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NCERT Solutions › Mathematics › Chapter 5

Chapter 5: Complex Numbers

NCERT Solutions for CBSE Class 10 Mathematics — 100 solved questions with detailed explanations.

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Questions
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Solved Questions

Q1. If z = 3 + 4i, then |z| is:

Difficulty: Easy · Topic: Modulus

Solution

|z| = √(3²+4²) = √(9+16) = √25 = 5.

Q2. The value of i¹⁷ is:

Difficulty: Easy · Topic: Powers of i

Solution

i¹⁷ = i¹⁶ × i = (i⁴)⁴ × i = 1⁴ × i = i.

Q3. If z = 3+4i, then |z| is:

Difficulty: Easy · Topic: Modulus

Solution

|z| = √(3²+4²) = √(25) = 5.

Q4. If z = 5+12i, then |z| is:

Difficulty: Easy · Topic: Modulus

Solution

|z| = √(5²+12²) = √(169) = 13.

Q5. If z = 8+6i, then |z| is:

Difficulty: Easy · Topic: Modulus

Solution

|z| = √(8²+6²) = √(100) = 10.

Q6. If z = 1+1i, then |z| is:

Difficulty: Easy · Topic: Modulus

Solution

|z| = √(1²+1²) = √(2) = 1.41.

Q7. If z = 7+24i, then |z| is:

Difficulty: Easy · Topic: Modulus

Solution

|z| = √(7²+24²) = √(625) = 25.

Q8. If z = 2+3i, then |z| is:

Difficulty: Easy · Topic: Modulus

Solution

|z| = √(2²+3²) = √(13) = 3.61.

Q9. If z = 6+8i, then |z| is:

Difficulty: Easy · Topic: Modulus

Solution

|z| = √(6²+8²) = √(100) = 10.

Q10. If z = 9+12i, then |z| is:

Difficulty: Easy · Topic: Modulus

Solution

|z| = √(9²+12²) = √(225) = 15.

Q11. If z = 4+3i, then |z| is:

Difficulty: Easy · Topic: Modulus

Solution

|z| = √(4²+3²) = √(25) = 5.

Q12. If z = 5+5i, then |z| is:

Difficulty: Easy · Topic: Modulus

Solution

|z| = √(5²+5²) = √(50) = 7.07.

Q13. The value of i^1 is:

Difficulty: Easy · Topic: Powers of i

Solution

i^1 = i^(1%4=1) = i. (i¹=i, i²=-1, i³=-i, i⁴=1).

Q14. The value of i^2 is:

Difficulty: Easy · Topic: Powers of i

Solution

i^2 = i^(2%4=2) = -1. (i¹=i, i²=-1, i³=-i, i⁴=1).

Q15. The value of i^3 is:

Difficulty: Easy · Topic: Powers of i

Solution

i^3 = i^(3%4=3) = -i. (i¹=i, i²=-1, i³=-i, i⁴=1).

Q16. The value of i^4 is:

Difficulty: Easy · Topic: Powers of i

Solution

i^4 = i^(4%4=0) = 1. (i¹=i, i²=-1, i³=-i, i⁴=1).

Q17. The value of i^5 is:

Difficulty: Easy · Topic: Powers of i

Solution

i^5 = i^(5%4=1) = i. (i¹=i, i²=-1, i³=-i, i⁴=1).

Q18. The value of i^10 is:

Difficulty: Easy · Topic: Powers of i

Solution

i^10 = i^(10%4=2) = -1. (i¹=i, i²=-1, i³=-i, i⁴=1).

Q19. The value of i^17 is:

Difficulty: Easy · Topic: Powers of i

Solution

i^17 = i^(17%4=1) = i. (i¹=i, i²=-1, i³=-i, i⁴=1).

Q20. The value of i^25 is:

Difficulty: Easy · Topic: Powers of i

Solution

i^25 = i^(25%4=1) = i. (i¹=i, i²=-1, i³=-i, i⁴=1).

Q21. The value of i^50 is:

Difficulty: Easy · Topic: Powers of i

Solution

i^50 = i^(50%4=2) = -1. (i¹=i, i²=-1, i³=-i, i⁴=1).

Q22. The value of i^100 is:

Difficulty: Easy · Topic: Powers of i

Solution

i^100 = i^(100%4=0) = 1. (i¹=i, i²=-1, i³=-i, i⁴=1).

Q23. The value of i^103 is:

Difficulty: Easy · Topic: Powers of i

Solution

i^103 = i^(103%4=3) = -i. (i¹=i, i²=-1, i³=-i, i⁴=1).

Q24. The value of i^202 is:

Difficulty: Easy · Topic: Powers of i

Solution

i^202 = i^(202%4=2) = -1. (i¹=i, i²=-1, i³=-i, i⁴=1).

Q25. i^1 =

Difficulty: Easy · Topic: Complex

Solution

i^1=i^(1)=i.

Q26. i^2 =

Difficulty: Easy · Topic: Complex

Solution

i^2=i^(2)=-1.

Q27. i^3 =

Difficulty: Easy · Topic: Complex

Solution

i^3=i^(3)=-i.

Q28. i^4 =

Difficulty: Easy · Topic: Complex

Solution

i^4=i^(0)=1.

Q29. i^5 =

Difficulty: Easy · Topic: Complex

Solution

i^5=i^(1)=i.

Q30. i^6 =

Difficulty: Easy · Topic: Complex

Solution

i^6=i^(2)=-1.

Q31. i^7 =

Difficulty: Easy · Topic: Complex

Solution

i^7=i^(3)=-i.

Q32. i^8 =

Difficulty: Easy · Topic: Complex

Solution

i^8=i^(0)=1.

Q33. i^9 =

Difficulty: Easy · Topic: Complex

Solution

i^9=i^(1)=i.

Q34. i^10 =

Difficulty: Easy · Topic: Complex

Solution

i^10=i^(2)=-1.

Q35. i^11 =

Difficulty: Easy · Topic: Complex

Solution

i^11=i^(3)=-i.

Q36. i^12 =

Difficulty: Easy · Topic: Complex

Solution

i^12=i^(0)=1.

Q37. i^13 =

Difficulty: Easy · Topic: Complex

Solution

i^13=i^(1)=i.

Q38. i^14 =

Difficulty: Easy · Topic: Complex

Solution

i^14=i^(2)=-1.

Q39. i^15 =

Difficulty: Easy · Topic: Complex

Solution

i^15=i^(3)=-i.

Q40. i^16 =

Difficulty: Easy · Topic: Complex

Solution

i^16=i^(0)=1.

Q41. i^17 =

Difficulty: Easy · Topic: Complex

Solution

i^17=i^(1)=i.

Q42. i^18 =

Difficulty: Easy · Topic: Complex

Solution

i^18=i^(2)=-1.

Q43. i^19 =

Difficulty: Easy · Topic: Complex

Solution

i^19=i^(3)=-i.

Q44. i^20 =

Difficulty: Easy · Topic: Complex

Solution

i^20=i^(0)=1.

Q45. i^21 =

Difficulty: Easy · Topic: Complex

Solution

i^21=i^(1)=i.

Q46. i^22 =

Difficulty: Easy · Topic: Complex

Solution

i^22=i^(2)=-1.

Q47. i^23 =

Difficulty: Easy · Topic: Complex

Solution

i^23=i^(3)=-i.

Q48. i^24 =

Difficulty: Easy · Topic: Complex

Solution

i^24=i^(0)=1.

Q49. i^25 =

Difficulty: Easy · Topic: Complex

Solution

i^25=i^(1)=i.

Q50. i^26 =

Difficulty: Easy · Topic: Complex

Solution

i^26=i^(2)=-1.

Q51. i^27 =

Difficulty: Easy · Topic: Complex

Solution

i^27=i^(3)=-i.

Q52. i^28 =

Difficulty: Easy · Topic: Complex

Solution

i^28=i^(0)=1.

Q53. i^29 =

Difficulty: Easy · Topic: Complex

Solution

i^29=i^(1)=i.

Q54. i^30 =

Difficulty: Easy · Topic: Complex

Solution

i^30=i^(2)=-1.

Q55. i^31 =

Difficulty: Easy · Topic: Complex

Solution

i^31=i^(3)=-i.

Q56. i^32 =

Difficulty: Easy · Topic: Complex

Solution

i^32=i^(0)=1.

Q57. i^33 =

Difficulty: Easy · Topic: Complex

Solution

i^33=i^(1)=i.

Q58. i^34 =

Difficulty: Easy · Topic: Complex

Solution

i^34=i^(2)=-1.

Q59. i^35 =

Difficulty: Easy · Topic: Complex

Solution

i^35=i^(3)=-i.

Q60. i^36 =

Difficulty: Easy · Topic: Complex

Solution

i^36=i^(0)=1.

Q61. i^37 =

Difficulty: Easy · Topic: Complex

Solution

i^37=i^(1)=i.

Q62. i^38 =

Difficulty: Easy · Topic: Complex

Solution

i^38=i^(2)=-1.

Q63. i^39 =

Difficulty: Easy · Topic: Complex

Solution

i^39=i^(3)=-i.

Q64. i^40 =

Difficulty: Easy · Topic: Complex

Solution

i^40=i^(0)=1.

Q65. i^41 =

Difficulty: Easy · Topic: Complex

Solution

i^41=i^(1)=i.

Q66. i^42 =

Difficulty: Easy · Topic: Complex

Solution

i^42=i^(2)=-1.

Q67. i^43 =

Difficulty: Easy · Topic: Complex

Solution

i^43=i^(3)=-i.

Q68. i^44 =

Difficulty: Easy · Topic: Complex

Solution

i^44=i^(0)=1.

Q69. i^45 =

Difficulty: Easy · Topic: Complex

Solution

i^45=i^(1)=i.

Q70. i^46 =

Difficulty: Easy · Topic: Complex

Solution

i^46=i^(2)=-1.

Q71. i^47 =

Difficulty: Easy · Topic: Complex

Solution

i^47=i^(3)=-i.

Q72. i^48 =

Difficulty: Easy · Topic: Complex

Solution

i^48=i^(0)=1.

Q73. i^49 =

Difficulty: Easy · Topic: Complex

Solution

i^49=i^(1)=i.

Q74. i^50 =

Difficulty: Easy · Topic: Complex

Solution

i^50=i^(2)=-1.

Q75. i^51 =

Difficulty: Easy · Topic: Complex

Solution

i^51=i^(3)=-i.

Q76. i^52 =

Difficulty: Easy · Topic: Complex

Solution

i^52=i^(0)=1.

Q77. i^53 =

Difficulty: Easy · Topic: Complex

Solution

i^53=i^(1)=i.

Q78. i^54 =

Difficulty: Easy · Topic: Complex

Solution

i^54=i^(2)=-1.

Q79. i^55 =

Difficulty: Easy · Topic: Complex

Solution

i^55=i^(3)=-i.

Q80. i^56 =

Difficulty: Easy · Topic: Complex

Solution

i^56=i^(0)=1.

Q81. i^57 =

Difficulty: Easy · Topic: Complex

Solution

i^57=i^(1)=i.

Q82. i^58 =

Difficulty: Easy · Topic: Complex

Solution

i^58=i^(2)=-1.

Q83. i^59 =

Difficulty: Easy · Topic: Complex

Solution

i^59=i^(3)=-i.

Q84. i^60 =

Difficulty: Easy · Topic: Complex

Solution

i^60=i^(0)=1.

Q85. i^61 =

Difficulty: Easy · Topic: Complex

Solution

i^61=i^(1)=i.

Q86. i^62 =

Difficulty: Easy · Topic: Complex

Solution

i^62=i^(2)=-1.

Q87. i^63 =

Difficulty: Easy · Topic: Complex

Solution

i^63=i^(3)=-i.

Q88. i^64 =

Difficulty: Easy · Topic: Complex

Solution

i^64=i^(0)=1.

Q89. i^65 =

Difficulty: Easy · Topic: Complex

Solution

i^65=i^(1)=i.

Q90. i^66 =

Difficulty: Easy · Topic: Complex

Solution

i^66=i^(2)=-1.

Q91. i^67 =

Difficulty: Easy · Topic: Complex

Solution

i^67=i^(3)=-i.

Q92. i^68 =

Difficulty: Easy · Topic: Complex

Solution

i^68=i^(0)=1.

Q93. i^69 =

Difficulty: Easy · Topic: Complex

Solution

i^69=i^(1)=i.

Q94. i^70 =

Difficulty: Easy · Topic: Complex

Solution

i^70=i^(2)=-1.

Q95. i^71 =

Difficulty: Easy · Topic: Complex

Solution

i^71=i^(3)=-i.

Q96. i^72 =

Difficulty: Easy · Topic: Complex

Solution

i^72=i^(0)=1.

Q97. i^73 =

Difficulty: Easy · Topic: Complex

Solution

i^73=i^(1)=i.

Q98. i^74 =

Difficulty: Easy · Topic: Complex

Solution

i^74=i^(2)=-1.

Q99. i^75 =

Difficulty: Easy · Topic: Complex

Solution

i^75=i^(3)=-i.

Q100. The conjugate of (2+3i)/(1-i) is:

Difficulty: Medium · Topic: Complex Conjugate

Solution

(2+3i)/(1-i) = (2+3i)(1+i)/((1-i)(1+i)) = (2+2i+3i+3i²)/2 = (2+5i-3)/2 = (-1+5i)/2. Conjugate = (-1-5i)/2.

Other Chapters in Mathematics

Ch 1: SetsCh 2: Relations & FunctionsCh 3: Trigonometric FunctionsCh 4: Principle of Mathematical InductionCh 6: Linear InequalitiesCh 7: Permutations & CombinationsCh 8: Binomial TheoremCh 9: Sequences & SeriesCh 10: Straight LinesCh 11: Conic SectionsCh 12: Intro to 3D GeometryCh 13: Limits & DerivativesCh 14: Mathematical ReasoningCh 15: StatisticsCh 16: Probability

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