Exercise 5.1 Class 11 Math Solutions - Unit 5 Partial Fractions, Punjab Board
Step-by-step solutions to Exercise 5.1, the opening exercise of Unit 5: Partial Fractions, for Class 11 Punjab Board students. Covers identifying proper and improper rational fractions, polynomial long division, and partial fraction decomposition for denominators containing non-repeated linear factors (Case 1) and repeated linear factors (Case 2) worked out question by question.
Exercise Overview
Exercise 5.1 introduces Unit 5: Partial Fractions in Class 11 Mathematics by establishing the foundational theory of rational fraction decomposition. A rational fraction is an algebraic quotient P(x) / Q(x) where P(x) and Q(x) are polynomials with Q(x) ≠ 0. The exercise reinforces the vital distinction between proper rational fractions (where degree of P(x) < degree of Q(x)) and improper rational fractions (where degree of P(x) ≥ degree of Q(x)), demonstrating how polynomial long division must first be applied to transform improper fractions into a polynomial quotient plus a proper rational remainder.
The core algebraic techniques in Exercise 5.1 focus on two fundamental cases of partial fraction decomposition: Case 1, where the denominator Q(x) consists purely of non-repeated linear factors of the form (ax + b), decomposed as A/(a₁x + b₁) + B/(a₂x + b₂); and Case 2, where the denominator contains repeated linear factors of the form (ax + b)ᵏ, decomposed into terms with increasing powers up to degree k. Students utilize both the zero-factor method (direct root substitution) and equating coefficients of like powers of x to solve for undetermined constants.
In Punjab Board HSSC Part 1 examinations, Exercise 5.1 is an essential chapter section frequently appearing in Section B short questions and Section C long questions across Lahore, Rawalpindi, Gujranwala, Faisalabad, and Multan boards. These step-by-step solutions provide students with structured mathematical clarity, helping them avoid sign errors and secure maximum exam scores.