Exercise 1.5 Class 11 Math Solutions - Unit 1 Complex Numbers, Punjab Board
Step-by-step solutions to Exercise 1.5, the concluding exercise of Unit 1: Complex Numbers, for Class 11 Punjab Board students. Covers the polar coordinates system, polar form representation, multiplication and division in polar form, and real-world electrical circuit applications (voltage, current, impedance) worked out question by question.
Exercise Overview
Exercise 1.5 completes Unit 1 of Class 11 Mathematics by exploring the geometric and trigonometric representations of complex numbers in polar form. Every complex number z = x + iy can be uniquely expressed in polar coordinates as z = r(cos θ + i sin θ), where r = |z| = √(x² + y²) represents the modulus and θ = Arg(z) is the principal argument. Determining the correct argument angle requires careful quadrant analysis to adjust the reference angle θ = tan⁻¹(|y/x|).
The exercise guides students through fundamental operations in polar form, proving that multiplying complex numbers multiplies their moduli and adds their arguments (r₁ cis θ₁ · r₂ cis θ₂ = r₁r₂ cis(θ₁ + θ₂)), while division divides their moduli and subtracts their arguments. Furthermore, the exercise features practical applications in electrical engineering, modeling AC circuits where sinusoidal alternating currents (I), voltages (V), and complex impedances (Z = R + jX) are solved via Ohm's law V = I · Z.
In Punjab Board HSSC Part 1 examinations, Exercise 1.5 is a prominent source of conceptual MCQs, argument calculations in short questions, and applied word problems in Section C long questions across all nine boards, including Lahore, Rawalpindi, and Faisalabad. These step-by-step solutions equip students with solid conceptual clarity and calculation precision.