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Chapter Analysis
Intermediate10 pages • EnglishQuick Summary
The chapter on Binomial Theorem in the Class 11 Mathematics textbook introduces the Binomial Theorem for positive integral indices, offering a simplified method to expand expressions like (a + b)^n. It discusses the use of Pascal's Triangle to determine binomial coefficients and includes the historical significance of mathematicians such as Blaise Pascal in its development. Through examples, the theorem is applied to solve typical problems involving binomial expansions, illustrating its utility in simplifying complex calculations.
Key Topics
- •Introduction to Binomial Theorem
- •Pascal's Triangle and Binomial Coefficients
- •Application of the Binomial Theorem
- •Expansion of Binomials with Positive Integral Exponents
- •Special Cases and Examples of Binomial Expansion
- •Historical Perspective on Binomial Theorem
Learning Objectives
- ✓Understand the concept and derivation of the binomial theorem.
- ✓Apply the binomial theorem to expand expressions.
- ✓Use Pascal's triangle to determine binomial coefficients.
- ✓Recognize the historical contributions to binomial theorem development.
- ✓Develop problem-solving skills using binomial expansions.
- ✓Explore real-world applications of binomial theorem.
Questions in Chapter
Expand using binomial theorem: (1–2x)^5
Page 133
Evaluate: (96)^3
Page 133
Using Binomial Theorem, indicate which number is larger (1.1)^10000 or 1000.
Page 133
Show that 9n+1 – 8n – 9 is divisible by 64, whenever n is a positive integer.
Page 134
Prove that a^n – b^n has a factor of (a – b) whenever n is a positive integer.
Page 134
Additional Practice Questions
Prove using the binomial theorem that (3x + 4)^3 = 27x^3 + 108x^2 + 144x + 64.
mediumAnswer: Using the binomial theorem, expand (3x + 4)^3 as: 3C0 * (3x)^3 + 3C1 * (3x)^2 * 4 + 3C2 * (3x) * 4^2 + 3C3 * 4^3. Calculating each term yields: 27x^3 + 108x^2 + 144x + 64.
If (1 + x)^6 = 1 + 6x + 15x^2 + 20x^3 + 15x^4 + 6x^5 + x^6, verify the computation of the coefficients using Pascal's triangle.
easyAnswer: The coefficients correspond to the 6th row of Pascal's triangle, which is 1, 6, 15, 20, 15, 6, 1, verifying that the expansion matches the given function.
Use the binomial theorem to estimate the value of (1.02)^5.
mediumAnswer: Approximate (1 + 0.02)^5 using the first three terms: 1 + 5(0.02) + 10(0.02)^2. This gives: 1 + 0.1 + 0.004 = 1.104.
Explain why (1 - x)^n = (1 + (-x))^n for n a positive integer.
easyAnswer: Substituting -x in place of b in the expansion (a + b)^n, the negative signs follow algebraic rules similarly to positive terms, demonstrating equivalence.
Expand and simplify the expression (2x - y)^3 using the binomial theorem.
hardAnswer: (2x - y)^3 = 3C0*(2x)^3 - 3C1*(2x)^2*y + 3C2*(2x)*y^2 - 3C3*y^3 = 8x^3 - 12x^2y + 6xy^2 - y^3.
If n is a positive integer, show that (1 - 1/n)^n approaches 1/e as n becomes large.
hardAnswer: The expression (1 - 1/n)^n is a simplification of the exponential limit definition for e, confirming that the limit tends towards 1/e.
Using the binomial theorem, find the middle term in the expansion of (a + b)^10.
mediumAnswer: The middle term in the expansion of (a + b)^10 is the 6th term, given by C(10,5)(a^5)(b^5).
Prove that for any positive integer n, (1 + i)^n + (1 - i)^n is a real number.
mediumAnswer: Expanding using the binomial theorem, imaginary terms cancel out, leaving a sum of only real terms, hence it is real.