If z is a complex number and l 1- l 2- l 3- m 1- m 2- m 3 are all real- thenl1z-m1 barz- m- 1 z-l- 1 barz- m- 1 z-l- 11- 2 z - m - 2 bar z - m - 2 z - l - 2 bar z - m - 2 z - l - 21- 3 z-m- 3 barz- m- 3 z-1- 3 barz- m- 3 z-l- 3lendvmatrix is equal toA- - z -2B- l 1 l 2 l 3- m 1 m 2 m 32- z -2C- 0D- 3

The correct option is C 0l_1 amp; m_1 amp; 0 l_2 amp; m_2 amp; 0 l_3 amp; m_3 amp; 0 × z amp; z̅ amp; 1 z̅ amp; z amp; z 1 am ...

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Byju's Answer

Standard XII

Mathematics

Modulus of a Complex Number

If z is a com...

Question

If z is a complex number and $l_{1}, l_{2}, l_{3}, m_{1}, m_{2}, m_{3}$ are all real, then
$∣ ∣ ∣ ∣ \begin{matrix} l_{1} z + m_{1} ¯ z & m_{1} z + l_{1} ¯ z & m_{1} z + l_{1} l_{2} z + m_{2} ¯ z & m_{2} z + l_{2} ¯ z & m_{2} z + l_{2} l_{3} z + m_{3} ¯ z & m_{3} z + l_{3} ¯ z & m_{3} z + l_{3} \end{matrix} ∣ ∣ ∣ ∣$ is equal to

{| z |}^{2}

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3

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(l_{1} l_{2} l_{3} + m_{1} m_{2} m_{3})^{2} {| z |}^{2}

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0

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Solution

The correct option is D $0$
$∣ ∣ ∣ ∣ \begin{matrix} l_{1} & m_{1} & 0 l_{2} & m_{2} & 0 l_{3} & m_{3} & 0 \end{matrix} ∣ ∣ ∣ ∣ \times ∣ ∣ ∣ ∣ \begin{matrix} z & ¯ z & 1 ¯ z & z & z 1 & z & ¯ z \end{matrix} ∣ ∣ ∣ ∣ = 0$

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If z is a complex number and l1,l2,l3,m1,m2,m3 are all real, then ∣∣ ∣∣l1z+m1¯zm1z+l1¯zm1z+l1l2z+m2¯zm2z+l2¯zm2z+l2l3z+m3¯zm3z+l3¯zm3z+l3∣∣ ∣∣ is equal to

The correct option is D 0∣∣ ∣∣l1m10l2m20l3m30∣∣ ∣∣×∣∣ ∣∣z¯z1¯zzz1z¯z∣∣ ∣∣=0

The correct option is D $0$
$∣ ∣ ∣ ∣ \begin{matrix} l_{1} & m_{1} & 0 l_{2} & m_{2} & 0 l_{3} & m_{3} & 0 \end{matrix} ∣ ∣ ∣ ∣ \times ∣ ∣ ∣ ∣ \begin{matrix} z & ¯ z & 1 ¯ z & z & z 1 & z & ¯ z \end{matrix} ∣ ∣ ∣ ∣ = 0$