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Complex Numbers — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSComplex Numbers4 Marks
Question
If $\text{z}_1=2-\text{i}, \ \text{z}_2=1+\text{i},$ Find $\Big|\frac{\text{z}_1+\text{z}_2+1}{\text{z}_1-\text{z}_2+\text{i}}\Big|.$
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Answer

If $\text{z}=\text{x}+\text{iy}$ then $|\text{z}|=\sqrt{\text{x}^2+\text{y}^2}$ We have, $\text{z}_1=2-\text{i}, \ \text{z}_2=1+\text{i}$ $\text{z}_1+\text{z}_2=2-\text{i}+1+\text{i}$ $=3$ And $\text{z}_1-\text{z}_2=2-\text{i}-1-\text{i}$ $=1-2\text{i}$ $\Big|\frac{\text{z}_1+\text{z}_2+1}{\text{z}_1-\text{z}_2+\text{i}}\Big|=\frac{3+1}{1-2\text{i}+\text{i}}$ $=\frac{4}{1-\text{i}}$ $=\frac{4}{1-\text{i}}\times\frac{1+\text{i}}{1+\text{i}}$ $=\frac{4(1+\text{i})}{1^2+1^2}$ $=\frac{4(1+\text{i})}{2}$ $=2{(1+\text{i})}$ $\therefore\Big|\frac{\text{z}_1+\text{z}_2+1}{\text{z}_1-\text{z}_2+\text{i}}\Big|=|2(1+\text{i})| $ $=|2||1+\text{i}| \ \big(\therefore|\text{z}_1\text{z}_2|=|\text{z}_1|\times|\text{z}_2|\big)$ $=2\times\sqrt{1^2+1^2}$ $=2\times\sqrt{2}$ $=2\sqrt{2}$

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