Test Index

AP DSC School Assistant Math Model Test 1

Section: Subject Specific

Question : 113 of 160

Marks: +1, -0

(\bar{z^{-1}})(\bar{z})

= ?

Solution:

Concept:

Complex conjugate and multiplicative inverse: For a complex number

z = x + iy

, its conjugate is

\bar{z}=x-iy

and its inverse is

z^{-1} = \frac{1}{z}

. The product

(\bar{z^{-1}})(\bar{z})

simplifies using the property

z \cdot \bar{z} = |z|^2

Explanation:

Let

z = x + iy

, where

x

and

y

are real numbers. Then

\bar{z} = x - iy

The multiplicative inverse is:

z^{-1} = \frac{1}{z} = \frac{1}{x+iy}

Rationalising the denominator:

z^{-1} = \frac{x-iy}{(x+iy)(x-iy)} = \frac{x-iy}{x^2+y^2}

Now take the conjugate of

z^{-1}

\bar{z^{-1}} = \frac{x+iy}{x^2+y^2}

Multiply

\bar{z^{-1}}

\bar{z}

(\bar{z^{-1}})(\bar{z}) = \frac{x+iy}{x^2+y^2} \cdot (x-iy) = \frac{(x+iy)(x-iy)}{x^2+y^2} = \frac{x^2 + y^2}{x^2 + y^2} = 1

Answer:

The correct option is A (1).

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