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4-1.Complex numbers
medium
$\left[i^{18}+\left(\frac{1}{i}\right)^{25}\right]^{3}$ का मान ज्ञात कीजिए
A
$2-2 i$
B
$2-2 i$
C
$2-2 i$
D
$2-2 i$
Solution
$\left[i^{18}+\left(\frac{1}{i}\right)^{25}\right]^{3}$
$=\left[i^{4 \times 4+2}+\frac{1}{i^{4 \times 6+1}}\right]^{3}$
$=\left[\left(i^{4}\right)^{4} \cdot i^{2}+\frac{1}{\left(i^{4}\right)^{6} \cdot i}\right]^{3}$
$=\left[i^{2}+\frac{1}{i}\right]^{3} \quad\left[i^{4}=1\right]$
$=\left[-1+\frac{1}{i} \times \frac{i}{i}\right]^{3} \quad\left[i^{2}=-1\right]$
$=\left[-1+\frac{i}{i^{2}}\right]^{3}$
$=[-1-i]^{3}$
$=(-1)^{3}[1+i]^{3}$
$=-\left[1^{3}+i^{3}+3 \cdot 1 \cdot i(1+i)\right]$
$=-\left[1+i^{3}+3 i+3 i^{2}\right]$
$=-[1-i+3 i-3]$
$=-[-2+2 i]$
$=2-2 i$
Standard 11
Mathematics
