Q153 / 270 · NDA 2023 (Phase 2) · Mathematics

What is the value of $\sqrt{12+5 i}+\sqrt{12-5 i}$, where $i=\sqrt{-1}$ ?

  • A
    24
  • B
    25
  • Correct answer
    C
    $5 \sqrt{2}$
  • D
    $5(\sqrt{2}-1)$

Solution

Let $\sqrt{12+5 \mathrm{i}}=\mathrm{x}+\mathrm{iy} \Rightarrow 12+5 \mathrm{i}=(\mathrm{x}+\mathrm{iy})^2$ $\Rightarrow \mathrm{x}^2-\mathrm{y}^2=12$...(i) and $2 x y=5$...(ii) On solving (i) and (ii), we get $x=\frac{5}{\sqrt{2}}, \quad y=\frac{1}{\sqrt{2}}$ So, $\sqrt{12+5 \mathrm{i}}=\frac{5}{\sqrt{2}}+\frac{\mathrm{i}}{\sqrt{2}}$ Similarly, $\sqrt{12-5 i}=\frac{5}{\sqrt{2}}-\frac{i}{\sqrt{2}}$ Now, $\sqrt{12+5 i}+\sqrt{12-5 i}=\frac{5}{\sqrt{2}}+\frac{\mathrm{i}}{\sqrt{2}}+\frac{5}{\sqrt{2}}-\frac{\mathrm{i}}{\sqrt{2}}$ $=5 \sqrt{2}$

Hence, the correct answer is option: 'C'.

Relevant Questions