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NCERT Solutions for Class 11 Math Ch-7: Binomial Theorem
NCERT Solutions for Class 11 Maths Chapter 7 Binomial Theorem covers the expansion of binomial expressions raised to a positive integral power, the general term, and the identification of the middle term in a binomial expansion.
Overview of Topics & Subtopics
Important Questions and Answers
Q1: State the Binomial Theorem for a positive integer n.
Answer: The Binomial Theorem states that for any positive integer n, (a + b)^n = Σ (from k=0 to n) of [nCk · a^(n−k) · b^k], which expands into a sum of (n+1) terms, each involving a binomial coefficient nCk.
Q2: What is the formula for the general term (the (r+1)th term) in the expansion of (a + b)^n?
Answer: The general term, denoted Tr+1, in the binomial expansion of (a + b)^n is given by Tr+1 = nCr · a^(n−r) · b^r, which allows any specific term in the expansion to be found directly without expanding the entire expression.
Q3: How is the middle term determined in a binomial expansion, and does it differ based on whether n is even or odd?
Answer: If n is even, there is exactly one middle term, the ((n/2)+1)th term. If n is odd, there are two middle terms, the ((n+1)/2)th and the ((n+3)/2)th terms, since the total number of terms (n+1) is even in that case.
Q4: What is Pascal's Triangle, and how is it related to binomial coefficients?
Answer: Pascal's Triangle is a triangular arrangement of numbers where each number is the sum of the two numbers directly above it. Each row of Pascal's Triangle directly gives the binomial coefficients (nC0, nC1, nC2, ..., nCn) for the expansion of (a + b)^n.
Q5: What is the sum of the binomial coefficients in the expansion of (1 + x)^n?
Answer: The sum of all the binomial coefficients in the expansion of (1 + x)^n is 2^n, obtained by substituting x = 1 into the expansion, giving nC0 + nC1 + nC2 + ... + nCn = 2^n.
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