Exercise 1.4 Class 11 Math Solutions - Unit 1 Complex Numbers, Punjab Board
Step-by-step solutions to Exercise 1.4, Unit 1: Complex Numbers, for Class 11 Punjab Board students. Covers the three cube roots of unity, four fourth roots of unity, and their fundamental algebraic properties worked out question by question.
Exercise Overview
Exercise 1.4 of Class 11 Mathematics covers one of the most core and elegant algebraic topics in Unit 1: the roots of unity. By solving x³ - 1 = 0 and x⁴ - 1 = 0, students derive the three cube roots of unity (1, ω, ω²) and the four fourth roots of unity (1, -1, i, -i). The exercise establishes essential identities, including 1 + ω + ω² = 0, ω³ = 1, and the reciprocal nature of fourth roots of unity.
The exercise features diverse problem types: finding cube roots of arbitrary numbers such as -1, 8, -27, and 64, evaluating expressions involving high powers of ω (e.g. ω²⁸ + ω²⁹ + 1), and proving symmetrical algebraic identities such as x³ + y³ = (x + y)(x + ωy)(x + ω²y). Systematic substitution using fundamental omega identities is essential to simplify complex cyclic relations without algebraic errors.
In Punjab Board HSSC Part 1 examinations, Exercise 1.4 is among the most heavily tested sections of Unit 1 for Section B short questions and Section C extensive proofs across Lahore, Rawalpindi, Faisalabad, Gujranwala, Multan, and other affiliated boards. Master these step-by-step solutions to ensure absolute confidence and top marks on board exams.