Unit 8 Class 11 Math Notes – Mathematical Induction and Binomial Theorem (Punjab Board New Book)
Unit 8 (Chapter 8) Class 11 Math Notes – Mathematical Induction and Binomial Theorem: step-by-step solutions of Exercise 8.1, Exercise 8.2, Exercise 8.3 and Exercise 8.4 for the Punjab Board new book (1st Year). Tap an exercise below to open its PDF solution, then view it online or download it.
This unit is part of our Class 11 maths notes for the Punjab Board new book, where you can open the solutions of the other units.
Exercise Wise Solutions of Unit 8 Class 11 Math
What Each Exercise of Unit 8 Covers
Exercise 8.1 – Mathematical Induction
Exercise 8.1 has 12 questions on proof by induction: sums of series, divisibility (for example that \(5^n-2^n\) is divisible by 3), inequalities such as \(4^n>3^n+2^n\), and word problems on compound interest, savings and loan balances.
Exercise 8.2 – Binomial Theorem for a Positive Integral Index
Exercise 8.2 has 10 questions on binomial expansions with a positive integer index: expanding brackets, finding \((0.97)^3\), a specific term, a coefficient, the term independent of \(x\), the middle term(s), and the sum of binomial coefficients.
Exercise 8.3 – Binomial Theorem for a Rational Index
Exercise 8.3 has 10 questions on expansions with a negative or fractional index: writing the expansion and its validity, finding the coefficient of \(x^n\), small-\(x\) approximations and sums of infinite series.
Exercise 8.4 – Applications of the Binomial Theorem
Exercise 8.4 has 14 questions on applications: approximating roots and powers, remainders and last digits, divisibility, revenue growth, compound interest and tournament matches.
Key Formulas of Unit 8 Class 11 Math
- Induction: prove the statement for \(n=1\), assume it for \(n=k\), then prove it for \(n=k+1\).
- General term: \(T_{r+1}={}^{n}C_r\,a^{n-r}b^r\).
- Middle term: one term when \(n\) is even, two terms when \(n\) is odd.
- For a rational index and \(|x|<1\): \((1+x)^n=1+nx+\dfrac{n(n-1)}{2!}x^2+\dfrac{n(n-1)(n-2)}{3!}x^3+\cdots\)
Related Class 11 Math Notes
Continue with Unit 7 Class 11 Math Notes and Unit 9 Class 11 Math Notes, or go back to the Class 11 maths notes hub for all 14 units.
FAQs About Unit 8 Class 11 Math
How many exercises are there in Unit 8?
Unit 8 has four exercises: Exercise 8.1, 8.2, 8.3 and 8.4.
What is the difference between Exercise 8.2 and Exercise 8.3?
Exercise 8.2 uses a positive whole-number index, so the expansion ends. Exercise 8.3 uses a negative or fractional index, so the expansion is an infinite series valid for \(|x|<1\).
Are these notes for the new book?
Yes. These solutions are for the new Class 11 mathematics book by PECTAA (2025 edition onwards), used by the nine BISE Punjab boards: Lahore, Gujranwala, Faisalabad, Multan, Rawalpindi, Sargodha, Sahiwal, Bahawalpur and Dera Ghazi Khan.