JEE Maths›Complex Numbers & Quadratic Equations›Visual Solution
Visual SolutionPYQ 2023 · Jan Session 1Standard
Question
If z=1+2i1−iz = \frac{1+2i}{1-i}, then Im(z)\text{Im}(z) equals:
(A)32\frac{3}{2}
(B)12\frac{1}{2}
(C)−12-\frac{1}{2}
(D)−32\frac{-3}{2}
Solution Path
Rationalize z=1+2i1−iz = \dfrac{1+2i}{1-i} by multiplying by 1+i1+i\dfrac{1+i}{1+i} to get z=−12+32iz = -\dfrac{1}{2} + \dfrac{3}{2}i, so Im(z)=32\text{Im}(z) = \dfrac{3}{2}.
01Question Setup
1/4
Find Im(z)\text{Im}(z) where z=1+2i1−iz = \dfrac{1+2i}{1-i}.
Im(z)=  ?\text{Im}(z) = \;?
02Rationalize
2/4
Multiply numerator and denominator by the conjugate 1+i1+i. Denominator becomes 22, numerator expands to −1+3i-1+3i.
(1+2i)(1+i)2=−1+3i2\dfrac{(1+2i)(1+i)}{2} = \dfrac{-1+3i}{2}
03Extract Im(z)KEY INSIGHT
3/4
z=−12+32iz = -\dfrac{1}{2} + \dfrac{3}{2}i, so the imaginary part is 32\dfrac{3}{2}.
Im(z)=32\text{Im}(z) = \dfrac{3}{2}
04Final Answer
4/4
Im(z)=32\text{Im}(z) = \dfrac{3}{2}.
32\boxed{\dfrac{3}{2}}
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