The modulus of (1-i)/(3+i)+(4i)/(5) is (A) sqrt5 unit (B) (sqrt11)/(5) unit (C) (sqrt5)/(5) unit (D) (sqrt12)/(5) unit

Simplify Expression: First, we need to simplify the expression (1-i)/(3+i). To do this, we multiply the numerator and the denominator by the conjugate of the denominator to remove the imaginary part from the denominator. (1-i)/(3+i) * (3-i)/(3-i) = (3 - i - 3i + i^2)/(9 - i^2)Numerator and Denominator Simplification: Simplify the numerator and denominator of the resulting fraction. (3 - i - 3i + i^2)/(9 - i^2) = (3 - 4i - 1)/(9 + 1) = (2 - 4i)/10 = 1/5 - (2/5)iAddition of Terms: Now, we add the second term (4i)/(5) to the simplified first term. (1/5 - (2/5)i) + (4/5)i = 1/5 + (2/5)iModulus Calculation: To find the modulus of the complex number 1/5 + (2/5)i, we use the formula for the modulus of a complex number a + bi, which is sqrt(a^2 + b^2). Modulus = sqrt((1/5)^2 + (2/5)^2) = sqrt(1/25 + 4/25) = sqrt(5/25)Final Modulus Simplification: Simplify the square root to find the modulus. sqrt(5/25) = sqrt(1/5) = sqrt(1)/sqrt(5) = 1/sqrt(5) = sqrt(5)/5

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The modulus of $\newline$ $\frac{1-i}{3+i}+\frac{4i}{5}$ is $\newline$ (A) $\sqrt{5}$ unit $\newline$ (B) $\frac{\sqrt{11}}{5}$ unit $\newline$ (C) $\frac{\sqrt{5}}{5}$ unit $\newline$ (D) $\frac{\sqrt{12}}{5}$ unit

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Algebra 2

Sum of finite series starts from 1

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Q. The modulus of

\newline

\frac{1-i}{3+i}+\frac{4i}{5}

\newline

(A)

\sqrt{5}

unit

\newline

(B)

\frac{\sqrt{11}}{5}

unit

\newline

(C)

\frac{\sqrt{5}}{5}

unit

\newline

(D)

\frac{\sqrt{12}}{5}

unit

Simplify Expression: First, we need to simplify the expression $(1-i)/(3+i)$ . To do this, we multiply the numerator and the denominator by the conjugate of the denominator to remove the imaginary part from the denominator. $\newline$ $(1-i)/(3+i) \cdot (3-i)/(3-i) = (3 - i - 3i + i^2)/(9 - i^2)$
Numerator and Denominator Simplification: Simplify the numerator and denominator of the resulting fraction. $\newline$ $(3 - i - 3i + i^2)/(9 - i^2) = (3 - 4i - 1)/(9 + 1) = (2 - 4i)/10 = \frac{1}{5} - \frac{2}{5}i$
Addition of Terms: Now, we add the second term $(\frac{4i}{5})$ to the simplified first term. $(\frac{1}{5} - \frac{2}{5}i) + \frac{4}{5}i = \frac{1}{5} + \frac{2}{5}i$
Modulus Calculation: To find the modulus of the complex number $\frac{1}{5} + \left(\frac{2}{5}\right)i$ , we use the formula for the modulus of a complex number $a + bi$ , which is $\sqrt{a^2 + b^2}$ . $\newline$ Modulus = $\sqrt{\left(\frac{1}{5}\right)^2 + \left(\frac{2}{5}\right)^2} = \sqrt{\frac{1}{25} + \frac{4}{25}} = \sqrt{\frac{5}{25}}$
Final Modulus Simplification: Simplify the square root to find the modulus. $\sqrt{\frac{5}{25}} = \sqrt{\frac{1}{5}} = \frac{\sqrt{1}}{\sqrt{5}} = \frac{1}{\sqrt{5}} = \frac{\sqrt{5}}{5}$