}}$
(c) $\log \left(x^{2}+y^{2}\right)$
(d) $\log x$

Uma Kumari

Numerade Educator

Problem 56

If $z_{1}, z_{2}$ and $z_{3}$ are complex numbers such that $\left|z_{1}\right|=\left|z_{2}\right|=\left|z_{3}\right|=1$ and $z_{1}+z_{2}+z_{3}=0$, then $\left|\frac{1}{z_{1}}+\frac{1}{z_{2}}+\frac{1}{z_{3}}\right|$ is
(a) equal to 0
(b) equal to 1
(c) greater than 1 and less than 3
(d) equal to 3

Uma Kumari

Numerade Educator

Problem 57

The equation $z \bar{z}+a \bar{z}+\bar{a} z+b=0, b \in R$ represents a circle if
(a) $\left|\mathrm{a}^{2}\right|=\mathrm{b}$
(b) $\left|\mathrm{a}^{2}\right|>\mathrm{b}$
(c) $\left|\mathrm{a}^{2}\right|<\mathrm{b}$
(d) None of these

Uma Kumari

Numerade Educator

Problem 58

If $\cos \alpha+\cos \beta+\cos \gamma=\sin \alpha+\sin \beta+\sin \gamma=0$ then $\cos (\beta+\gamma)+\cos (\alpha+\beta)+\cos (\alpha+\gamma)$ is equal to
(a) 1
(b) 0
(c) $-1$
(d) $\frac{1}{2}$

Uma Kumari

Numerade Educator

Problem 59

If $|z|=3$ then the points representing $-2+4 z$ lie on
(a) the circle of radius 4 and centre at $(-2,0)$
(b) the circle of radius 5 and centre at origin
(c) the circle of radius 12 and centre at $(-2,0)$
(d) the circle passing through the points $(-2,0),(2,0)$ and $(0,2)$

Uma Kumari

Numerade Educator

Problem 60

If $a=\frac{\sqrt{3}+\mathrm{i}}{2}$, then the value of $1+\mathrm{a}^{3}+\mathrm{a}^{6}+\mathrm{a}^{9}+\mathrm{a}^{12}+\mathrm{a}^{15} \mathrm{is}$
(a) 1
(b) 0
(c) $1+\mathrm{i}$
(d) $\frac{\sqrt{3}+\mathrm{i}}{2}$

Anas Venkitta

Numerade Educator

Problem 61

The minimum value of $\mathrm{n}$, so that the two nth roots of unity subtend an angle $\frac{\pi}{6}$ at the centre is
(a) 36
(b) 3
(c) 6
(d) 12

Uma Kumari

Numerade Educator

Problem 62

The points representing the complex number $z$ in the Argand plane such that $|z|=2$ and $|z-1-i|-|z+1+i|=0$ are
(a) $\pm 2 \mathrm{i}$
(b) $\pm \sqrt{2}(1+\mathrm{i})$
(c) $\pm \sqrt{2}(-1+i)$
(d) $\pm 2$

Uma Kumari

Numerade Educator

Problem 63

The complex numbers $\mathrm{z}_{1}, \mathrm{z}_{2}$ and $\mathrm{z}_{3}$ satisfying $\frac{\mathrm{z}_{1}-\mathrm{z}_{3}}{\mathrm{z}_{2}-\mathrm{z}_{3}}=\frac{1-\mathrm{i} \sqrt{3}}{2}$ are the vertices of a triangle which is
(a) a right angled isosceles triangle
(b) an equilateral triangle
(c) an obtuse angled isosceles triangle
(d) None of the above

Uma Kumari

Numerade Educator

Problem 64

If $\left|z+\frac{9}{z}\right|=6$, then the greatest value of $|z|$ is
(a) 3
(b) $3+\sqrt{18}$
(c) $3-\sqrt{18}$
(d) $6+\sqrt{8}$

Anas Venkitta

Numerade Educator

Problem 65

If $\left|z_{1}\right|=\left|z_{2}\right|$ and $\arg \left(z_{1}\right)+\arg \left(z_{2}\right)=0$, then.
(a) $\mathrm{z}_{2}=\mathrm{z}_{1}$
(b) $\mathrm{z}_{1} \mathrm{z}_{2}$ is purely imaginary
(c) $\mathrm{z}_{1} \mathrm{z}_{2}=1$
(d) $\mathrm{z}_{1} \mathrm{z}_{2}$ is real

Uma Kumari

Numerade Educator

Problem 66

The value of $\frac{(-1+i \sqrt{3})^{15}}{(1-i)^{20}}+\frac{(-1-i \sqrt{3})^{15}}{(1+i)^{20}}$ is
(a) 32
(b) 64
(c) $-64$
(d) $-32$

Anas Venkitta

Numerade Educator

Problem 67

If $\mathrm{x}=\mathrm{i}^{i}$ where $\mathrm{i}=\sqrt{-1}$, then
(a) $x<0$
(b) $0<x<1$
(c) $1<\mathrm{x}<\mathrm{e}$
(d) $\mathrm{e}<\mathrm{x}<\pi$

Anas Venkitta

Numerade Educator

Problem 68

Let a and $\mathrm{b} \in \mathrm{R}$ such that $0<\mathrm{a}<1,0<\mathrm{b}<1$. The values of $\mathrm{a}$ and $\mathrm{b}$ such that the complex numbers $\mathrm{z}_{1}=-\mathrm{a}+\mathrm{i}, \mathrm{z}_{2}=-1$ $+\mathrm{bi}$ and $\mathrm{z}_{\mathrm{a}}=0$ form an equilateral triangle are
(a) $a=b=2-\sqrt{3}$
(b) $a=2-\sqrt{3}, b=2+\sqrt{3}$
(c) $a=\sqrt{3}, b=-\sqrt{3}$
(d) $2,2 \sqrt{3}$

Uma Kumari

Numerade Educator

Problem 69

In the argand plane, a vector $\overline{\mathrm{OA}}$, where $\mathrm{O}$ represents the origin and $\mathrm{A}$ represents the complex number $(1+2 \mathrm{i})$ is turned in the clockwise direction through an angle $\frac{\pi}{4}$ and then stretched $\sqrt{2}$ times. The complex number represent-
ing the new vector is
$\begin{array}{llll}\text { (a) }(3+\mathrm{i}) & \text { (b) }(\sqrt{2}(1+\sqrt{3} \mathrm{i}) & \text { (c) } \sqrt{2}(-2+\mathrm{i}) & \text { (d) } \frac{\sqrt{2}}{2}(1+\sqrt{3} \mathrm{i})\end{array}$

Uma Kumari

Numerade Educator

Problem 70

If $z_{k}=\cos \frac{\pi}{5^{k}}+i \sin \frac{\pi}{5^{k}} \quad k=1,2,3, \ldots$ then $z_{1} z_{2} z_{y}, \ldots \infty$ is
(a) i
(b) 0
(c) $1+\mathrm{i}$
(d) $\frac{(1+\mathrm{i})}{\sqrt{2}}$

Uma Kumari

Numerade Educator

Problem 71

The value of $\sum_{k=0}^{10}\left(\sin \frac{2 \mathrm{k} \pi}{10}-\mathrm{i} \cos \frac{2 \mathrm{k} \pi}{10}\right)$ is
(a) 0
(b) i
(c) $-1$
(d) $-\mathrm{i}$

Uma Kumari

Numerade Educator

Problem 72

Given that real parts of $\sqrt{5+12 \mathrm{i}}$ and $\sqrt{5-12 \mathrm{i}}$ are negative, the number $\frac{\sqrt{5+12 \mathrm{i}+\sqrt{5-12 \mathrm{i}}}}{\sqrt{5+12 \mathrm{i}}-\sqrt{5-12 \mathrm{i}}}$ reduces to
(a) $\frac{3}{2} \mathrm{i}$
(b) $-\frac{3}{2} \mathrm{i}$
(c) $-3+\frac{2}{5} i$
(d) None of these

Uma Kumari

Numerade Educator

Problem 73

The value of $\sum_{k=0}^{10}\left(\sin \frac{2 k \pi}{10}-i \cos \frac{2 k \pi}{10}\right)$ is
(a) 0
(b) i
(c) $-1$
(d) $-\mathrm{i}$

Uma Kumari

Numerade Educator

Problem 74

$\ln (\ln (\cos \mathrm{e}+\mathrm{i} \sin \mathrm{e}))$ is equal to
(a) e
(b) $1+\frac{\pi}{2} \mathrm{i}$
(c) $\mathrm{e} \times \frac{\pi}{2} \times \mathrm{i}$
(d) $\mathrm{e}+\frac{\pi}{2} \mathrm{i}$

Uma Kumari

Numerade Educator

Problem 75

If $\mathrm{z}_{1} \mathrm{z}_{2}$ and $\mathrm{z}_{3}$ are the roots of the equation $\mathrm{az}^{3}+\mathrm{bz}^{2}+\mathrm{cz}+\mathrm{d}=0$ such that $\mathrm{z}_{1}, \mathrm{z}_{2}, \mathrm{z}_{3}$ form the vertices of form an equilateral triangle, then
(a) $b^{2}=4 a c$
(b) $b^{2}=3 a c$
(c) $(b+c)^{2}=2 a d$
(d) $\quad b^{2}=3 \operatorname{acd}$

Uma Kumari

Numerade Educator

Problem 76

If the points $\mathrm{A}$ and $\mathrm{B}$ represent the complex numbers $\frac{1}{2}+3 \mathrm{i}$ and $-3+\frac{1}{2} \mathrm{i}$ in the Argand plane, the area of the triangle OAB where $\mathrm{O}$ is the origin, is
(a) $\frac{1}{4}$
(b) $\frac{37}{4}$
(c) $\frac{\sqrt{37}}{3}$
(d) $\frac{37}{0}$

Uma Kumari

Numerade Educator

Problem 77

If $(\sqrt{3}+\mathrm{i})^{x}=2^{x}$, the non-zero solution for $x$ is
(a) 5
(b) 12
(c) 0
(d) 4

Uma Kumari

Numerade Educator

Problem 78

The points representing the values of $(8+6 \mathrm{i})^{113}$
(i) lie on the circle whose centre is at $(0,0)$ and radius $\sqrt[3]{10}$
(ii) lie on a straight line passing through the origin
(iii) are the vertices of an equilateral triangle
(iv) are the vertices of a triangle having circum centre at $(0,0)$ Then, out of the above statements
(a) (i) and (ii) are true
(b) (i), (iii) and (iv) are true
(c) (i), (ii) and (iii) are true
(d) all the statements are true

Uma Kumari

Numerade Educator

Problem 79

The length of the tangent segment drawn from the point represented by $-1+\mathrm{i}$ to the circle $|\mathrm{z}-(3+4 \mathrm{i})|=2$ is
(a) $\sqrt{21}$
(b) $5+\sqrt{2}$
(c) $7-\sqrt{2}$
(d) $\sqrt{24}$

Uma Kumari

Numerade Educator

Problem 80

The complex number on $|z+3-3 i|=2$ having least absolute value is
(a) $(-1+i)$
(b) $\frac{-1+\mathrm{i}}{\sqrt{2}}$
(c) $(3-\sqrt{2})(-1+i)$
(d) $3 \sqrt{2}(-1+i)$

Uma Kumari

Numerade Educator

Problem 81

If a and b are two real numbers such that $\mathrm{b}<0$ and $\mathrm{a}>0$, then $\sqrt{\mathrm{a}} \cdot \sqrt{\mathrm{b}}$ is equal to
(a) $\sqrt{\mathrm{a}|\mathrm{b}|} \cdot \mathrm{i}$
(b) $-\sqrt{|a| b}$
(c) $\mathrm{i} \sqrt{|\mathrm{a}| \mathrm{b}}$
(d) $\sqrt{a|b|}$

Anas Venkitta

Numerade Educator

Problem 82

The value of $\sum_{n=1}^{17}\left(i^{n}+i^{n+1}\right)$ is
(a) i
(b) $\mathrm{i}=1$
(c) -i
(d) 0

Uma Kumari

Numerade Educator

Problem 83

The value of $6\left[\cos 50^{\circ}+\mathrm{i} \sin 50^{\circ}\right] \div 3\left(\cos 20^{\circ}+\mathrm{i} \sin 20^{\circ}\right)$ in polar form is
(a) $2\left(\cos 30^{\circ}+\mathrm{i} \sin 30^{\circ}\right)$
(b) $5\left(\cos 30^{\circ}+i \sin 30^{\circ}\right)$
(c) $2\left(\cos 30^{\circ}-\mathrm{i} \sin 30^{\circ}\right)$
(d) $3\left(\cos 70^{\circ}+i \sin 70^{\circ}\right)$

Anas Venkitta

Numerade Educator

Problem 84

$\mathrm{z}=\frac{\cos \theta+\mathrm{i} \sin \theta}{\cos \theta-\mathrm{i} \sin \theta}, \frac{\pi}{4}<\theta<\frac{\pi}{2}$, then arg (z) is
(a) $4 \theta$
(b) $(2 \theta-\pi)$
(c) $\pi+2 \theta$
(d) $2 \theta$

Anas Venkitta

Numerade Educator

Problem 85

If $(a+i b)^{5}=(\alpha+i \beta)$, then $(b+i a)^{5}$ is equal to
(a) $\beta+\mathrm{i} \alpha$
(b) $\alpha-\mathrm{i} \beta$
(c) $\beta-\mathrm{i} \alpha$
(d) $-\alpha-i \beta$

Uma Kumari

Numerade Educator

Problem 86

If $\alpha$ and $\beta$ are the complex cube roots of unity, then $\alpha^{7}+\beta^{7}=$
(a) $\frac{1}{\alpha \beta}$
(b) $\frac{-1}{\alpha \beta}$
(c) $\alpha \beta$
(d) $\alpha-\beta$

Anas Venkitta

Numerade Educator

Problem 87

If $\frac{1+2 \mathrm{i}}{2+\mathrm{i}}=\mathrm{r}[\cos \theta+\mathrm{i} \sin \theta]$, then
(a) $\mathrm{r}=1 ; \theta=\tan ^{-1} \frac{3}{4}$
(b) $\mathrm{r}=\sqrt{5} ; \theta=\tan ^{-1} \frac{4}{3}$
(c) $\mathrm{r}=1 ; \theta=\tan ^{-1} \frac{4}{3}$
(d) $\mathrm{r}=1, \theta=\tan ^{-1}\left(\frac{-3}{4}\right)$

Anas Venkitta

Numerade Educator

Problem 88

If $\omega$ is a complex cube root of unity, then the product of $\left(1-\omega+\omega^{2}\right)\left(1-\omega^{2}+\omega^{4}\right)\left(1-\omega^{4}+\omega^{8}\right) \ldots$ to 20 factors is equal to
(a) $4^{10}$
(b) $4^{20}$
(c) $(4 \omega)^{10}$
(d) $\left(\frac{4}{\omega}\right)^{10}$

Uma Kumari

Numerade Educator

Problem 89

If arg $(z)<0$, then $\arg (-z)-\arg (z)$ is equal to
(a) $\pi / 2$
(b) $-\pi / 2$
(c) $-\pi$
(d) $\pi$

Uma Kumari

Numerade Educator

Problem 90

If $x+i y=(1-i \sqrt{3})^{160}$ then $(x, y)$ is
(a) $\left(2^{99}, 2^{99} \sqrt{3}\right)$
(b) $\left(2^{100}, 2 \sqrt{3}\right)$
(c) $\left(-2^{99}, 2^{99} \sqrt{3}\right)$
(d) $\left(-2^{100}, 2 \sqrt{3}\right)$

Anas Venkitta

Numerade Educator

Problem 91

If $|z| \geq 3$, the least value of $\left|Z+\frac{1}{Z}\right|$ is
(a) $\frac{10}{3}$
(b) $\frac{8}{3}$
(c) $\frac{5}{3}$
(d) $\frac{4}{3}$

Uma Kumari

Numerade Educator

Problem 92

If $\omega$ is a complex cube root of unity then the value of $\left(3+3 \omega+5 \omega^{2}\right)^{5}-\left(2+4 \omega+2 \omega^{2}\right)^{3}$ is equal to
(a) 1
(b) 3
(c) 2
(d) 0

Anas Venkitta

Numerade Educator

Problem 93

If the vertices of a triangle are $(\sqrt{7}+\mathrm{i} \sqrt{3}), \mathrm{i}(\sqrt{7}+\mathrm{i} \sqrt{3})$ and $\sqrt{7}+\mathrm{i} \sqrt{3}+\mathrm{i}(\sqrt{7}+\mathrm{i} \sqrt{3})$, then the area of the
triangle is
(a) $\sqrt{21}$
(b) 10
(c) 5
(d) 8

Uma Kumari

Numerade Educator

Problem 94

If $z$ and $w$ are complex numbers satisfying $\arg \left(\frac{z-2 i}{z+2}\right)=\frac{\pi}{4}$ and $\arg \left(\frac{w-2 i}{w+2}\right)=\frac{\pi}{2}$ respectively, then, the intersection of the locus of $z$ and the locus of $w$ is the
(a) straight line joining $(0,2)$ and $(-2,0)$
(b) set containing the points $(0,2)$ and $(-2,0)$
(c) circle passing through the points $(0,2)$ and $(-2,0)$
(d) empty set

Uma Kumari

Numerade Educator

Problem 95

If $z=x+i y$ and $z^{1 / 3}=a-i b$ and $\frac{x}{a}-\frac{y}{b}=k\left(a^{2}-b^{2}\right)$, then the value of $k$ is given by
(a) 4
(b) 2
(c) 1
(d) $\frac{1}{4}$

Uma Kumari

Numerade Educator

Problem 96

$2 \cos \theta=x+\frac{1}{x}$ then the value of $x^{6}-\frac{1}{x^{6}}$ is
(a) $2 \mathrm{i} \sin ^{6} \theta$
(b) $2 \sin 2 \theta$
(c) i $\sin 6 \theta$
(d) $2 \mathrm{i} \sin 6 \theta$

Anas Venkitta

Numerade Educator

Problem 97

If $z_{r}=\cos \frac{2 r \pi}{4}+i \sin \frac{2 r \pi}{4}$ where $r=0,1,2,3$ then $\frac{z_{0}+z_{1}}{z_{2}+z_{3}}$ equals
(a) 1
(b) $-1$
(c) i
(d) $-\mathrm{i}$

Uma Kumari

Numerade Educator

Problem 98

If $\omega$ is a complex cube root of unity, then the value of $\omega^{\frac{1}{2}} \times \omega^{\frac{1}{4}} \times \omega^{\frac{1}{6}} \times \omega^{\frac{1}{18}} \times \omega^{\frac{1}{54}} \ldots \infty$ is
(a) 0
(b) 1
(c) $\omega$
(d) $\omega^{2}$

Anas Venkitta

Numerade Educator

Problem 99

Let $z_{1}, z_{2}, z_{3}$ be the vertices of an equilateral triangle. If $\frac{z_{1}-z_{2}}{z_{3}-z_{2}}=z$ then $1+z+z^{2}$ is equal to
(a) 0
(b) $2 \omega$
(c) $\omega$
(d) $-2 \omega^{2}$

Uma Kumari

Numerade Educator

Problem 100

$\left[\frac{1+\cos \frac{\pi}{8}+i \sin \frac{\pi}{8}}{1+\cos \frac{\pi}{8}-i \sin \frac{\pi}{8}}\right]^{8}$ is equal to
(a) $1+\mathrm{i}$
(b) $1-i$
(c) 1
(d) $-1$

Uma Kumari

Numerade Educator

Problem 101

If $z=\cos \theta+i \sin \theta$ then $\frac{z^{2}-1}{z^{2}+1}$ is equal to
(a) $\mathrm{i}+\tan \theta$
(b) $\mathrm{i}+\cot \theta$
(c) $\mathrm{i} \tan \theta$
(d) $\cot \theta$

Anas Venkitta

Numerade Educator

Problem 102

The value of $(1+i)^{8}-(1-i)^{8}$ is
(a) 0
(b) 16
(c) 32
(d) $\sqrt{2}(1+i)$

Anas Venkitta

Numerade Educator

Problem 103

If $z=2+t+i \sqrt{3-t^{2}}$ where, $t$ is real, the locus of the points $z$ for different values of $t$ is the
(a) circle with centre $(3,0)$ and radius $2 .$
(b) line segment joining the points $(0,1)$ and $(0,-1)$
(c) circle passing through the points $(1,0),(0,1)$ and $(0,-1)$
(d) circle of radius $\sqrt{3}$ passing through $(2-\sqrt{3}, 0)$ and $(2+\sqrt{3}, 0)$.

Uma Kumari

Numerade Educator

Problem 104

If the first term and the common difference of a GP are each equal to $x+\sqrt{5-x^{2}}$ i, then the modulus of its nth term is
(a) $5^{n}$
(b) $5^{n-1}$
(c) $5^{\frac{n-1}{2}}$
(d) $5^{\frac{n}{2}}$

Uma Kumari

Numerade Educator

Problem 105

If $|z|=1$ and $w=\frac{z-1}{z+1}$ where $z \neq-1$ then $\operatorname{Re}($ w)
(a) 0
(b) $\frac{-1}{|z+1|^{2}}$
(c) $\frac{\sqrt{2}}{|z+1|^{2}}$
(d) 2

Uma Kumari

Numerade Educator

Problem 106

If $z_{1}$ and $z_{2}$ are two complex numbers such that $\left|z_{1}+z_{2}\right|=\left|z_{1}\right|+\left|z_{2}\right|$, then arg $\left(z_{1} \omega\right)-\arg \left(z_{2}\right.$ i), where $\omega$ is the complex cube root of unity, is
(a) 0
(b) $\frac{\pi}{2}$
(c) $\frac{\pi}{3}$
(d) $\frac{\pi}{6}$

Uma Kumari

Numerade Educator

Problem 107

The area of the triangle whose vertices are represented by $\frac{1-\sqrt{3} \mathrm{i}}{2}, \frac{-1+\sqrt{3} \mathrm{i}}{2}$ and $\frac{-1-\sqrt{3} \mathrm{i}}{2}$ is
(a) $\frac{1+\sqrt{3}}{2}$
(b) $\frac{\sqrt{3}}{2}$
(c) $\frac{1-\sqrt{3}}{2}$
(d) $\frac{3}{2} \pi$

Uma Kumari

Numerade Educator

Problem 108

The maximum value of $\left|z_{1}-z_{2}\right|$ satisfying the conditions $\left|z_{1}\right|=10$ and $\left|z_{2}+3+4 i\right|=5$ is
(a) 20
(b) 0
(c) 10
(d) 15

Ekaveera Kumar

Numerade Educator

Problem 109

If arg $\left(\frac{z-1}{z+1}\right)=\frac{\pi}{2}$, where $z$ is a complex number, locus of $z$ is
(a) $|\mathrm{z}|=1, \operatorname{Im}(\mathrm{z})>0$
(b) $|\mathrm{z}|=1, \operatorname{Im}(\mathrm{z})<0$
(c) $|z|=1$
(d) $|z-1|=1, \operatorname{Im}(z)>0$

Uma Kumari

Numerade Educator

Problem 110

We can draw two tangents from the point $\mathrm{P}$ to the circle $|\mathrm{z}+5+2 \mathrm{i}|=2$. The right choice for the coordinates of $\mathrm{P}$ from the following is
(a) $(-7,-2)$
(b) $(-6,-4)$
(c) $(-4,-3)$
(d) $(-6,-3)$

Uma Kumari

Numerade Educator

Problem 111

Statement 1 All the points on the line $\alpha \bar{z}+\bar{\alpha}_{z}+\beta=0(\beta \neq 0)$ have equal arguments. and
Statement 2
Slope of a line is an absolute constant

P Krishnamurthy

P Krishnamurthy

Numerade Educator

Problem 112

Statement 1 The function $\sec ^{-1}\left(\frac{i}{|z|^{2}+z-\bar{z}-4}\right)$ is defined in the region bounded by the circle $|z|=2$ and the lines $\operatorname{In}(z)=\pm \frac{1}{2}$ and
Statement 2
$\sec ^{-1} x$ is defined if $|x| \geq 1$

Uma Kumari

Numerade Educator

Problem 113

Statement 1
$\arg [(-1+i \sqrt{3})(-2+2 i)(i)]=-\frac{\pi}{12}$
and
Statement 2
$\arg \left(z_{1} z_{2} z_{3}\right)=\arg z_{1}+\arg z_{2}+\arg z_{3}$

Uma Kumari

Numerade Educator

Problem 114

The circle $|\mathrm{z}|=2$ maps onto the circle
(a) $|\mathrm{w}-5|=6$
(b) $|w|=5$
(c) $|\mathrm{w}+5|=6$
(d) $|w-5|=3$

Uma Kumari

Numerade Educator

Problem 115

The line $\arg z=\frac{\pi}{4}$ is mapped onto
(a) line passing through $(5,0)$ with slope 1
(b) line passing through $(5,0)$ with slope $-1$
(c) segment of the line passing through $(5,0)$ with slope $-1$ above the U-axis of the $\mathrm{W}$ plane
(d) segment of the line passing through $(5,0)$ with slope $-1$ below the u-axis.

Uma Kumari

Numerade Educator

Problem 116

The line $\mathrm{y}=\mathrm{x}$ in the $\mathrm{Z}$ plane is mapped onto
(a) the line $\mathrm{v}+\mathrm{u}=5$ in the $\mathrm{W}$ plane
(b) the line $\mathrm{v}=\mathrm{u}$ in the $\mathrm{W}$ plane
(c) the line $\mathrm{u}+\mathrm{v}=0$ in the $\mathrm{W}$ plane
(d) None of the above

WM

William Mead

Numerade Educator

Problem 117

If $x, y, a, b$ are real numbers such that $(x+i y) y / 5=a+i b$ and $p=\frac{x}{a}-\frac{y}{b}$ then
(a) $\mathrm{a}-\mathrm{b}$ is a factor of $\mathrm{p}$
(b) $(a+b)$ is a factor of $p$
(c) $\mathrm{a}+\mathrm{ib}$ is a factor of $\mathrm{p}$
(d) $a-i b$ is a factor of $p$

Uma Kumari

Numerade Educator

Problem 118

Let $\mathrm{z}=\frac{\cos \theta+\mathrm{i} \sin \theta}{\cos \theta-\mathrm{i} \sin \theta}, \frac{\pi}{4}<\theta<\frac{\pi}{2}$. Then $\arg (\mathrm{z})$ is
(a) $2 \theta$
(b) $2 \theta-\pi$
(c) $\pi+2 \theta$
(d) $2 \theta-2 \pi$

Anas Venkitta

Numerade Educator

Problem 119

The distinct complex numbers $z_{1}, z_{2}, z_{3}$ are in AP. Then,
(a) they lie on a circle
(b) they are collinear
(c) $z_{1}\left|z_{2}-z_{3}\right|-z_{2}\left|z_{3}-z_{1}\right|+z_{3}\left|z_{1}-z_{2}\right|=0$
(d) $\mathrm{z}_{1}\left|\mathrm{z}_{2}-\mathrm{z}_{3}\right|-\mathrm{z}_{2}\left(\mathrm{z}_{3}-\mathrm{z}_{1}\right)+\mathrm{z}_{3}\left|\mathrm{z}_{1}-\mathrm{z}_{2}\right|=1$

Uma Kumari

Numerade Educator

Problem 120

$\begin{array}{ll}\text { Column I } & \text { Column II }\end{array}$
(a) Cube roots of unity form
(p) origin
(b) The equation $|\mathrm{z}-1|^{2}+|\mathrm{z}+1|^{2}=2$ represents
(q) a triangle
(c) If $z=(k+3)+i \sqrt{5-k^{2}}$, then locus of $z$ represents
(r) a pair of straight lines
(d) If $|z+\bar{z}|=|z-\bar{z}|$, then locus of $z$ is
(s) a circle

Uma Kumari

Numerade Educator

Problem 121

The product of two complex numbers is $5+7 \mathrm{i}$. If the sum and the product of their real parts are 5 and 6 respectively, find the complex numbers.

Uma Kumari

Numerade Educator

Problem 122

If $\cos \alpha+\cos \beta+\cos \gamma=0=\sin \alpha+\sin \beta+\sin \gamma_{\cdot}$, prove that
(i) $\cos 3 \alpha+\cos 3 \beta+\cos 3 \gamma=3 \cos (\alpha+\beta+\gamma)$
(ii) $\sin 3 \alpha+\sin 3 \beta+\sin 3 \gamma=3 \sin (\alpha+\beta+\gamma)$
(iii) $\cos 2 \alpha+\cos 2 \beta+\cos 2 \gamma=0$
(iv) $\sin 2 \alpha+\sin 2 \beta+\sin 2 \gamma=0$

Uma Kumari

Numerade Educator

Problem 123

If $\frac{\lambda}{\left|z_{2}-z_{3}\right|}=\frac{m}{\left|z_{3}-z_{1}\right|}=\frac{n}{\left|z_{1}-z_{2}\right|}$ where $\ell, m, n$ are real and $z_{1}, z_{2}, z_{3}$ are complex numbers prove that $\frac{\lambda^{2}}{z_{n}-z_{2}}+\frac{m^{2}}{z_{n}-z_{.}}+\frac{n^{2}}{z_{-z}}=0$

Uma Kumari

Numerade Educator

Problem 124

(i) If $n$ is an odd integer $>3$ but $n$ is not a multiple of 3 , prove that $\left(x^{3}+x^{2}+x\right)$ is a factor of $(x+1)^{n}-x^{n}-1$
(ii) Show that the polynomial $x^{4 p}+x^{4_{q}+1}+x^{4 z+2}+x^{4 n+3}$ where $\mathrm{p}, \mathrm{q}, \mathrm{r}, \mathrm{s}$ are positive integers, is divisible by $\left(\mathrm{x}^{3}+\mathrm{x}^{2}+\mathrm{x}+1\right)$

P Krishnamurthy

P Krishnamurthy

Numerade Educator

Problem 125

Sum to infinity, the series $1+\mathrm{k} \cos \theta+\mathrm{k}^{2} \cos 2 \theta+\mathrm{k}^{3} \cos 3 \theta+\ldots+\infty,|\mathrm{k}|<1$.

P Krishnamurthy

P Krishnamurthy

Numerade Educator

Problem 126

Prove that for two complex numbers $\mathrm{z}_{1}$ and $\mathrm{z}_{2},\left|\frac{\mathrm{z}_{1}-\mathrm{z}_{2}}{1-\frac{3}{2}}\right|<1$ if $\left|\mathrm{z}_{1}\right|<1$ and $\left|\mathrm{z}_{2}\right|<1$.

Uma Kumari

Numerade Educator

Problem 127

Indicate the regions in the complex plane represented by
(i) $|z+1|^{2}+|z-1|^{2}=4$
(ii) $\frac{\pi}{6} \leq \arg z \leq \frac{\pi}{4}$
(iii) $\log _{\operatorname{cosec} \frac{\pi}{4}} \frac{\left|\mathrm{z}^{2}\right|+|\mathrm{z}|+4}{3|\mathrm{z}|-1}>2$

Uma Kumari

Numerade Educator

Problem 128

$\mathrm{A} \mathrm{BC}$ is an equilateral triangle with its vertices $\mathrm{A}, \mathrm{B}, \mathrm{C}$ representing the complex numbers $\mathrm{z}_{1}, \mathrm{z}_{2}, \mathrm{z}_{3}$ in the Argand plane. Prove that
(i) $\frac{1}{z_{2}-z_{3}}+\frac{1}{z_{3}-z_{1}}+\frac{1}{z_{1}-z_{2}}=0$
(ii) $z_{1}^{2}+z_{2}^{2}+z_{3}^{2}=z_{1} z_{2}+z_{2} z_{3}+z_{3} z_{1}$
(iii) If $z_{0}$ is the circum centre of the triangle $\mathrm{ABC}, \mathrm{z}_{1}^{2}+\mathrm{z}_{2}^{2}+\mathrm{z}_{3}^{2}=3 \mathrm{z}_{0}^{2}$

AG

Archana Goyal

Numerade Educator

Problem 129

Factorize the polynomial $(x-2)^{8}-256$ completely.

Uma Kumari

Numerade Educator

Problem 130

$\mathrm{z}_{1}, \mathrm{z}_{2}, \mathrm{z}_{3}$ are points on the circle of radius 5 and centre at origin such that $\left|z_{1}-z_{2}\right|=\left|z_{2}-z_{3}\right|=\left|z_{3}-z_{1}\right| .$ If $z_{1}=4+3 i$, find $z_{2}$ and $z_{3}$,

Uma Kumari

Numerade Educator

Problem 131

If $1, \alpha, \alpha^{2}, \alpha^{3}, \ldots, \alpha^{n-1}$ are the nth roots of unity, then the value of $(2-\alpha)\left(2-\alpha^{2}\right)\left(2-\alpha^{3}\right) \ldots\left(2-\alpha^{n-1}\right)$ is
(a) $2^{n}$
(b) $1-2^{n}$
(c) $2^{n}-1$
(d) 2

Uma Kumari

Numerade Educator

Problem 132

The equation whose roots are $\cos \frac{2 \pi}{7}, \cos \frac{4 \pi}{7}, \cos \frac{6 \pi}{7}$
(a) $x^{3}+x+1=0$
(b) $8 x^{3}+4 x^{2}-4 x+1=0$
(c) $2 x^{3}+3 x^{2}+x+1=0$
(d) $8 x^{3}-4 x^{2}+4 x+1=0$

Uma Kumari

Numerade Educator

Problem 133

Complex numbers $\mathrm{z}_{1}, \mathrm{z}_{2}, \mathrm{z}_{3}$ are the vertices $\mathrm{A}, \mathrm{B}, \mathrm{C}$ respectively of an isoceles right angled triangle right angled at $\mathrm{A} .$ Then, $\left(\mathrm{z}_{2}-\mathrm{z}_{3}\right)^{2}$ equals
(a) $2\left(z_{1}-z_{2}\right)\left(z_{1}-z_{3}\right)$
(b) $2\left(z_{2}-z_{1}\right)\left(z_{1}-z_{3}\right)$
(c) $3\left(z_{1}+z_{2}+z_{3}\right)$
(d) $\left(z_{2}-z_{1}\right)\left(z_{1}-z_{3}\right)$

Uma Kumari

Numerade Educator

Problem 134

The greatest value of the modulli of the complex numbers satisfying the relation $\left|z-\frac{4}{z}\right|=2$ is
(a) $\sqrt{2}$
(b) $\sqrt{3}+1$
(c) $\sqrt{5}+1$
(d) $\sqrt{5}-1$

Uma Kumari

Numerade Educator

Problem 135

Let $\mathrm{z}=\mathrm{x}+\mathrm{iy}, \mathrm{w}=\mathrm{u}+\mathrm{iv}$ where $\mathrm{x}, \mathrm{y}, \mathrm{u}, \mathrm{v}$ are real. If $\mathrm{w}=2 \mathrm{iz}$ and $\mathrm{z}$ moves such that $\arg \mathrm{z}=\frac{\pi}{4}$, the locus of $\mathrm{w}$ is
(a) Straight line
(b) circle
(c) ellipse
(d) parabola

Uma Kumari

Numerade Educator

Problem 136

Let $z=x+i y, w=u+$ iv where $x, y, u, v$ are real. If $w=\frac{2}{z}$ and $z$ moves along the circle $|z-2 i|=2 .$ Then the locus
of $w .$
(a) perpendicular bisector of the line joining $(0,-1)$ and $(0,0)$
(b) circle with centre $(0,0)$ radius 1
(c) circle with centre $(0,0)$ radius 2
(d) straight line passing through $(0,2)$

Uma Kumari

Numerade Educator

Problem 138

A point in the region $|z-3| \leq 2$ such that arg $z_{0}$ is a maximum.
(a) $\frac{5}{3}+\frac{2 \sqrt{5}}{3} \mathrm{i}$
(b) $\frac{1}{3}+\frac{2}{3} \mathrm{i}$
(c) $5+\frac{2}{3}$ i
(d) $\frac{\sqrt{5}}{3}+\frac{2}{3} \mathrm{i}$

Uma Kumari

Numerade Educator

Problem 139

If $e^{i \theta}=\cos \theta+i \sin \theta$, then for $\Delta A B C, e^{i A} e^{i B} e^{i C}$ equals
(a) $-\mathrm{i}$
(b) 1
(c) $-1$
(d) $3 \mathrm{e}^{\text {it }}$

Anas Venkitta

Numerade Educator

Problem 140

If $1, \omega, \omega^{2}$ are the cube roots of unity, then the value of $\left(1+5 \omega^{2}+\omega^{4}\right)\left(1+5 \omega+\omega^{2}\right)$ is
(a) 16
(b) 64
(c) 28
(d) 18

Anas Venkitta

Numerade Educator

Problem 141

If $|z|=r$ and $\arg (z)=\frac{\pi}{4}$, then $\left|z+\frac{1}{z}\right|$ equals
(a) $\mathrm{r}+\frac{1}{\mathrm{r}}$
(b) $\sqrt{r^{2}+\frac{1}{r^{2}}}$
(c) $\sqrt{\mathrm{r}+1 / \mathrm{r}}$
(d) $\sqrt{r^{2}+1 / r^{2}-2}$

Uma Kumari

Numerade Educator

Problem 142

If $\left|z_{1}\right|=\left|z_{2}\right|=\left|z_{3}\right|=\left|z_{4}\right|$, then the points representing the complex numbers $z_{1}, z_{2}, z_{3}$ and $z_{4}$ are
(a) the vertices of a square
(b) the vertices of a rhombus
(c) the vertices of a cyclic quadrilateral
(d) collinear

Uma Kumari

Numerade Educator

Problem 143

If $\alpha \neq 1$ is an nth root of unity then $1+3 \alpha+5 \alpha^{2}+\ldots$ to $\mathrm{n}$ terms is equal to
(a) $\frac{n}{(1-\alpha)}$
(b) $\frac{2 n}{(1-\alpha)}$
(c) $\frac{-n}{2(1-\alpha)}$
(d) $\frac{-2 n}{(1-\alpha)}$

Uma Kumari

Numerade Educator

Problem 144

If $n$ is odd, then the value of the expression $\left(\frac{\sqrt{3}+i}{\sqrt{3}-i}\right)^{\prime}+1$ is
(a) $\omega$
(b) $\omega^{2}$
(c) 0
(d) $-1$

Uma Kumari

Numerade Educator

Problem 145

If $\mathrm{n}$ is odd, the value of $\frac{(1+\mathrm{i})^{2 \mathrm{n}+1}}{(1-\mathrm{i})^{2 \mathrm{n}-1}}$, is
(a) 0
(b) 2
(c) $-2$
(d) 4

Uma Kumari

Numerade Educator

Problem 146

If $z$ is a complex number such that $|z|=1$, then the least value of $|z+1|+|z-1|$ is
(a) 1
(b) 0
(c) 2
(d) None of these

Uma Kumari

Numerade Educator

Problem 147

Common roots of the equations $z^{3}+2 z^{2}+2 z+1=0$ and $z^{1985}+z^{100}+1=0$ are
(a) $1, \omega, \omega^{2}$,
(b) $-\omega,-\omega^{2}$
(c) $\omega, \omega^{2}$
(d) $-\omega, \omega^{2}$

Uma Kumari

Numerade Educator

Problem 148

The values of $\frac{\left(\cos 30^{\circ}+i \sin 30^{\circ}\right)\left(\cos 45^{\circ}+i \sin 45^{\circ}\right)}{\left(\cos 5^{\circ}+i \sin 5^{\circ}\right)\left(\cos 10^{\circ}+i \sin 10^{\circ}\right)\left(\cos 15^{\circ}+i \sin 15^{\circ}\right)}$ is
(a) $\frac{\sqrt{2}}{2}(1+\mathrm{i})$
(b) $1+\mathrm{i}$
(c) $e^{\frac{i \pi}{3}}+e^{\frac{i n}{6}}$
(d) $(1+i) \frac{2}{5}$

Uma Kumari

Numerade Educator

Problem 149

If i stands for $\sqrt{-1}$, then, for positive integers $\mathrm{n}_{1}, \mathrm{n}_{2}$, the value of the expression $(1+\mathrm{i})^{\mathrm{n}_{1}}+\left(1+\mathrm{i}^{3}\right)^{\mathrm{n}_{1}}+\left(1+\mathrm{i}^{5}\right)^{\mathrm{n}_{2}}+\left(1+\mathrm{i}^{7}\right)^{\mathrm{n}_{2}}$
is a real number if and only if
(a) $\mathrm{n}_{1}=\mathrm{n}_{2}+1$
(b) $n_{1}=n_{2}-1$
(c) $\mathrm{n}_{1}=\mathrm{n}$,
(d) $\mathrm{n}$, and $\mathrm{n}_{2}$ can be any positive integers

Uma Kumari

Numerade Educator

Problem 150

If three complex numbers $\mathrm{z}_{1}, \mathrm{z}$ and $\mathrm{z}_{2}$ are in A.P. such that $\left|\mathrm{z}_{1}-1\right|=\left|\mathrm{z}_{1}+1\right|$ and $\left|\mathrm{z}_{2}-\mathrm{i}\right|=\left|\mathrm{z}_{2}+\mathrm{i}\right|$ then $|\mathrm{z}|$ is equal to
(a) $\frac{\left|z_{1}\right|+\left|z_{2}\right|}{2}$
(b) $\frac{1}{2} \sqrt{\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}}$
(c) $\frac{\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}}{4}$
(d) $\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}+\left|z_{1} z_{2}\right|$

Uma Kumari

Numerade Educator

Problem 151

If $\left|z_{1}+z_{2}\right|=\left|z_{1}-z_{2}\right|$ and $\left|z_{2}\right|=\sqrt{3}\left|z_{1}\right|$, then the absolute value of the difference of the arguments of $z_{1}+z_{2}$ and $\mathrm{z}_{1}$ is
(a) 0
(b) $\frac{\pi}{2}$
(c) $\frac{\pi}{3}$
(d) $\frac{\pi}{6}$

Uma Kumari

Numerade Educator

Problem 152

The square root of $\frac{x^{2}}{y^{2}}-\frac{y^{2}}{x^{2}}-2\left(\frac{y}{x}-\frac{x}{y}_{i}\right)+2 i-1$ is
(a) $\pm\left(\frac{x}{y}+i\left(\frac{y}{x}+1\right)\right)$
(b) $\pm\left(\frac{x}{y}-\frac{y}{x}+i\right)$
(c) $\pm\left(\frac{x}{y}-i\left(\frac{y}{x}+1\right)\right)$
(d) $\pm\left[\left(\frac{x}{y}-\frac{y}{x}\right) i+1\right]$

Uma Kumari

Numerade Educator

Problem 153

Let $\mathrm{f}: \mathrm{C} \rightarrow \mathrm{R}$ be a function (C-set of complex numbers, $\mathrm{R}$ -set of real numbers) defined by $\mathrm{f}(\mathrm{z})=|\mathrm{z}|+\frac{1}{2}$ then
(a) $\mathrm{f}$ is one-one and onto
(b) $\mathrm{f}$ is one-one but not onto
(c) $\mathrm{f}$ is onto but not one-one
(d) $\mathrm{f}$ is neither one-one nor onto

Uma Kumari

Numerade Educator

Problem 154

If $x^{2}+x+1=0$ and $n$ is a multiple of 3, then the value of $\left(x+\frac{1}{x}\right)\left(x^{2}+\frac{1}{x^{2}}\right)\left(x^{3}+\frac{1}{x^{3}}\right) \ldots . .\left(x^{n}+\frac{1}{x^{n}}\right)$
(a) 1
(b) 0
(c) $2^{n / 3}$
(d) $\omega^{2}-1$

Uma Kumari

Numerade Educator

Problem 155

If $a$ and $b$ are positive numbers such that $\frac{a z+0}{a z-b}$ is pure imaginary, then $|z|$ is equal to
(a) 1
(b) $\frac{a}{b}$
(c) $\frac{b}{a}$
(d) $\frac{a+b}{a-b}$

Uma Kumari

Numerade Educator

Problem 156

One of the factors of $x^{88}+x^{69}+x^{50}+x^{43}$ is
(a) $x+1$
(b) $x+i$
(c) $x-i$
(d) All the above

Uma Kumari

Numerade Educator

Problem 157

If $z=r e^{i \theta}$ and $w=e^{i z}$ then $(\ln |w|)^{2}+(\arg w)^{2}$
(a) 0
(b) 2
(c) $2|z|^{2}$
(d) $|z|^{2}$

Uma Kumari

Numerade Educator

Problem 158

Given $z=\cos \frac{2 \pi}{13}+i \sin \frac{2 \pi}{13}$, the equation whose roots are $\alpha$ and $\beta$ where, $\alpha=1+z+z^{3}+z^{5}+z^{7}+z^{9}+z^{11}$ and
$\beta=1+z^{2}+z^{4}+z^{6}+z^{8}+z^{10}+z^{12}$, is
(a) $4 x^{2}-4 x+1=0$
(b) $4 x^{2}-4 x+\sec ^{2} \frac{\pi}{13}=0$
(c) $x^{2}+x+\operatorname{cosec}^{2} \frac{\pi}{13}=0$
(d) $4 x^{2}+x+\sec ^{2} \frac{\pi}{13}=0$

P Krishnamurthy

P Krishnamurthy

Numerade Educator

Problem 159

In the solution of the equation $z^{6}+19 z^{3}-216=0$, number of roots having negative imaginary part is
(a) 4
(b) 2
(c) 1
(d) 6

Uma Kumari

Numerade Educator

Problem 160

Argument of solutions of the equation $\mathrm{z}^{2}+\mathrm{z}|\mathrm{z}|+\left|\mathrm{z}^{2}\right|=0$ is
(a) $\frac{\pi}{3}$
(b) $\frac{2 \pi}{3}$
(c) $\frac{4 \pi}{3}$
(d) Both
(b) and (c)

Uma Kumari

Numerade Educator

Problem 161

The point on the circle $|z-5 i|=4$ having the greatest argument is
(a) $\left(\frac{-1}{5}, \frac{9}{5}\right)$
(b) $\left(\frac{-2}{5}, \frac{9}{5}\right)$
(c) $\left(\frac{-12}{5}, \frac{9}{5}\right)$
(d) $\left(\frac{-12}{5}, \frac{3}{5}\right)$

Uma Kumari

Numerade Educator

Problem 162

If three distinct complex numbers $z_{1}, z_{z}, z_{3}$ are collinear, then $\sum z_{1}\left(\overline{z_{2}}-\overline{z_{3}}\right)=$
(a) $\mathrm{z}_{1} \mathrm{z}_{2} \mathrm{z}$
(b) $\mathrm{z}_{1}+\mathrm{z}_{2}+\mathrm{z}_{2}$
(c) 0
(d) 1

Uma Kumari

Numerade Educator

Problem 163

If complex numbers $\mathrm{A}=\mathrm{a}+\mathrm{b} \omega+\mathrm{c} \omega^{2}, \mathrm{~B}=a \omega+\mathrm{b} \omega^{2}+\mathrm{c}$ and $\mathrm{C}=a \omega^{2}+\mathrm{b}+\mathrm{c} \omega$, represent the vertices of a triangle,
then the area of the triangle formed by $\mathrm{A}, \mathrm{B}, \mathrm{C}$ is, (where $\mathrm{r}$ is the radius of the circumcircle of the triangle)
(a) $\frac{3 \sqrt{3}}{4} \mathrm{r}^{2}$
(b) $\frac{\sqrt{3}}{4} \mathrm{r}^{2}$
(c) $\frac{2 \sqrt{3}}{4} \mathrm{r}^{2}$
(d) $\frac{3 \sqrt{3}}{2} \mathrm{r}^{2}$

AG

Archana Goyal

Numerade Educator

Problem 164

If $z$ be any point on the circle $|z-1|=1$, then, $\frac{z-2}{z}$ equals
(a) $\mathrm{i} \tan (\arg \mathrm{z})$
(b) $\tan (\arg z)$
(c) $\mathrm{i} \arg (\mathrm{z})$
(d) i $\sin (\arg z)$

Uma Kumari

Numerade Educator

Problem 165

$\sum_{k=1}^{13}\left(\sin \frac{2 k \pi}{13}-i \cos \frac{2 k \pi}{13}\right)$ equals
(a) $-\mathrm{i}$
(b) 0
(c) $-\mathrm{i}-1$
(d) $1+\mathrm{i}$

Uma Kumari

Numerade Educator

Problem 166

Let $' z^{\prime}$ be a complex number such that $z+\frac{1}{z}=1$ and $a=z^{2007}+\frac{1}{z^{2007}}$ and $b$ is the last digit of the number $2^{2^{n}}+1$ where $\mathrm{n}$ is an integer $>1$. Then the value of $\left(a^{2}+b^{2}\right)$ is
(a) 23
(b) 13
(c) 53
(d) 1

Uma Kumari

Numerade Educator

Problem 167

If $n$ is a positive integer and $(1+x)^{n}=a_{0}+a_{1} x+a_{2} x^{2}+\ldots a_{n} x^{n}$, the value of $a_{0}-a_{2}+a_{4}-a_{6} \ldots . .-a_{102}$
(a) 0
(b) 1
(c) $-1$
(d) 2
$\sqrt{2}$

Uma Kumari

Numerade Educator

Problem 168

Value of $\left(\frac{\sqrt{3}+1}{\sqrt{3}-\mathrm{i}}\right)+3$ is equal to
(a) 0
(b) 2
(c) $-2$
(d) 4

Anas Venkitta

Numerade Educator

Problem 169

If $z$ satisfies $\left|\frac{z-12}{z-8 i}\right|=\frac{5}{3}$ and $\left|\frac{z-4}{z-8}\right|=1$, then $|z|$ is equal to
(a) $\sqrt{3}$
(b) 1
(c) 2
(d) $5 \sqrt{13}$

Uma Kumari

Numerade Educator

Problem 170

$\mathrm{z}_{1}$ and $\mathrm{z}_{2}$ are two complex numbers in which $\mathrm{z}_{2}$ is not uni-modular. If $\frac{2 \mathrm{t}_{2}^{2}}{\mathrm{z}_{1}-2 \mathrm{z}_{2}}$ is unimodular, $\left|\mathrm{z}_{1}\right|$ is equal to
(a) $\underline{1}$
(b) 2
(c) 3
(d) 4

Uma Kumari

Numerade Educator

Problem 171

Statement 1 $\mathrm{z}_{1}$ and $\mathrm{z}_{2}$ are two complex numbers such that $\left|\mathrm{z}_{1}\right|=\left|\mathrm{z}_{2}\right|=1 .$ Then, $\left|\frac{1}{z_{1}}+\frac{1}{z_{2}}\right|=\left|z_{1}+z_{2}\right|$
and
Statement 2 For any two complex numbers satisfying the conditions $\left|z_{1}\right|=\left|z_{2}\right|=r,\left|\frac{1}{z_{1}}+\frac{1}{z_{2}}\right|=\left|z_{1}+z_{2}\right|$.

Uma Kumari

Numerade Educator

Problem 172

Statement 1 The line $\mathrm{y}=\mathrm{x}$ represents the locus of points $\mathrm{z}$ in the Argand plane satisfying the condition $\arg \mathrm{z}=\frac{\pi}{4}$. and
Statement 2
The line $y=x$ makes an angle $45^{\circ}$ with the positive direction of the $x$ -axis.

Uma Kumari

Numerade Educator

Problem 173

Statement 1 The points $\mathrm{A}, \mathrm{B}, \mathrm{C}$ in the Argand plane represented by the complex numbers $\mathrm{z}_{1}, \mathrm{z}_{2}, \mathrm{z}_{3}$ respectively form an isosceles triangle $\mathrm{ABC}$ with $\angle \mathrm{BAC}=120^{\circ} .$ Then, $\mathrm{z}_{3}=(1-\omega) \mathrm{z}_{1}+\omega \mathrm{z}_{2}$
OR $z_{3}=\left(1-\omega^{2}\right) z_{1}+\omega^{2} z_{2}$ where $\omega$ denotes a complex cube root of unity. and
Statement 2
If $\omega$ denotes a complex cube root of unity, $1+\omega+\omega^{2}=0$.

Gaurav Kalra

Numerade Educator

Problem 174

Statement 1 Roots of the equation $\mathrm{z}^{2}-2 \mathrm{iz}-5=0$ are complex. and
Statement 2 For any quadratic equation $a x^{2}+b x+c=0$ where $a, b, c$ are real or complex, if the discriminant $\left(b^{2}-4 a c\right)$ is a perfect square, the corresponding roots of the quadratic equation are rational.

Uma Kumari

Numerade Educator

Problem 175

Statement 1
Sum of the squares of the reciprocals of the roots of the equation $x^{9}-1=0$ is zero. and
Statement 2
Sum of the roots of the equation $x^{9}-1=0$ is zero.

Uma Kumari

Numerade Educator

Problem 176

Statement 1
Sum of the squares of the reciprocals of the roots of the equation $x^{9}-1=0$ is zero. and
Statement 2
Sum of the roots of the equation $x^{9}-1=0$ is zero.

Uma Kumari

Numerade Educator

Problem 177

Statement 1
$\operatorname{Im}\left\{\frac{(3+2 i)}{(5+3 i)(1-5 i)}+\frac{(3-2 i)}{(5-3 i)(1+5 i)}\right\}=0$
and
Statement 2
For any non-zero complex number $\mathrm{z},(\mathrm{z}+\overline{\mathrm{z}})$ is a real number.

Uma Kumari

Numerade Educator

Problem 178

Statement 1 The point $\mathrm{z}$ lies on the circle $|\mathrm{z}-2-\mathrm{i}|=1$. The point $\mathrm{z}_{0}$ on this circle with maximum argument is given by $\mathrm{z}_{0}=2$ $(\cos \theta+\mathrm{i} \sin \theta)$ where $\tan \theta=\frac{4}{5}$.
and
Statement 2 Point $z$ on the circle $|z-2-i|=1$ nearest to the origin has modulus $(\sqrt{5}-1)$

Uma Kumari

Numerade Educator

Problem 179

Statement 1 Let the points $\mathrm{A}, \mathrm{B}, \mathrm{P}$ in a plane be represented by the complex numbers $\mathrm{z}_{1}, \mathrm{z}_{2}$ and $\mathrm{z}$ respectively. If arg $\left(\frac{\mathrm{z}-\mathrm{z}_{1}}{\mathrm{z}-\mathrm{z}_{2}}\right)=\alpha$, then $\mathrm{z}$ lies on a circle.
and
Statement 2 $\mathrm{z}_{0}$ is a fixed point and $\mathrm{z}$ is any point in the complex plane such that $\left|\mathrm{z}-\mathrm{z}_{0}\right|=\mathrm{r}$. Then, $\mathrm{z}$ lies on the circle centered at $\mathrm{z}_{0}$ and whose radius is $r$.

Gaurav Kalra

Numerade Educator

Problem 180

The 5 values of $(1+\mathrm{i})^{3 / 5}$ are given by $2^{3 / 10}\left[\cos \left(\frac{3 \pi}{20}+\frac{6 \mathrm{k} \pi}{5}\right)+\mathrm{i} \sin \left(\frac{3 \pi}{20}+\frac{6 \mathrm{k} \pi}{5}\right)\right]$, where $\mathrm{k}=0,1,2,3,4$
and
Statement 2 If $z=r(\cos \alpha+i \sin \alpha)$ and $n$ is a positive integer, the $n$ values of $z^{1 / n}$ are given by $r^{1 / n}\left(\cos \left(\frac{\alpha}{n}+\frac{2 k \pi}{n}\right)+i \sin \left(\frac{\alpha}{n}+\frac{2 k \pi}{n}\right)\right.$
$\mathrm{k}=0,1,2, \ldots \ldots, \mathrm{n}-1$

Uma Kumari

Numerade Educator

Problem 181

The circle $a z \bar{z}+g(z+\bar{z})-i f(z-\bar{z})+c=0$, where $a, c \neq 0$ is mapped into
(a) a circle in the $w$ - plane, which does not pass through the origin
(b) a circle in the $w$ plane passing through the origin
(c) a straight line through the origin
(d) a straight line not through the origin

Uma Kumari

Numerade Educator

Problem 182

The image of $x^{2}+y^{2}=2 x$ is
(a) a circle through the origin of the $w$ plane
(b) a straight line parallel to the imaginary axis of the $\mathrm{w}$ plane
(c) a straight line parallel to the real axis of the $\mathrm{w}$ -plane
(d) a straight line through the origin of the $w$ plane

Uma Kumari

Numerade Educator

Problem 183

Let $w_{1}=\frac{1}{z_{1}}, i=1,2,3,4$. Then, $\frac{\left(w_{1}-w_{2}\right)\left(w_{5}-w_{4}\right)}{\left(w_{1}-w_{4}\right)\left(w_{3}-w_{2}\right)}$ is equal to
(a) $\frac{\left(z_{1}-z_{2}\right)\left(z_{3}-z_{4}\right)}{\left(z_{1}-z_{4}\right)\left(z_{3}-z_{2}\right)}$
(b) $-\frac{\left(z_{1}-z_{2}\right)}{\left(z_{3}-z_{4}\right)}$
(c) $-\frac{\left(z_{1}-z_{2}\right)\left(z_{3}-z_{4}\right)}{\left(z_{1}-z_{4}\right)\left(z_{3}-z_{2}\right)}$
(d) $\frac{z_{1} z_{2}\left(z_{1}-z_{4}\right)}{z_{3} z_{4}\left(z_{2}-z_{3}\right)}$

Uma Kumari

Numerade Educator

Problem 184

If the conic is a hyperbola and the origin lies on the hyperbola then its eccentricity is
(a) $\frac{\sqrt{17}}{4}$
(b) $\frac{\sqrt{17}}{3}$
(c) $\frac{9}{\sqrt{17}}$
(d) $\frac{16}{\sqrt{15}}$

Saurabh Chandra

Saurabh Chandra

Numerade Educator

Problem 185

If the conic is an ellipse and the origin lies on the ellipse, then its eccentricity is
(a) $\frac{\sqrt{17}}{9}$
(b) $\frac{4}{\sqrt{17}}$
(c) $\frac{3}{\sqrt{17}}$
(d) $\frac{15}{18}$

Goutam Chand

Numerade Educator

Problem 186

If the conic is an ellipse and the origin lies outside the ellipse, then
(a) $0<\mathrm{e}<\frac{\sqrt{17}}{9}$
(b) $\frac{4}{\sqrt{17}}<\mathrm{e}<1$
(c) $\frac{\sqrt{17}}{9}<\mathrm{e}<1$
(d) $\frac{\sqrt{17}}{9}<\mathrm{e}<\frac{4}{\sqrt{17}}$

Dilip Paruchuri

Dilip Paruchuri

Numerade Educator

Problem 187

if $\mathrm{z}_{1}$ is a real number, one of the relations below is true
(a) $\mathrm{n}^{2}-\mathrm{npq}+\mathrm{mq}^{2}=0$
(b) $\mathrm{m}^{2}-\mathrm{npq}+\mathrm{nq}^{2}=0$
(c) $\mathrm{n}^{2}+\mathrm{npq}-\mathrm{mq}^{2}=0$
(d) $\mathrm{m}^{2}+\mathrm{npq}+\mathrm{nq}^{2}=0$

Uma Kumari

Numerade Educator

Problem 188

if $\mathrm{z}_{1}$ is is real number, $\mathrm{z}_{2}=$
(a) $\frac{1}{\mathrm{~m}}[(\mathrm{n}+\mathrm{pq})+\mathrm{in}]$
(b) $\frac{1}{\mathrm{n}}[(\mathrm{n}+\mathrm{pq})+\mathrm{im}]$
(c) $\frac{1}{\mathrm{q}}\left[(\mathrm{n}-\mathrm{pq})-\mathrm{iq}^{2}\right]$
(d) $\frac{1}{p}\left[(n-p q)-i n^{2}\right]$

Uma Kumari

Numerade Educator

Problem 189

if $z_{1}=2 z_{2}$, one of the relations below is true
(a) $9\left(\mathrm{p}^{2}+\mathrm{q}^{2}\right)=16\left(\mathrm{~m}^{2}+\mathrm{n}^{2}\right)^{2}$
(b) $81\left(\mathrm{p}^{2}+\mathrm{q}^{2}\right)^{2}=4\left(\mathrm{~m}^{2}-\mathrm{n}^{2}\right)^{2}$
(c) $4\left(\mathrm{p}^{2}-\mathrm{q}^{2}\right)^{2}=81\left(\mathrm{~m}^{2}+\mathrm{n}^{2}\right)$
(d) $4\left(\mathrm{p}^{2}+\mathrm{q}^{2}\right)^{2}=81\left(\mathrm{~m}^{2}+\mathrm{n}^{2}\right)$

Uma Kumari

Numerade Educator

Problem 190

$\mathrm{A}\left(\mathrm{z}_{1}\right), \mathrm{B}\left(\mathrm{iz}_{1}\right)$ form a triangle with the origin. For $\triangle \mathrm{OAB}$,
(a) circumcentre is $\frac{z_{1}(1-i)}{2}$
(b) circumcentre is $\frac{z_{1}(1+i)}{2}$
(c) orthocenter is at the origin
(d) the altitudes through $\mathrm{O}, \mathrm{A}$ and $\mathrm{B}$ are in the ratio $1: \sqrt{2}: \sqrt{2}$

Uma Kumari

Numerade Educator

Problem 191

If $\mathrm{k}$ is a positive integer, maximum number of real roots of the equation $\mathrm{x}^{2 k}-1=0$
(a) cannot exceed 3
(b) cannot exceed 2
(c) is equal to 3
(d) is equal to 2

Uma Kumari

Numerade Educator

Problem 192

If $z=x+i y$, then $\left|\frac{2 z-31}{z+i}\right|=p$ represents a circle if
(a) $\mathrm{P}=2$
(b) $\mathrm{P}=3$
(c) $\mathrm{p} \neq 2$
(d) $\mathrm{p}=1$

P Krishnamurthy

P Krishnamurthy

Numerade Educator

Problem 193

If $\alpha$ and $\beta$ are the roots of the equation $x^{2}+x+1=0$,
(a) $\alpha^{200}+\beta^{200}=1$
(b) $\alpha^{19}+\beta^{k}=2 \alpha$
(c) $\alpha^{19}+\beta^{B}=2 \beta$
(d) $1+\alpha^{n}+\beta^{2 n}=3$ if $n$ is a multiple of 3

Uma Kumari

Numerade Educator

Problem 194

If the points $\mathrm{z}, \mathrm{i} z,(\mathrm{z}-\mathrm{i} z)$ represent the vertices of a triangle in the Argand plane, then,
(a) area of the triangle $=\frac{1}{2}|\mathrm{z}|^{2}$
(b) orthocenter is at $(1-\mathrm{i}) \mathrm{z}$
(c) circumcentre is at $\frac{1}{2}(1-i) z$
(d) centroid is at the origin

P Krishnamurthy

P Krishnamurthy

Numerade Educator

Problem 195

Consider the equation $x^{11}-1=0$ Let $\alpha=\cos \frac{2 \pi}{11}+I \sin \frac{2 \pi}{11}$ and let $\beta=\alpha^{3}$. Then,
(a) the roots of the equation may be represented as $1, \alpha, \alpha^{2}, \alpha^{3}, \alpha^{4}, \alpha^{5}, \frac{1}{\alpha}, \frac{1}{\alpha^{2}}, \frac{1}{\alpha^{3}}, \frac{1}{\alpha^{4}}$ and $\frac{1}{\alpha^{5}}$
(b) $\alpha^{2}=\beta^{3}$
(c) $\alpha^{4}=\beta^{5}$
(d) the roots of the equation may be represented as $1, \beta, \beta^{2}, \beta^{3}, \beta^{4}, \beta^{5}, \frac{1}{\beta}, \frac{1}{\beta^{2}}, \frac{1}{\beta^{3}}, \frac{1}{\beta^{4}}$ and $\frac{1}{\beta^{5}}$

Uma Kumari

Numerade Educator

Problem 196

Let $z_{1}=5+12 i$ and $z_{2}$ is another complex number satisfying the relation $\left|z_{2}\right|=1$. Then,
(a) maximum value of $\left|z_{1}+z_{2}\right|$ is 15
(b) minimum value of $\left|\mathrm{z}_{1}+\mathrm{z}_{2}\right|$ is 12
(c) maximum value of $\left|z_{1}+z_{2}\right|$ is 14
(d) minimum value of $\left|\mathrm{z}_{1}+\mathrm{z}_{2}\right|$ is 11

Uma Kumari

Numerade Educator

Problem 197

If $z=\frac{2-i}{3+i}+4 i$
(a) $|z|=\frac{5}{\sqrt{2}}$
(b) $|z|=\frac{\sqrt{5}}{2}$
(c) $\arg z=\tan ^{-1} 7$
(d) $\arg z=\tan ^{-1}\left(\frac{1}{7}\right)$

Uma Kumari

Numerade Educator

Problem 198

Column I $\quad$ Column II
(a) If $\alpha$ is root $x^{5}=1$ with $\alpha \neq 0$, then the value
(p) 0 of $\alpha^{101}+\alpha^{102}+\ldots+\alpha^{230}$ is
(b) The maximum and minimum value of $|z|$ where $z$
(q) 1 represents the curve $z=\frac{3}{2+\cos \theta+i \sin \theta}$ are
respectively equal to
(c) Let $z=x+$ iy satisfies the equation
(r) 2 $z^{2}+|z|^{2}=0$, then values of $y$ are
(d) If $\omega$ is the complex cube root of unity,
(s) 3 then $\left|\begin{array}{ccc}1 & 1+i+\omega^{2} & \omega^{2} \\ 1-i & -1 & \omega^{2}-1 \\ -i & -i+\omega-1 & -1\end{array}\right|$ equals

Uma Kumari

Numerade Educator

Problem 199

Column I Column II
(a) If $2+\sqrt{3} \mathrm{i}$ is a root of the equation $\mathrm{x}^{4}-4 \mathrm{x}^{2}+8 \mathrm{x}+$
(p) $2+2 \mathrm{i}$
$35=0$ then the other roots are
(b) Conjugate of $(1+\mathrm{i})(2-3 \mathrm{i})\left(\frac{-11}{26}+\frac{3 \mathrm{i}}{26}\right)$ is equal to
(q) $2-\sqrt{3}$ i
(c) The complex number $\mathrm{z}$ satisfying the relation
(r) $-2-i$ $|z-1|=|z-3|=|z-i|$ is
(d) Modulus of the complex number $\frac{(1+i)(2-i)}{\frac{7}{5}-\frac{1}{5} i}$ is
(s) $-2+\mathrm{i}$

Ekaveera Kumar

Numerade Educator

Problem 200

Locus of the complex number z satisfying Column I Column II
(a) $|z-1|=|z+i|$ is
(p) Circle
(b) $|z-4 i|+|z+4 i|=10$
(q) Ellipse
(c) $|z+1|=\sqrt{3}|z-1|$
(r) $\mathrm{y}$ -axis
(d) $|\mathrm{iz}-1|+|\mathrm{z}-\mathrm{i}|=2$
(s) Straight line

Ekaveera Kumar

Numerade Educator