If Z_1 and Z_2 are conjugate complex numbers, | vedclass

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(A) Given $Z_1$ and $Z_2$ are conjugate complex numbers. Let $Z_1 = a + ib$, then $Z_2 = a - ib$.$(A) Z_1 Z_2 = (a + ib)(a - ib) = a^2 + b^2 = |Z_1|^2

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$A-3, B-1, C-2, D-4$

Vedclass · Answer

(A) Given $Z_1$ and $Z_2$ are conjugate complex numbers. Let $Z_1 = a + ib$, then $Z_2 = a - ib$.$(A) Z_1 Z_2 = (a + ib)(a - ib) = a^2 + b^2 = |Z_1|^2$. Thus, $A-3$.$(B) Z_1 + Z_2 = (a + ib) + (a - ib) = 2a$. If $Z_1 + Z_2 = 0$, then $2a = 0 \Rightarrow a = 0$. This represents the imaginary axis. Thus, $B-1$.$(C) 	ext{Im}(Z_1) = b$. Also, $	ext{Im}(-Z_2) = 	ext{Im}(-(a - ib)) = 	ext{Im}(-a + ib) = b$. Thus, $	ext{Im}(Z_1) = 	ext{Im}(-Z_2)$. So, $C-2$.$(D) 	ext{Re}(Z_1) = a$ and $	ext{Re}(Z_2) = a$. Thus, $	ext{Re}(Z_1) = 	ext{Re}(Z_2)$. So, $D-4$.The correct matching is $A-3, B-1, C-2, D-4$.

Column-$I$	Column-$II$
$A. Z_1 Z_2$	$1. \text{imaginary axis}$
$B. Z_1 + Z_2 = 0$	$2. \text{Im}(-Z_2)$
$C. \text{Im}(Z_1)$	$3. \|Z_1\|^2$
$D. \text{Re}(Z_1)$	$4. \text{Re}(Z_2)$

If $Z_1$ and $Z_2$ are conjugate complex numbers, match the items in Column-$I$ with Column-$II$:
Column-$I$ Column-$II$
$A. Z_1 Z_2$ $1. \text{imaginary axis}$
$B. Z_1 + Z_2 = 0$ $2. \text{Im}(-Z_2)$
$C. \text{Im}(Z_1)$ $3. |Z_1|^2$
$D. \text{Re}(Z_1)$ $4. \text{Re}(Z_2)$

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If $Z_1$ and $Z_2$ are conjugate complex numbers, match the items in Column-$I$ with Column-$II$:Column-$I$Column-$II$$A. Z_1 Z_2$$1. \text{imaginary axis}$$B. Z_1 + Z_2 = 0$$2. \text{Im}(-Z_2)$$C. \text{Im}(Z_1)$$3. |Z_1|^2$$D. \text{Re}(Z_1)$$4. \text{Re}(Z_2)$