Question 7, Exercise 1.2

Solutions of Question 7 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.

Question 7(i)

Separate into real and imaginary parts 2+3i5−2i.

Solution

2+3i5−2i=2+3i5−2i×5+2i5+2iby rationalizing=10−6+15i+4i25+4=4+19i29=429+1929i Real part =429
Imaginary part =1929

Question 7(ii)

Separate into real and imaginary parts (1+2i)21−3i.

Solution

(1+2i)21−3i=1−4+4i1−3i=−3+4i1−3i=−3+4i1−3i×1+3i1+3iby rationalizing=−3−12+4i−9i1+9=−15−5i10=−32−12i Real part =−32
Imaginary part =−12

Question 7(iii)

Separate into real and imaginary parts 1−i(1+i)2.

Solution

1−i(1+i)2=1−i1−1+2i=1−i2i×−2i−2i=1−i2i×−2i−2i=−2−2i4=−12−12i Real part =−12
Imaginary part =−12

Question 7(iv)

Separate into real and imaginary parts (2a−bi)−2.

Solution

(2a−bi)−2=1(2a−bi)2=1(4a2−b2)−4abi=1(4a2−b2)−4abi×(4a2−b2)+4abi(4a2−b2)+4abi=(4a2−b2)+4abi(4a2−b2)2+16a2b2=(4a2−b2)+4abi(4a2−b2)2+16a2b2=(4a2−b2)+4abi16a4+b4−8a2b2+16a2b2=(4a2−b2)+4abi(4a2+b2)2=4a2−b2(4a2+b2)2+4abi(4a2+b2)2 Real part =4a2−b24a2+b22
Imaginary part =4ab(4a2+b2)2

Question 7(v)

Separate into real and imaginary parts (3−4i4−3i)−2.

Solution

(3−4i4−3i)−2=(4−3i3−4i)2=16−9−24i9−16−24i=7−24i−7−24i=7−24i−7−24i×−7+24i−7+24i=−(49+576−336i)49+576=−(625−336i)625=−625625+336625i=−1+336625i Real part =−1
Imaginary part =336625

Question 7(vi)

Separate into real and imaginary parts (4−5i2+3i)2.

Solution

(4−5i2+3i)2=16−25−40i4−9+12i=−9−40i−5+12i=−9−40i−5+12i×−5−12i−5−12i=45−480+200i+108i25+144=−435+308i169=435169+308i169 Real part =435169
Imaginary part =308169

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