Practice Questions on Complex Numbers Solutions

This set of practice questions focuses on complex numbers, including finding conjugates, adding, subtracting, and multiplying complex numbers. It also investigates the truth of statements involving operations with given complex numbers.

• Complex numbers
• Conjugates
• Addition
• Subtraction
• Multiplication

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1. CHAPTER 28 Complex Numbers Solutions: Practice Questions 28.5

2. 28 Practice Questions 28.5 1. Find the conjugate of each of the following complex numbers: 8i 4 + 5i (i) (v) Conjugate: 8i Conjugate: 4 5i 3i + 4 7 6i (vi) (ii) Conjugate: 4 + 3i Conjugate: 7 + 6i (iii) 9 + i Conjugate: 9 i 5 2i (iv) Conjugate: 5 + 2i

3. 28 Practice Questions 28.5 2. If z1 = 2 + i, z2= 3 2i and z3 = 1 + i, find the value of: z1+z3 (i) z1+z3=2+i+1-i =2+1+i-i =3 z2+z1 (ii) z2+z1=3-2i+2-i =3+2-2i-i =5-3i

4. 28 Practice Questions 28.5 2. If z1 = 2 + i, z2= 3 2i and z3 = 1 + i, find the value of: (iii) z1- z3 z1-z3=2-i-(1-i) =2-i-1+i =2-1-i+i =1

5. 28 Practice Questions 28.5 2. If z1 = 2 + i, z2= 3 2i and z3 = 1 + i, find the value of: (iv) z1z1 z1z1=(2+i)(2-i) =2(2-i)+i(2-i) =4-2i+2i-i2 =4-(-1) =4+1 =5

6. 28 Practice Questions 28.5 2. If z1 = 2 + i, z2= 3 2i and z3 = 1 + i, find the value of: z1z2 (v) z1z2=(2-i)(3+2i) =2(3+2i)-i(3+2i) =6+4i-3i-2i2 =6+i-2(-1) =6+2+i =8+i

7. 28 Practice Questions 28.5 2. If z1 = 2 + i, z2= 3 2i and z3 = 1 + i, find the value of: (vi) z2z3 z2z3=(3+2i)(1-i) =3(1-i)+2i(1-i) =3-3i+2i-2i2 =3-i-2(-1) =3+2-i =5-i

8. 28 Practice Questions 28.5 3. Given z1 = 2 + 3i and z2= 1 2i,investigate whether the following statements are true: z1+z2= z1+z2 (i) ( ) R.H.S: z1=2-3i + = 2 3 + + L.H.S: 1 2 z z i i 1 2 =2-1+3i-2i z2=-1+2i =1+i z1+z2=2-3i+(-1+2i) =2-1-3i+2i z1+z2=1-i =1-i z1+z2= z1+z2 Hence, this statement is true.

9. 28 Practice Questions 28.5 3. Given z1 = 2 + 3i and z2= 1 2i,investigate whether the following statements are true: z1 z2= z1 z2 (ii) z1 z2=(2+3i)(-1-2i) L.H.S: =2(-1-2i)+3i(-1-2i) =-2-4i-3i-6i2 =-2-7i-6(-1) =-2+6-7i =4-7i z1 z2=4+7i

10. 28 Practice Questions 28.5 3. Given z1 = 2 + 3i and z2= 1 2i,investigate whether the following statements are true: z1 z2= z1 z2 (ii) R.H.S: z1=2-3i z2=-1+2i z1 z2=(2-3i)(-1+2i) =2(-1+2i)-3i(-1+2i) =-2+4i+3i-6i2 =-2+7i-6(-1) =-2+6+7i =4+7i z1 z2= z1 z2 Hence, this statement is true.

11. 28 Practice Questions 28.5 4. Divide the following complex numbers and write your answer in the form a + bi: 2 3i (i) 2 3i Conjugate of 3i= 3i 2 3i -3i =-6i -9i2 -6i -9(-1) =-6i 9 -3i = =-2i 3

12. 28 Practice Questions 28.5 4. Divide the following complex numbers and write your answer in the form a + bi: -5 6i (ii) -5 6i Conjugate of 6i= 6i -5 6i -6i 30i -36i2 -6i = 30i = -36(-1) =30i 36 =5i 6

13. 28 Practice Questions 28.5 4. Divide the following complex numbers and write your answer in the form a + bi: -7 -2i (iii) -7 -2i Conjugate of 2i = 2i -7 -2i 2i =-14i -4i2 -14i -4(-1) 2i = =-14i 4 =-7 2i

14. 28 Practice Questions 28.5 4. Divide the following complex numbers and write your answer in the form a + bi: 2+3i 4i (iv) 2+3i 4i Conjugate of 4i= 4i 2+3i 4i =-8i-12i2 -16i2 =-8i-12(-1) -16(-1) -4i -4i =-8i+12 16 =12-8i 16 =3-2i 4 =3 4-1 2i

15. 28 Practice Questions 28.5 4. Divide the following complex numbers and write your answer in the form a + bi: -1-i -6i (v) -1-i -6i Conjugate = 6i -1-i -6i 6i 6i =-6i-6(i)2 -36i2 =-6i-6(-1) -36(-1) =6 36-6 36i =-6i+6 36 =1 6-1 6i

16. 28 Practice Questions 28.5 4. Divide the following complex numbers and write your answer in the form a + bi: -1-3i -2i (vi) -1-3i -2i Conjugate = 2i -1-3i -2i 2i 2i =-i+3 =-2i-6i2 -4i2 2 =-2i-6(-1) -4(-1) =-i 2+3 2 =-2i+6 =3 2-1 2i 4

17. 28 Practice Questions 28.5 5. Divide the following complex numbers and write your answer in the form a + bi: 2+7i 4+3i (i) 2+7i 4+3i 2+7i 4+3i 4-3i =2(4-3i)+7i(4-3i) 4(4-3i)+3i(4-3i) Conjugate of 4 + 3i = 4 3i 4-3i =8+21+22i 16+9 =29+22i 25 8-6i+28i-21i2 16- 12i + 12i -9i2 = =29 25+22 =8+22i-21(-1) 16-9(-1) 25i

18. 28 Practice Questions 28.5 5. Divide the following complex numbers and write your answer in the form a + bi: -3+5i 6-4i (ii) -3+5i 6-4i -3+5i 6-4i =-3(6+4i)+5i(6+4i) 6(6+4i)-4i(6+4i) Conjugate of 6 4i = 6 + 4i 6+4i 6+4i =-18-20+18i 52 =-38+18i 52 =-18-12i+30i+20i2 36+ 24i - 24i -16i2 =-19 26+9 =-18+18i+20(-1) 36-16(-1) 26i

19. 28 Practice Questions 28.5 5. Divide the following complex numbers and write your answer in the form a + bi: -6-5i 7-3i (iii) -6-5i 7-3i -6-5i 7-3i =-6(7+3i)-5i(7+3i) 7(7+3i)-3i(7+3i) Conjugate of 7 3i = 7 + 3i 7+3i 7+3i =-42+15-53i 49+9 =-27-53i 58 =-42-18i-35i-15i2 49+ 21i - 21i -9i2 =-27 58-53 =-42-53i-15(-1) 49-9(-1) 58i

20. 28 Practice Questions 28.5 5. Divide the following complex numbers and write your answer in the form a + bi: -1-8i 7-i (iv) -1-8i 7-i -1-8i 7-i =-1(7+i)-8i(7+i) 7(7+i)-i(7+i) Conjugate of 7 i = 7 + i 7+i 7+i =-7+8-57i 49+1 =1-57i 50 =-7-i-56i-8i2 49+ 7i - 7i -i2 =1 50-57 =-7-57i-8(-1) 49-(-1) 50i

21. 28 Practice Questions 28.5 5. Divide the following complex numbers and write your answer in the form a + bi: -5-9i 6+2i (v) -5-9i 6+2i Conjugate of 6 + 2i = 6 2i -5-9i 6+2i =-5(6-2i)-9i(6-2i) 6(6-2i)+2i(6-2i) 6-2i 6-2i =-30-18-44i 36+4 =-48-44i 40 =-30+10i-54i+18i2 36- 12i + 12i -4i2 =-6 5-11 =-30-44i+18(-1) 36-4(-1) 10i

22. 28 Practice Questions 28.5 5. Divide the following complex numbers and write your answer in the form a + bi: -2-3i -5-i (vi) -2-3i -5-i Conjugate of 5 i= 5 + i -2-3i -5-i =-2(-5+i)-3i(-5+i) -5(-5+i)-i(-5+i) -5+i -5+i =10+3+13i 25+1 =13+13i 26 =10-2i+15i-3i2 25- 5i + 5i -i2 =1 2+1 =10+13i-3(-1) 25-(-1) 2i

23. 28 Practice Questions 28.5 6. z1 = 2 + i, z2= 3 2i, z3= 4 5iand z4= 1 i. Find the value of each of the following: z1 z4 (i) =2+i -1-i z1 z4 Conjugate of 1 i= 1 + i 2+i -1-i -1+i =2(-1+i)+i(-1+i) -1(-1+i)-i(-1+i) -1+i =-2-1+i 1+1 =-3+i =-2+2i-i+i2 1-i+i-i2 2 =-3 2+1 =-2+i+(-1) 1-(-1) 2i

24. 28 Practice Questions 28.5 6. z1 = 2 + i, z2= 3 2i, z3= 4 5iand z4= 1 i. Find the value of each of the following: z4 z2 (ii) =-1-i 3-2i z4 z2 Conjugate of 3 2i = 3 + 2i -1-i 3-2i 3+2i =-1(3+2i)-i(3+2i) 3(3+2i)-2i(3+2i) 3+2i =-3+2-5i 9+4 =-1-5i 13 =-3-2i-3i-2i2 9+6i-6i-4i2 =-1 13-5 =-3-5i-2(-1) 9-4(-1) 13i

25. 28 Practice Questions 28.5 6. z1 = 2 + i, z2= 3 2i, z3= 4 5iand z4= 1 i. Find the value of each of the following: z1+ z2 z4 z1+ z2 z4 5-i -1-i -1+i =5(-1+i)-i(-1+i) -1(-1+i)-i(-1+i) (iii) -1+i =2+i+3-2i -1-i =-5+5i+i-i2 1-i+i-i2 =2+3+i-2i -1-i =-5+6i-(-1) 1-(-1) =-5+1+6i 1+1 =-4+6i 2 =-2+3i =5-i -1-i Conjugate of 1 i= 1 + i

26. 28 Practice Questions 28.5 6. z1 = 2 + i, z2= 3 2i, z3= 4 5iand z4= 1 i. Find the value of each of the following: z3- z2 z1 z3- z2 z1 -7-3i 2+i =-7(2-i)-3i(2-i) 2(2-i)+i(2-i) =-14+7i-6i+3i2 4-2i+2i-i2 =-14+i+3(-1) 4-(-1) =-14+i-3 4+1 =-17+i 5 =-17 5i 2-i 2-i (iv) =-4-5i-(3-2i) 2+i =-4-5i-3+2i 2+i =-4-3-5i+2i 2+i =-7-3i 2+i 5+1 Conjugate of 2 + i = 2 i

27. 28 Practice Questions 28.5 7. If z = 2 + 4i and w = 3 2i,find the value of the following: z w (i) z w=-2+4i -3-2i Conjugate of -3-2i =-3+2i -2+4i -3-2i -3+2i -3+2i =-2(-3+2i)+4i(-3+2i) -3(-3+2i)-2i(-3+2i) =6-8-16i 9+4 =-2-16i 13 =6-4i-12i+8i2 9-6i+6i-4i2 =6-16i+8(-1) 9-4(-1) =-2 13-16 13i

28. 28 Practice Questions 28.5 7. If z = 2 + 4i and w = 3 2i,find the value of the following: z w (ii) z w=-2-4i -3-2i -2-4i -3-2i -3+2i =-2(-3+2i)-4i(-3+2i) -3(-3+2i)-2i(-3+2i) The conjugate of 3 2i= 3 + 2i -3+2i =14+8i 9+4 =6-4i+12i-8i2 9-6i+6i-4i2 =6+8i-8(-1) 9-4(-1) =14 13+8 13i

29. 28 Practice Questions 28.5 7. If z = 2 + 4i and w = 3 2i,find the value of the following: -2i-4 -3-2i -3+2i =-2i(-3+2i)-4(-3+2i) -3(-3+2i)-2i(-3+2i) =6i-4i2+12-8i 9-6i+6i-4i2 =6i-4(-1)+12-8i 9-4(-1) =4+12+6i-8i 9+4 =16-2i 13 =16 13i iz w (iii) -3+2i iz w=i(-2+4i) -3-2i =-2i+4i2 -3-2i =-2i+4(-1) -3-2i =-2i-4 -3-2i Conjugate of -3-2i =-3+2i 13-2

30. 28 Practice Questions 28.5 7. If z = 2 + 4i and w = 3 2i,find the value of the following: z w (iv) =-2-4i -3+2i Conjugate of -3+2i =-3-2i z w -2-4i -3+2i -3-2i =-2(-3-2i)-4i(-3-2i) -3(-3-2i)+2i(-3-2i) -3-2i =6-8+16i 9+4 =6+4i+12i+8i2 9+ 6i - 6i -4i2 =-2+16i 13 =6+16i+8(-1) 9-4(-1) =-2 13+16 13i

31. 28 Practice Questions 28.5 8. Express each of the following in the form a + bi: (1+2i)-(2+i) (2-i)(3+i) (1+2i)-(2+i) (2-i)(3+i) =1+2i-2-i (2-i)(3+i) 1-2+2i-i 2(3+i)-i(3+i) (i) = -1+i = 6+2i-3i-i2 -1+i 6-i-(-1) =-1+i 6+1-i = =-1+i 7-i

32. 28 Practice Questions 28.5 8. Express each of the following in the form a + bi: (1+2i)-(2+i) (2-i)(3+i) (i) Conjugate of 7 i = 7 + i -1+i 7-i =-1(7+i)+i(7+i) 7(7+i)-i(7+i) -7-i+7i+i2 49+ 7i - 7i -i2 =-7+6i+(-1) 49-(-1) =-7-1+6i 49+1 7+i 7+i =-8+6i 50 =-4+3i 25 =-4 = 25+3 25i

33. 28 Practice Questions 28.5 8. Express each of the following in the form a + bi: (2+3i)-(1-2i) (3+2i)(4-i) (ii) (2+3i)-(1-2i) (3+2i)(4-i) =2+3i-1+2i (3+2i)(4-i) 2-1+3i+2i 3(4-i)+2i(4-i) = 1+5i 12+2+5i 1+5i = = 12-3i+8i-2i2 1+5i =1+5i 14+5i = 12+5i-2(-1)

34. 28 Practice Questions 28.5 8. Express each of the following in the form a + bi: (2+3i)-(1-2i) (3+2i)(4-i) (ii) Conjugate of 14 + 5i = 14 5i 1+5i 14+5i 14-5i =1(14-5i)+5i(14-5i) 14(14-5i)+5i(14-5i) 14-5i 14-5i+70i-25i2 196-70i+70i-25i2 =14+65i-25(-1) 196-25(-1) =14+25+65i 196+25 = =39+65i 221 =3 17+5 17i

35. 28 Practice Questions 28.5 9. Impedance to the flow of alternating current in an electrical circuit is measured in ohms. Z =V The impedance, Z, in a circuit is found by using the formula , where Vis the voltage (measured in volts) and Iis the current (measured in amperes). I Find the impedance when V = 1 8 0 4i volts and I = 0 3i amperes. (i) =1 8-0 4i -0 3i =0 54i-0 12i2 -0 09i2 =0 54i-0 12(-1) -0 09(-1) Z =V I Conjugate of -0 3i =0 3i 1 8-0 4i -0 3i 0 3i 0 3i =0 12+0 54i 0 09 =1 8(0 3i)-0 4i(0 3i) (-0 3i)(0 3i) =4 3+6i

36. 28 Practice Questions 28.5 9. Impedance to the flow of alternating current in an electrical circuit is measured in ohms. Z =V The impedance, Z, in a circuit is found by using the formula , where Vis the voltage (measured in volts) and Iis the current (measured in amperes). I Find the current when V = 1 6 0 3i volts and Z = 1 5 + 8i ohms. (ii) Z =V I IZ =V I =V Z Z=1 6-0 3i 1 5+8i I =V Conjugate of 1 5+8i =1 5-8i

37. 28 Practice Questions 28.5 9. Impedance to the flow of alternating current in an electrical circuit is measured in ohms. Z =V The impedance, Z, in a circuit is found by using the formula , where Vis the voltage (measured in volts) and Iis the current (measured in amperes). I Find the current when V = 1 6 0 3i volts and Z = 1 5 + 8i ohms. (ii) v 1 6-0 3i 1 5+8i 1 5-8i 1 5-8i =1 6(1 5-8i)-0.3i(1 5-8i) 1 5(1 5-8i)+8i(1 5-8i) =2 4-2 4-13 25i 2 25+64 =-13 25i 66 25 =2 4-12 8i-0 45i+2 4i2 2 25- 12i + 12i -64i2 =-0 2i =2 4-13 25i+2 4(-1) 2 25-64(-1)