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Question 1 :
The locus of the centre of a circle which touches: the circle {tex} \left| z - z _ { 1 } \right| = a {/tex} and {tex} \left| z - z _ { 2 } \right| = b {/tex} externally {tex} ( z , z _ { 1 } \& z _ { 2 } {/tex} are complex numbers) will be

Question 2 :
If {tex} z = i \log ( 2 - \sqrt { - 3 } ) , {/tex} then {tex} \cos z = {/tex}

Question 3 :
If {tex} z = 3 / ( 2 + \cos \theta + i \sin \theta ) , {/tex} then locus of {tex} z {/tex} is

Question 4 :
If {tex} k > 0 , | z | = | w | = k {/tex} and {tex} \alpha = \frac { z - \bar { w } } { k ^ { 2 } + z \bar { w } } , {/tex} then {tex} \operatorname { Re } ( \alpha ) {/tex} equals

Question 5 :
The roots of the equation {tex} t ^ { 3 } + 3 a t ^ { 2 } + 3 b t + c = 0 {/tex} are {tex} z _ { 1 } , z _ { 2 } , z _ { 3 } {/tex} which represent the vertices of an equilateral triangle, then

Question 6 :
If &alpha;, &beta; are non-real roots of ax² + bx + c = 0 (a, b, c &isin; R), then-<br>

Question 7 :
If {tex} z _ { 1 } , z _ { 2 } , z _ { 3 } {/tex} are three complex numbers and<br>{tex} A = \left| \begin{array} { l l l } { \arg z _ { 1 } } & { \arg z _ { 2 } } & { \arg z _ { 3 } } \\ { \arg z _ { 2 } } & { \arg z _ { 3 } } & { \arg z _ { 1 } } \\ { \arg z _ { 3 } } & { \arg z _ { 1 } } & { \arg z _ { 2 } } \end{array} \right| {/tex}<br>then {tex} A {/tex} is divisible by

Question 8 :
If {tex} z {/tex} lies on the circle {tex} | z | = 1 , {/tex} then 2{tex} / z {/tex} lies on a

Question 9 :
If {tex} z _ { 1 } {/tex} is a root of the equation {tex} a _ { 0 } z ^ { n } + a _ { 1 } z ^ { n - 1 } + \cdots + a _ { n - 1 } z + a _ { n } = 3, {/tex} where {tex} \left| a _ { i } \right| < 2 {/tex} for {tex} i = 0,1 , \ldots , n . {/tex} Then

Question 10 :
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 11 :
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 12 :
The value of {tex} z {/tex} satisfying the equation log {tex} z {/tex} + log {tex} z^2 {/tex}+........+log {tex} z^n {/tex}={tex}0 {/tex} is

Question 13 :
If z₁ = 1 + 2i and z₂ = 3 + 5i, then Re $\lbrack\overline{z_{2}}z_{1}/z_{2}\rbrack$ is equal to

Question 14 :
If sec α and cosec α are the roots of the equation x² − px + q = 0, then

Question 15 :
If α, β be the roots of the quadratic equation ax² + bx + c = 0 and k be a real number, then the condition so that α &lt; k &lt; β is given by