Question:
A complex number is given by \(Z = \frac{1+2i}{1-(1-i)^2}\). What is the modulus of \(Z\)?
एक सम्मिश्र संख्या \(Z = \frac{1+2i}{1-(1-i)^2}\) द्वारा दी गई है। \(Z\) का मापांक क्या है?
Options:
- 4
- 2
- 1
- 1/2
- 4
- 2
- 1
- 1/2
Correct Answer: c)
Explanation:
Simplify the denominator:
\[(1 – i)^2 = 1 – 2i + i^2 = 1 – 2i – 1 = -2i\]
Denominator \(= 1 – (-2i) = 1 + 2i\).
Therefore:
\[Z = \frac{1 + 2i}{1 + 2i} = 1\]
The modulus is:
\[|Z| = |1| = 1\]
Hence, the correct option is (c).
हर को सरल करने पर:
\[(1 – i)^2 = 1 – 2i – 1 = -2i\]
हर \(= 1 – (-2i) = 1 + 2i\)।
अतः सम्मिश्र संख्या:
\[Z = \frac{1 + 2i}{1 + 2i} = 1\]
मापांक:
\[|Z| = |1| = 1\]
अतः सही विकल्प (c) है।