Explore popular questions from Complex Numbers and Quadratic Equations for JEE Advanced. This collection covers Complex Numbers and Quadratic Equations previous year JEE Advanced questions hand picked by experienced teachers.

## Chemistry

Complex Numbers and Quadratic Equations

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Q 1. Let {tex} z {/tex} and {tex} \omega {/tex} be two complex numbers such that {tex} | z | \leq 1, {/tex} {tex} | \omega | \leq 1 {/tex} and {tex} | z + i \omega | = | z - i \bar { \omega } | = 2 {/tex} then z equals

A

1 or {tex} i {/tex}

B

{tex} i {/tex} or {tex} - i {/tex}

1 or -1

D

{tex} i {/tex} or - 1

##### Explanation

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Q 2. Let {tex}\omega=\frac {-1}{2}+i\frac{\sqrt{3}}{2}{/tex} ,then value of the det.is

A

{tex}3\omega{/tex}

{tex}3\omega(\omega-1){/tex}

C

{tex}3\omega^2{/tex}

D

{tex}3\omega(1-\omega){/tex}

##### Explanation

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Q 3. If {tex} \frac { w - \bar { w } z } { 1 - z } {/tex} is purely real where {tex} w = \alpha + i \beta , \beta \neq 0 {/tex} and {tex} z \neq 1 {/tex} then the set of the values of {tex} z {/tex} is

A

{tex} \{ z: | z | = 1 \} {/tex}

B

{tex} \{ z: z = \overline z \} {/tex}

C

{tex} \{ z: z \neq 1 \} {/tex}

{tex} \{ z: | z | = 1 , z \neq 1 \} {/tex}

##### Explanation

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Q 4. Let complex numbers {tex} \alpha {/tex} and {tex} \frac { 1 } { \alpha } {/tex} lie on circles {tex} \left( x - x _ { 0 } \right) ^ { 2 } {/tex} {tex} + \left( y - y _ { 0 } \right) ^ { 2 } = r ^ { 2 } {/tex} and {tex} \left( x - x _ { 0 } \right) ^ { 2 } + \left( y - y _ { 0 } \right) ^ { 2 } = 4 r ^ { 2 } {/tex} respectively. If {tex} z _ { 0 } = x _ { 0 } + {/tex} {tex}iy _ { 0 } {/tex} satisfies the equation {tex} 2 \left| z _ { 0 } \right| ^ { 2 } = r ^ { 2 } + 2 , {/tex} then {tex} | \alpha | = {/tex}

A

{tex} \frac { 1 } { \sqrt { 2 } } {/tex}

B

{tex} \frac { 1 } { 2 } {/tex}

{tex} \frac { 1 } { \sqrt { 7 } } {/tex}

D

{tex} \frac { 1 } { 3 } {/tex}

##### Explanation

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Q 5. If {tex} a _ { 1 } , a _ { 2 } \ldots \ldots , a _ { n } {/tex} are positive real numbers whose product is a fixed number {tex} c , {/tex} then the minimum value of
{tex} a _ { 1 } + a _ { 2 } + \ldots \ldots + a _ { n - 1 } + 2 a _ { n } {/tex} is

{tex} n ( 2 c ) ^ { 1 / n } {/tex}

B

{tex} ( n + 1 ) c ^ { 1 / n } {/tex}

C

{tex} 2 n c ^ { 1 / n } {/tex}

D

{tex} ( n + 1 ) ( 2 c ) ^ { 1 / n } {/tex}