Exercise 5.2 Class 11 Math Solutions - Unit 5 Partial Fractions, Punjab Board

Step-by-step solutions to Exercise 5.2, the concluding exercise of Unit 5: Partial Fractions, for Class 11 Punjab Board students. Covers partial fraction decomposition for rational expressions containing non-repeated irreducible quadratic factors (Case 3) and repeated irreducible quadratic factors (Case 4) worked out question by question.

This page provides complete, step-by-step solved solutions for Exercise 5.2 of Unit 5: Partial Fractions for Class 11 Mathematics under the Punjab Board (PCTB / PECTAA). All questions involving the resolution of proper and improper rational fractions P(x)/Q(x) where Q(x) contains non-repeated irreducible quadratic factors (Case 3) or repeated irreducible quadratic factors (Case 4) are thoroughly worked out using linear numerators (Ax + B) and coefficient comparison for HSSC Part 1 board exam preparation.
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Exercise Overview

Exercise 5.2 concludes Unit 5: Partial Fractions in Class 11 Mathematics by addressing advanced partial fraction decomposition involving irreducible quadratic factors. A quadratic factor ax² + bx + c is irreducible over the set of real numbers if its discriminant is negative (b² - 4ac < 0), meaning it cannot be broken down into real linear factors.

The exercise investigates two critical scenarios: Case 3, where the denominator contains non-repeated irreducible quadratic factors requiring a linear numerator form (Ax + B)/(ax² + bx + c); and Case 4, where the denominator contains repeated irreducible quadratic factors of the form (ax² + bx + c)², requiring successive terms (Ax + B)/(ax² + bx + c) + (Cx + D)/(ax² + bx + c)². Because simple linear root substitution cannot determine all constants, students master the comprehensive method of expanding polynomial expressions and equating coefficients of corresponding powers of x to form solvable simultaneous algebraic equations.

In Punjab Board HSSC Part 1 examinations, Exercise 5.2 problems are premium targets for Section C long questions across Lahore, Rawalpindi, Faisalabad, Gujranwala, and Multan boards. These step-by-step solutions guide students through each algebraic expansion and linear system reduction, guaranteeing accuracy and top marks in board exams.

FAQs

Are these Exercise 5.2 solutions according to the Punjab Board syllabus?
Yes, these solutions strictly follow the official Punjab Curriculum and Textbook Board (PCTB / PECTAA) syllabus for Class 11 Mathematics, covering Case 3 and Case 4 irreducible quadratic factor decompositions for the 2026-2027 academic session.
Do these solutions cover every question in Exercise 5.2?
Yes, every textbook problem in Exercise 5.2 is solved with complete algebraic steps, including setting up linear numerators (Ax + B), expanding polynomial products, and systematically equating coefficients of like powers of x.
What is an irreducible quadratic factor?
An irreducible quadratic factor is a quadratic expression ax² + bx + c that cannot be factored into real linear factors (its discriminant b² - 4ac &lt; 0), such as (x² + 1) or (x² + 4).
How do you set up partial fractions for repeated irreducible quadratic factors like (ax² + bx + c)²?
For repeated quadratic factors (ax² + bx + c)², the partial fraction expansion is written as (Ax + B) / (ax² + bx + c) + (Cx + D) / (ax² + bx + c)², where undetermined coefficients A, B, C, and D are solved by equating corresponding coefficients.
Can students read these solutions online?
Yes, students can read all Exercise 5.2 solutions directly online through MaryamNotes.pk on any web browser without needing to download external files or install third-party apps.
Are these solutions useful for Punjab Board exams?
Yes, Case 3 and Case 4 partial fraction questions are among the most frequently tested Section C long questions in HSSC Part 1 Mathematics across all nine Punjab boards.
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 year.
Are these solutions free for students?
Yes, all Class 11 Mathematics Exercise 5.2 solutions on MaryamNotes.pk are available entirely free of cost for students and educators across Punjab.