Question

what is the moduli and arguments of this problam 3-i/2+i + 3+i/2-i​

Answer 1

Answer:

The modulus and argument of the given complex number are 2 units & 0 radian respectively.

Step-by-step-explanation:

We have given a complex number.

We have to find its modulus and argument.

The given complex number is

$\displaystyle{\sf\:z\:=\:\dfrac{3\:-\:i}{2\:+\:i}\:+\:\dfrac{3\:+\:i}{2\:-\:i}}$

$\displaystyle{\implies\sf\:z\:=\:\dfrac{(\:3\:-\:i\:)\:(\:2\:-\:i\:)\:+\:(\:3\:+\:i\:)\:(\:2\:+\:i\:)}{(\:2\:+\:i\:)\:(\:2\:-\:i\:)}}$

$\displaystyle{\implies\sf\:z\:=\:\dfrac{3\:(\:2\:-\:i\:)\:-\:i\:(\:2\:-\:i\:)\:+\:3\:(\:2\:+\:i\:)\:+\:i\:(\:2\:+\:i\:)}{2^2\:-\:(\:i\:)^2}}$

$\displaystyle{\implies\sf\:z\:=\:\dfrac{6\:-\:\cancel{3i}\:-\:\cancel{2i}\:+\:i^2\:+\:6\:+\:\cancel{3i}\:+\:\cancel{2i}\:+\:i^2}{4\:-\:(\:-\:1\:)}}$

$\displaystyle{\implies\sf\:z\:=\:\dfrac{6\:+\:6\:+\:2i^2}{4\:+\:1}}$

$\displaystyle{\implies\sf\:z\:=\:\dfrac{12\:-\:2}{5}}$

$\displaystyle{\implies\sf\:z\:=\:\cancel{\dfrac{10}{5}}}$

$\displaystyle{\implies\pink{\sf\:z\:=\:2\:+\:0i\:}}$

Now, we know that,

$\displaystyle{\sf\:|\:z\:|\:=\:\sqrt{a^2\:+\:b^2}}$

$\displaystyle{\implies\sf\:|\:z\:|\:=\:\sqrt{2^2\:+\:0^2}}$

$\displaystyle{\implies\sf\:|\:z\:|\:=\:\sqrt{4}}$

$\displaystyle{\implies\underline{\boxed{\red{\sf\:|\:z\:|\:=\:2\:units\:}}}}$

Now,

$\displaystyle{\sf\:\arg\:z\:(\:\theta\:)\:=\:\tan^{-1}\:\left(\:\dfrac{b}{a}\:\right)}$

$\displaystyle{\implies\sf\:\arg\:z\:=\:\tan^{-1}\:\left(\:\dfrac{0}{2}\:\right)}$

$\displaystyle{\implies\sf\:\arg\:z\:=\:\tan^{-1}\:0}$

$\displaystyle{\implies\underline{\boxed{\red{\sf\:\arg\:z\:=\:0\:rad\:}}}}$

∴ The modulus and argument of the given complex number are 2 units & 0 radians respectively.