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If z=5−2i1+i z=\frac{5-2i}{1+i} , which of the following options is equivalent to z z , where the imaginary number i i is such that i2=−1 i^2=-1 ?\newline(A) 0 0 \newline(B) i i \newline(C) 7(1−i)2 \frac{7(1-i)}{2} \newline(D) 3−7i2 \frac{3-7i}{2}

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Q. If z=5−2i1+i z=\frac{5-2i}{1+i} , which of the following options is equivalent to z z , where the imaginary number i i is such that i2=−1 i^2=-1 ?\newline(A) 0 0 \newline(B) i i \newline(C) 7(1−i)2 \frac{7(1-i)}{2} \newline(D) 3−7i2 \frac{3-7i}{2}
  1. Multiply by Conjugate: To remove the imaginary number from the denominator, we multiply the numerator and denominator by the complex conjugate of the denominator. The complex conjugate of (1+i)(1+i) is (1−i)(1-i).
  2. Expand Numerator and Denominator: Multiplying the numerator and denominator by the complex conjugate, we get:\newlinez=(5−2i)(1−i)(1+i)(1−i)z = \frac{(5 - 2i)(1 - i)}{(1 + i)(1 - i)}
  3. Simplify Expressions: Now we expand the numerator and denominator:\newlineNumerator: (5−2i)(1−i)=5−5i−2i+2i2(5 - 2i)(1 - i) = 5 - 5i - 2i + 2i^2\newlineDenominator: (1+i)(1−i)=1−i2(1 + i)(1 - i) = 1 - i^2
  4. Divide Numerator by Denominator: Since i2=−1i^2 = -1, we can simplify the expressions:\newlineNumerator: 5−5i−2i−2(−1)=5−5i−2i+2=7−7i5 - 5i - 2i - 2(-1) = 5 - 5i - 2i + 2 = 7 - 7i\newlineDenominator: 1−(−1)=1+1=21 - (-1) = 1 + 1 = 2
  5. Compare with Options: Now we divide the numerator by the denominator:\newlinez = (7−7i)/2(7 - 7i) / 2\newlineThis can be written as:\newlinez = 7/27/2 - (7/2)i(7/2)i
  6. Compare with Options: Now we divide the numerator by the denominator:\newlinez=7−7i2z = \frac{7 - 7i}{2}\newlineThis can be written as:\newlinez=72−(72)iz = \frac{7}{2} - \left(\frac{7}{2}\right)iWe compare the result with the given options:\newlinez=72−(72)iz = \frac{7}{2} - \left(\frac{7}{2}\right)i is equivalent to 7−7i2\frac{7 - 7i}{2}, which matches option C: 7(1−i)/27(1−i)/2.

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