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JEE Advanced

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.

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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}

Explanation