Exercise 8.3 Class 11 Math Solutions - Unit 8 Mathematical Inductions and Binomial Theorem, Punjab Board

Step-by-step solutions to Exercise 8.3, Unit 8: Mathematical Inductions and Binomial Theorem, for Class 11 Punjab Board students. Covers the infinite binomial series expansion (1 + x)ⁿ for rational, fractional, and negative indices under the validity condition |x| < 1, finding general terms, decimal approximations of roots, and proving sum identities worked out question by question.

This page provides complete, step-by-step solved solutions for Exercise 8.3 of Unit 8: Mathematical Inductions and Binomial Theorem for Class 11 Mathematics under the Punjab Board (PCTB / PECTAA). All questions on expanding (1 + x)ⁿ for negative integers and fractional indices, verifying the validity condition |x| < 1, computing general terms Tᵣ₊₁, evaluating decimal approximations for roots of real numbers, and demonstrating infinite series summation proofs are thoroughly worked out for HSSC Part 1 board exam preparation.
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Exercise Overview

Exercise 8.3 generalizes the Binomial Theorem to any rational index n (including negative integers and fractions) in Unit 8 of Class 11 Mathematics. When n is not a positive integer, the expansion of (1 + x)ⁿ generates an infinite series: (1 + x)ⁿ = 1 + nx + [n(n - 1)/2!]x² + [n(n - 1)(n - 2)/3!]x³ + ... + Tᵣ₊₁ + ... which is valid and convergent if and only if |x| < 1.

The exercise trains students in crucial mathematical applications: transforming binomial expressions (a + b)ⁿ into aⁿ(1 + b/a)ⁿ to establish convergence when |b/a| < 1, computing terms of specified orders, and approximating higher-order roots of numbers (such as evaluating √26 = √(25 + 1) = 5(1 + 1/25)^(1/2)) to designated decimal places. Additionally, students master the technique of recognizing standard binomial series patterns to sum infinite numerical series into compact closed-form radical values.

In Punjab Board HSSC Part 1 examinations, Exercise 8.3 is a consistent and high-frequency source for Section B short questions (validity conditions and initial term expansions) and Section C long questions (approximations and series summation proofs) across Lahore, Rawalpindi, Gujranwala, Faisalabad, and Multan boards. Mastering these step-by-step solutions gives students strong analytical capability in infinite series expansions.

FAQs

Are these Exercise 8.3 solutions according to the Punjab Board syllabus?
Yes, these solutions strictly adhere to the official Punjab Curriculum and Textbook Board (PCTB / PECTAA) syllabus for Class 11 Mathematics, covering binomial series for any rational index for the 2026-2027 academic session.
Do these solutions cover every question in Exercise 8.3?
Yes, every textbook problem from Exercise 8.3 is completely solved with step-by-step algebraic working, including infinite series expansions, finding specific coefficients, numerical root approximations, and sum of series proofs.
When is the binomial expansion (1 + x)ⁿ valid for negative or fractional n?
The infinite series expansion (1 + x)ⁿ = 1 + nx + [n(n - 1)/2!]x² + ... is valid if and only if the absolute value of x is strictly less than 1 (|x| < 1 or -1 < x < 1).
What is the general term formula in the binomial expansion for any rational index?
The general (r + 1)-th term is given by Tᵣ₊₁ = [n(n - 1)(n - 2)...(n - r + 1) / r!] xʳ.
Can students read these solutions online?
Yes, students can read all Exercise 8.3 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, approximating roots (like √26 or ∛65 to four decimal places) and proving that infinite series equal specific algebraic expressions are frequently tested in Section B short questions and Section C long questions.
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 8.3 solutions on MaryamNotes.pk are available entirely free of cost for students and teachers across Punjab.