Question 1 :
If {tex} ( \cos \theta + i \sin \theta ) ( \cos 2 \theta + i \sin 2 \theta ) \cdots ( \cos n \theta + i \sin n \theta ) = 1 {/tex} then the value of {tex} \theta {/tex} is, {tex} m \in N {/tex}

Question 2 :
Which of the following is equal to {tex} \sqrt [ 3 ] { - 1 } ? {/tex}

Question 3 :
If {tex} \alpha {/tex} is the {tex} n ^ { \text {th } } {/tex} root of unity, then {tex} 1 + 2 \alpha + 3 \alpha ^ { 2 } + \cdots {/tex} to {tex} n {/tex} terms equal to

Question 4 :
If {tex} |2 z - 1| = | z - 2 | {/tex} and {tex} z _ { 1 } , z _ { 2 } , z _ { 3 } {/tex} are complex numbers such that {tex} \left| z _ { 1 } - \alpha \right| < \alpha , \left| z _ { 2 } - \beta \right| < \beta , {/tex} then {tex} \left| \frac { z _ { 1 } + z _ { 2 } } { \alpha + \beta } \right| {/tex}

Question 5 :
Let {tex} z , w {/tex} be complex numbers such that {tex} \bar { z } + i \bar { w } = 0 {/tex} and arg {tex} z w = {/tex} {tex} \pi . {/tex} Then arg {tex} z {/tex} equals

Question 6 :
The greatest positive argument of complex number satisfying {tex} | z - 4 | = \operatorname { Re } ( z ) {/tex} is

Question 7 :
{tex} z _ { 1 } {/tex} and {tex} z _ { 2 } {/tex} are two distinct points in an Argand plane. If {tex} a \left| z _ { 1 } \right| = b \left| z _ { 2 } \right| {/tex} (where {tex} a , b \in R ) , {/tex} then the point {tex} \left( a z _ { 1 } / b z _ { 2 } \right) + \left( b z _ { 2 } / a z _ { 1 } \right) {/tex} is a point on the

Question 8 :
Roots of the equations are {tex} ( z + 1 ) ^ { 5 } = ( z - 1 ) ^ { 5 } {/tex} are

Question 9 :
Consider the equation {tex} 10 z ^ { 2 } - 3 i z - k = 0 , {/tex} where {tex} z {/tex} is a complex variable and {tex} i ^ { 2 } = - 1 . {/tex} Which of the following statements is true?

Question 10 :
The polynomial {tex} x ^ { 6 } + 4 x ^ { 5 } + 3 x ^ { 4 } + 2 x ^ { 3 } + x + 1 {/tex} is divisible by where {tex}\mathrm w {/tex} is cube root of units

Question 11 :
The polynomial {tex} x ^ { 6 } + 4 x ^ { 5 } + 3 x ^ { 4 } + 2 x ^ { 3 } + x + 1 {/tex} is divisible by where {tex}\mathrm w {/tex} is cube root of units

Question 12 :
If {tex} z ( 1 + a ) = b + i c {/tex} and {tex} a ^ { 2 } + b ^ { 2 } + c ^ { 2 } = 1 , {/tex} then {tex} [ ( 1 + i z ) / ( 1 - i z )] = {/tex}

Question 13 :
{tex} P ( z ) {/tex} be a variable point in the Argand plane such that {tex} | z | = {/tex} minimum {tex} \{ | z - 1 | , | z + 1 | \} {/tex} then {tex} z + \bar { z } {/tex} will be equal to

Question 14 :
Let {tex} a {/tex} be a complex number such that {tex} | a | < 1 {/tex} and {tex} z _ { 1 } , z _ { 2 } , z _ { 3 } , \ldots {/tex} be the vertices of a polygon such that {tex} z _ { k } = 1 + a + a ^ { 2 } + \cdots + a ^ { k - 1 } {/tex} for all {tex} k = 1,2,3 , \ldots {/tex} then {tex} z _ { 1 } , z _ { 2 } , \ldots {/tex} lie within the circle

Question 15 :
If {tex} A \left( z _ { 1 } \right) , B \left( z _ { 2 } \right) , C \left( z _ { 3 } \right) {/tex} are the vertices of the triangle {tex} A B C {/tex} such that {tex} \left( z _ { 1 } - z _ { 2 } \right) / \left( z _ { 3 } - z _ { 2 } \right) = ( 1 / \sqrt { 2 } ) + ( i / \sqrt { 2 } ) , {/tex} the triangle {tex} A B C {/tex} is