Q. Find the value of the coefficient of x^4 in the expansion of (x - 2)^6.
A.
15
B.
20
C.
30
D.
40
Solution
Using the binomial theorem, the coefficient of x^4 in (a + b)^n is given by nCk * a^(n-k) * b^k. Here, n=6, a=x, b=-2, and k=2. Thus, the coefficient is 6C2 * (1)^4 * (-2)^2 = 15 * 4 = 60.
Q. In the expansion of (a + b)^n, if the coefficient of a^3b^2 is 60, what is the value of n?
A.
5
B.
6
C.
7
D.
8
Solution
C(n,3) * b^2 = 60. For n = 5, C(5,3) = 10, which does not satisfy. For n = 6, C(6,3) = 20, which does not satisfy. For n = 7, C(7,3) = 35, which does not satisfy. For n = 8, C(8,3) = 56, which satisfies.
The Binomial Theorem is a crucial topic in mathematics that plays a significant role in various school and competitive exams. Understanding this theorem not only helps in solving complex problems but also enhances your ability to tackle objective questions effectively. Practicing MCQs related to the Binomial Theorem can significantly improve your exam preparation and boost your confidence in answering important questions.
What You Will Practise Here
Understanding the Binomial Theorem and its applications
Deriving the Binomial expansion formula
Identifying coefficients in binomial expansions
Solving problems involving positive and negative integer exponents
Exploring the concept of binomial coefficients
Applying the theorem in real-life scenarios and word problems
Analyzing patterns in binomial expansions through diagrams
Exam Relevance
The Binomial Theorem is frequently featured in CBSE, State Boards, NEET, and JEE exams. Students can expect questions that require them to apply the theorem to find specific coefficients or to expand binomial expressions. Common question patterns include multiple-choice questions that test conceptual understanding and application of the theorem in various contexts.
Common Mistakes Students Make
Confusing the terms of the binomial expansion with those of other algebraic expressions
Misapplying the formula for binomial coefficients
Overlooking the importance of signs in expansions involving negative exponents
Failing to simplify expressions correctly after applying the theorem
FAQs
Question: What is the Binomial Theorem? Answer: The Binomial Theorem provides a formula for expanding expressions raised to a power, expressed as (a + b)^n.
Question: How can I use the Binomial Theorem in exams? Answer: You can use it to solve problems involving expansions, coefficients, and applications in various mathematical contexts.
Start solving practice MCQs on the Binomial Theorem today to enhance your understanding and prepare effectively for your exams. Test your knowledge and boost your confidence with our objective questions!
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