Exercise 1.3 Class 11 Math Solutions - Unit 1 Complex Numbers, Punjab Board

Step-by-step solutions to Exercise 1.3, Unit 1: Complex Numbers, for Class 11 Punjab Board students. Covers the factorization of complex polynomials into linear factors and solving quadratic equations with complex roots by completing the square worked out question by question.

This page provides complete, step-by-step solved solutions for Exercise 1.3 of Unit 1: Complex Numbers for Class 11 Mathematics under the Punjab Board (PCTB / PECTAA). All questions on factoring real and complex polynomials into linear factors using i² = -1 and solving quadratic equations with complex roots by completing the square are fully solved to assist Intermediate Part 1 students with exam preparation.
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Exercise Overview

Exercise 1.3 of Class 11 Mathematics builds a crucial bridge between algebra and complex numbers by focusing on polynomial factorization over the complex field and solving quadratic equations with imaginary roots. Using the fundamental identity i² = -1, expressions of the form a² + b² that cannot be factored over real numbers can be rewritten as a² - (ib)² and factored into linear complex terms (a + ib)(a - ib).

The exercise features two primary types of problems: linear factorization of quadratic and higher-degree expressions, and solving quadratic equations ax² + bx + c = 0 where the discriminant is negative (b² - 4ac < 0) using the method of completing the square. Students learn to manipulate algebraic terms systematically, complete the square by adding (b/2a)², and extract complex conjugate roots with absolute precision.

In Punjab Board HSSC Part 1 examinations, Exercise 1.3 is a frequent source of Section B short questions and Section C long questions across all affiliated boards, including Lahore, Rawalpindi, Gujranwala, Faisalabad, and Multan. Practicing these step-by-step solutions ensures students develop mastery over quadratic techniques and complex algebraic factorization.

FAQs

Are these Exercise 1.3 solutions according to the Punjab Board syllabus?
Yes, these solutions strictly adhere to the Punjab Curriculum and Textbook Board (PCTB / PECTAA) syllabus for Class 11 Mathematics, covering complex polynomial factorization and quadratic equations for the 2026-2027 session.
Do these solutions cover every question in Exercise 1.3?
Yes, every textbook problem from Exercise 1.3 is solved in detail, including expressing polynomials as products of linear complex factors and solving quadratic equations with complex roots by completing the square.
Are the solutions step-by-step?
Yes, each solution provides clear algebraic steps, detailing how to convert sum of squares into difference of squares using i² = -1 and applying the method of completing the square systematically.
How do you factorize polynomials like a² + b² using complex numbers?
By rewriting a² + b² as a² - (i²b²) = a² - (ib)², the expression becomes a difference of two squares and factors directly into linear complex factors (a + ib)(a - ib).
Can students read these solutions online?
Yes, students can read all Exercise 1.3 solutions directly online on MaryamNotes.pk on any web browser without needing to download external files.
Are these solutions useful for Punjab Board exams?
Yes, questions on factorizing algebraic expressions into complex linear factors and solving quadratic equations with non-real roots frequently appear as short and long questions in HSSC Part 1 board exams.
Are these solutions based on the latest PCTB / PECTAA syllabus?
Yes, they are fully aligned with the updated Student Learning Outcomes (SLO) mandated by the Punjab Curriculum and Textbook Board (PECTAA) for the 2026-2027 academic session.
Are these solutions free for students?
Yes, all Class 11 Mathematics Exercise 1.3 solutions on MaryamNotes.pk are completely free for students and teachers across Punjab.