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Complex Numbers and Quadratic Equations — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsComplex Numbers and Quadratic Equations1 Mark
MCQ
The polar form of $(\text{i}^{25})^3$ is:
  • A
    $\cos\frac{\pi}{2}+\text{i}\sin\frac{\pi}{2}$
  • B
    $\cos\pi+\text{i}\sin\pi$
  • C
    $\cos\pi-\text{i}\sin\pi$
  • ✓
    $\cos\frac{\pi}{2}-\text{i}\sin\frac{\pi}{2}$
✓

Answer

Correct option: D.
$\cos\frac{\pi}{2}-\text{i}\sin\frac{\pi}{2}$
$(\text{i}^{25})^3=(\text{i})^{75}$
$=(\text{i})^{4\times18+3}$
$=(\text{i})^3$
$=-\text{i} \ (\because\text{i}^4=1)$
Let $\text{z}=0-\text{i}$
Since, the point $(0,−1)$ lies on the negative direction of imaginary axis.
Therefore, $\text{arg(z)}=\frac{-\pi}{2}$
Modulus, $\text{r}=|\text{z}|=|1|=1$
$\therefore$ Polar form $=\text{r}(\cos\theta+\text{i}\sin\theta)$
$=\cos\Big(\frac{-\pi}{2}\Big)+\text{i}\sin\Big(\frac{-\pi}{2}\Big)$
$=\cos\frac{\pi}{2}-\text{i}\sin\frac{\pi}{2}$

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