(A) Let the two complex numbers be $z_1$ and $z_2$ such that $|z_1| = 1$ and $|z_2| = 1$. Using the property of the modulus of the product of complex

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(A) Let the two complex numbers be $z_1$ and $z_2$ such that $|z_1| = 1$ and $|z_2| = 1$. Using the property of the modulus of the product of complex numbers,we have $|z_1 z_2| = |z_1| |z_2|$. Substituting the given values,we get $|z_1 z_2| = 1 \times 1 = 1$. Thus,the product $z_1 z_2$ is also a complex number with unit modulus.

Class 11 Mathematics 4-1.Complex numbers Conjugate and Modulus of complex numbers

Medium

The product of two complex numbers,each of unit modulus,is also a complex number of:

A
Unit modulus
B
Less than unit modulus
C
Greater than unit modulus
D
None of these

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Class 11

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4-1.Complex numbers

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Conjugate and Modulus of complex numbers

Find the real numbers $x$ and $y$ if $(x-iy)(3+5i)$ is the conjugate of $-6-24i$.

Medium

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

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If $|z_1| = |z_2|$ and $\text{amp}(z_1) + \text{amp}(z_2) = 0$,then

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The modulus of $\frac{1-i}{3+i}+\frac{4i}{5}$ is

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If $\frac{z - i}{z + i}$ $(z \ne -i)$ is a purely imaginary number,then $z \cdot \bar{z}$ is equal to

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

The product of two complex numbers,each of unit modulus,is also a complex number of:

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Find the real numbers $x$ and $y$ if $(x-iy)(3+5i)$ is the conjugate of $-6-24i$.

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)$

If $|z_1| = |z_2|$ and $\text{amp}(z_1) + \text{amp}(z_2) = 0$,then

The modulus of $\frac{1-i}{3+i}+\frac{4i}{5}$ is

If $\frac{z - i}{z + i}$ $(z \ne -i)$ is a purely imaginary number,then $z \cdot \bar{z}$ is equal to

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