Question 1 :
If $\cos { \left( \log { { i }^{ 4i } } \right) } =a+ib$, then
Question 2 :
Two complex numbers are represented by ordered pairs $z_1: (3,4)\ \&\ z_2: (4,5)$, which of the following is correct representation for $z_1\times z_2=$?
Question 3 :
Two complex numbers are represented by ordered pairs $z_1: (a,b)\ \&\ z_2: (c,d)$, when these two complex numbers are equal?
Question 4 :
The complex number $e^{i\theta }$ can be expressed in vector form by
Question 5 :
$\displaystyle \frac{3+2i sin \theta}{1-2 i sin \theta}$ will be purely imaginary, if $\theta$ =
Question 7 :
A complex number is represented by an ordered pair $(a,b)$, which of the following is true for $a$ and $b$?
Question 9 :
The complex number system, denoted by $C$, is the set of all ordered pairs of real numbers (that is, $R \times R$) with the operation (denoted by $\times$) which satisfy multiplication
Question 10 :
The least positive integer $n$ such that $\left ( \dfrac{1-i}{1+i} \right )^{2n}=1$ is .....
Question 12 :
If $z$ is a unimodular complex number, then its multiplicative inverse is,
Question 13 :
The real part of $\left[ 1 + \cos \left( \dfrac { \pi } { 5 } \right) + i \sin \left( \dfrac { \pi } { 5 } \right) \right] ^ { - 1 }$ is
Question 15 :
Inequality $a + i b > c + i d$ can be explained only when :
Question 16 :
$z_{1}$ and $z_{2}$ are two non-zero complex numbers such that $|z_{1}|=|z_{2}|$ and $argz_{1}+argz_{2}=\pi$, then $z_{2}$ equals
Question 19 :
The sum of two complex numbers $a + ib$ and $c+ id$ is purely imaginary if
Question 20 :
The complex number system, denoted by $C$, is the set of all ordered pairs of real numbers (that is, $R \times R$) with the operations of addition (denoted by $+$) which satisfy
Question 21 :
The locus of complex number z such that z is purely real and real part is equal to - 2 is
Question 28 :
A complex number is represented by an ordered pair $z: (3,4)$, which of the following is true for $z$?
Question 30 :
If $i^2$ $= -1$, then find the odd one out of the following expressions.
Question 31 :
If ${ Z }_{ 1 }={ a+i } ,a\neq 0\quad and\quad { Z }_{ 2 }={ 1+bi } ,b\neq 0$ are such that ${ z }_{ 1 }=\bar { { z }_{ 2 } } $ then-
Question 33 :
The roots of the equation ${ \left( z+\alpha \beta \right) }^{ 3 }={ \alpha }^{ 3 }$ represent the vertices of a triangle, one of whose sides is of length
Question 34 :
Two complex numbers are represented by ordered pairs $z_1: (2,4)\ \&\ z_2: (-4,5)$, which of the following is real part for $z_1\times z_2=$?
Question 35 :
Which of the the following is correct representation of the complex number: $(a,b)$
Question 38 :
Two complex numbers are represented by ordered pairs $z_1: a+ib\ \&\ z_2: c+id $, which of the following is correct representation for $z_1- z_2=$?
Question 40 :
Put the following in the form of A + iB :<br>$\dfrac{(3 \, - \, 2i)(2 \, + \, 3i)}{(1 \, + \, 2i)(2 \, - \, i)}$
Question 43 :
The real part of $(1 - \cos\theta + 2i \sin\theta)^{-1}$ is:
Question 44 :
If $i^2= -1$, then $1+ i^2+ i^4 +i^6+i^8 +.............to ( 2n +1)$ terms is equal to
Question 45 :
If $z = a + ib$ then its conjugate is $a - ib$. If $1, \omega, \omega^2$ are cube roots of unity then (i) $1 + \omega + \omega^2 = 0$ (ii) $\omega^3 = 1$ The conjugate of $\dfrac{6 - 3i}{7 + i}$ is
Question 46 :
Let $\left| z _ { i } \right| = i , i = 1,2,3,4$ and $\left| 16 z _ { 1 } z _ { 2 } z _ { 3 } + 9 z _ { 1 } z _ { 2 } z _ { 4 } + 4 z _ { 1 } z _ { 3 } z _ { 4 } + z _ { 2 } z _ { 3 } z _ { 4 } \right| = 48 ,$ then the value of $\left| \dfrac { 1 }{ \overline { z } _{ { 1 } } } +\dfrac { 4 }{ \overline { z } _{ { 2 } } } +\dfrac { 9 }{ \overline { z } _{ { 3 } } } +\dfrac { 16 }{ \overline { z } _{ { 4 } } } \right| .$
Question 47 :
If $i^2 = - 1$, then the value of $\displaystyle \sum^{200}_{n = 1} i^n $ is
Question 49 :
Let P$\left( x \right) ={ x }^{ 3 }-6{ x }^{ 2 }+Bx+C$ has 1+5i as a zero and B,C real number, then value of (B+C) is
Question 50 :
If ${z}_{1}=1+2i,\ {z}_{2}=2+3i,\ {z}_{3}=3+4i$, then ${z}_{1},\ {z}_{2}$ and ${z}_{3}$ are collinear.
Question 51 :
$\displaystyle\ \sin ^{ -1 }{ \left\{ \dfrac { 1 }{ i } (z-1) \right\} } $, where $z$ is nonreal, can be the angle of a triangle if<br>
Question 53 : If $A = (3 - 4i)$ and $B = (9 + ki)$, where $k$ is a constant. <div>If $AB - 15 = 60$, then the value of $k$ is</div>
Question 54 :
If $\displaystyle \sum_{k = 0}^{100} i^k = x + iy$, then the values of x and y are
Question 55 :
If $z_1z_2 \in C, z_1^2+z_2^2 \epsilon R, z_1(z_1^2-3z_2^2)=2$ and $z_2(3z_1^2-z_2^2)=11$, then the value of $z_1^2+z_2^2$ is
Question 56 :
Two complex numbers are represented by ordered pairs $z_1: (3,4)\ \&\ z_2: (4,5)$, which of the following is true for $z_1+z_2$?
Question 57 :
If z is complex number such that $z\ne0$ and $R_{e}(z)=0$, then
Question 58 :
$\displaystyle \left [ \left ( \cos \theta +i \sin \theta \right )\left ( \cos \theta -i\sin \theta \right ) \right ]^{-1}$
Question 60 :
If $z + \sqrt {2}|z + 1| + i = 0$ and $z = x + iy$, then
Question 61 : <div>Evaluate:</div>${ \left( \dfrac { cos\dfrac { \pi }{ 8 } -isin\dfrac { \pi }{ 8 } }{ cos\dfrac { \pi }{ 8 } +isin\dfrac { \pi }{ 8 } } \right) }^{ 4 }$<br/>
Question 62 :
If $ z = \dfrac {-1}{2} + i \dfrac {\sqrt3}{2} $, then $ 8 + 10z + 7z^2 $ is equal to :
Question 66 :
Let z be a complex number such that the imaginary part of z is nonzero and a = $z^2 + z + 1$ is real. Then a cannot take the value
Question 67 :
The value of the sum $\displaystyle\ \sum _{n=1}^{13}\left ( i^{n}+i^{n+1} \right ) $, where $\displaystyle\ i=\sqrt{-1} $
Question 68 :
Two complex numbers are represented by ordered pairs $z_1: (6,4)\ \&\ z_2: (4,-5)$, which of the following is real part of $z_1+z_2$?
Question 69 :
For each real $x$, let $f(x) = max \left \{x, x^{2}, x^{3}, x^{4}\right \}$, then $f(x)$ is
Question 70 :
If $z_1, z_2$ are two complex numbers such that $arg(z_1+z_2)=0$ and $Im(z_1z_2)=0$, then.
Question 71 :
Find the conjugates of the following complex numbers:<br><span>$ \dfrac{\left ( 3-2i \right )\left ( 2+3i \right )}{\left(1+2i\right) \left(2-i \right)}$</span>
Question 72 :
If $\alpha +i\beta =\tan^{-1}z$, $z=x+iy$ and $\alpha $ is constant then the locus of $z$ is
Question 73 :
Find the value of the expression <br>$\left(cos \frac{\pi}{2} + i sin \frac{\pi}{2}\right)\left(cos \frac{\pi}{2^2} + i sin \frac{\pi}{2^2}\right) ... $ to $\infty$
Question 75 :
If $\displaystyle z =-5+2\sqrt{-4} $, then the value of $\displaystyle z^{2}+10z+41 $ is equal to<br>
Question 76 :
If $z=\cos { \theta } +i\sin { \theta } $, then
Question 77 :
The solution of the equation |z|z = 1 + 2i is
Question 79 :
If $x + i y = \dfrac{3}{2 + cos \theta + i sin \theta}$, then $x^2 + y^2$ is equal to
Question 80 :
Consider the complex numbers $z=\dfrac{(1 - i \sin \theta) }{(1 + i \cos \theta)},$<br>The value of $\theta$ for which z is purely imaginary real are,
Question 81 :
Assertion: The product of the real part of the roots of the equation ${ z }^{ 2 }+4z+4i=0$ and $2-2\sqrt { 2 } $
Reason: The roots of equation is given as $\displaystyle2\left( -1\pm \sqrt [ 4 ]{ 2 } { e }^{ -i\frac { \pi }{ 8 } } \right) $ and then take the product of real part.
Question 82 :
$i \, \log \left(\dfrac{x - i}{x + i}\right)$ is equal to
Question 83 :
Number of roots of the equation $z^{10} - z^5 - 992 = 0$ where real parts are negative is
Question 86 :
<br>The locus of $z$ such that $\dfrac { { z }^{ 2 } }{ z-1 } $ is always real is<br>
Question 87 :
The real part of ${ \left( 1-\cos { \theta } +i\sin { \theta } \right) }^{ -1 }$ is
Question 88 :
Find the product and write the answer in standard form.<br/>$\left( 2-4i \right) \left( 3+7i \right) $
Question 94 :
$ | \frac{z_1 - 2z_2}{2 - z_1\bar{z}_2} | = 1$ and $|z_2| \neq 1$ then the value of $|z_1|$ is
Question 95 :
The two complex numbers satisfying the equation $\displaystyle z\bar{z}-\left ( 1+i \right )z-\left ( 3+2i \right )\bar{z}+\left ( 1+5i \right )=0$ are
Question 96 :
If $\dfrac{x+3i}{2+iy}=1-i$, then the value of $\left ( 5x-7y \right )^2$ is
Question 97 :
If $w = \dfrac { z } { z - \dfrac { 1 } { 3 } i }$ and $| w | = 1$ then $z$ lies on
Question 100 :
Two complex numbers are represented by ordered pairs $z_1: (2,4)\ \&\ z_2: (-4,5)$, which of the following is imaginary part of $z_1\times z_2=$?
