Crash Course Notes: Binomial Theorem - Maths
Topic Importance in JEE
| Metric | Value | Remarks |
|---|---|---|
| Total Questions (2017-2024) | 10 | At least one question has been included each year in JEE. |
| Weightage | 4.26% | Represents the proportion of questions from this topic. |
Yearly Question Distribution
| Year | Topic Area | Concepts Covered | Number of Questions | Difficulty Level | Key Focus Areas |
|---|---|---|---|---|---|
| 2024 | Coefficient of the expansion $(1+x)^n$ | Put $x=1$ | 1 | Easy | Focus on evaluating total sum of coefficients |
| 2023 | Binomial expansion | $\displaystyle\sum_{r=1}^n {n \choose 2r-1} = 2^{n-1}$ | 2 | Average | Emphasis on odd term coefficients and symmetry |
| 2022 | Combination/ Coefficient of the expansion $(1-x)^n$ | Sum of series $\displaystyle\sum_{i=0}^n {n \choose i} = 2^n$ Multiply the coefficients | 2 | Average | Use identity and sign alternation for coefficient manipulation |
| 2021 | To obtained reminder | $x = {(x-1)+1}^n$ or ${[x+1]-1}^n$ | 1 | Average | Apply binomial expansion to simplify expressions for remainder |
| 2020 | Binomial Expansion | General Terms | 1 | Average | Identify and extract general term using binomial formula |
| 2019 | Binomial Expansion | General Terms | 1 | Average | Focus on term structure and position in expansion |
| 2018 | Coefficient of Binomial Expansion | Sum of coefficients of odd degree terms | 1 | Easy | Use substitution to isolate odd-degree contributions |
Related Video
Table of Contents
- Binomial Theorem
1.1. Definition and Expansion
1.2. Key Concepts
1.3. Applications - Binomial Coefficients
2.1. Properties
2.2. Examples - Summary of Binomial Theorem
1. Binomial Theorem
1.1. Definition and Expansion
The Binomial Theorem is an algebraic expression that expands powers of a binomial (e.g., $ (x + y)^n $) into a sum of terms involving binomial coefficients.
- Formula:
$$ (x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k $$ - Explanation:
Each term in the expansion is of the form $ \binom{n}{k} x^{n-k} y^k $, where $ \binom{n}{k} $ is the binomial coefficient. - Non-negative Exponents:
The exponents $ b $ and $ c $ in $ a x^b y^c $ satisfy $ b + c = n $.
1.2. Key Concepts
- Binomial Coefficient:
$ \binom{n}{k} $ is calculated as $ \dfrac{n!}{k!(n-k)!} $. - Symmetry:
$ \binom{n}{k} = \binom{n}{n-k} $. - Special Cases:
- $ (x + y)^0 = 1 $
- $ (x + y)^1 = x + y $
- $ (x + y)^2 = x^2 + 2xy + y^2 $
1.3. Applications
- Divisibility Problems:
Example: $ (1 + \alpha)^n - 1 $ is divisible by $ \alpha $. - R-f Factor Relationship:
A relationship between the remainder and the divisor in polynomial division. - Calculus:
Used in Taylor series expansions and analyzing function behavior. - Negative/Rational Exponents:
The theorem extends to non-integer exponents, e.g., $ (1 + x)^r $ for rational $ r $.
2. Binomial Coefficients
2.1. Properties
- Symmetry:
$ \binom{n}{k} = \binom{n}{n-k} $. - Sum of Coefficients:
$ \sum_{k=0}^{n} \binom{n}{k} = 2^n $. - Recursive Relationship:
$ \binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k} $. - Combinatorial Interpretation:
Represents the number of ways to choose $ k $ elements from $ n $.
2.2. Examples
- Divisibility Example:
$ (1 + \alpha)^n - 1 $ is divisible by $ \alpha $. - General Expansion:
For $ (x + y)^n $, the coefficients follow the pattern of Pascal’s Triangle. - Negative Exponents:
$ (1 + x)^{-1} = 1 - x + x^2 - x^3 + \dots $ (infinite series).
3. Summary of Binomial Theorem
- Core Concept:
Expands $ (x + y)^n $ into a sum of terms with binomial coefficients. - Formula:
$$ (x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k $$ - Key Components:
- Binomial Coefficients: $ \binom{n}{k} $
- Exponents: $ b + c = n $
- Applications: Divisibility, calculus, series expansions.
- Special Cases:
- $ n = 0 $: Result is 1.
- $ n = 1 $: Result is $ x + y $.
- $ n = 2 $: Result is $ x^2 + 2xy + y^2 $.
Practice Problems
##### If $(1+a x)^{n}=1+8 x+24 x^{3}+\ldots$, then the values of $a$ and $n$ are
1. [x] 2,4
2. [ ] 2, 3
3. [ ] 3,6
4. [ ] 1,2
##### The coefficient of $x^{n}$ in the expansion of $(1+x)^{2 n}$ and $(1+x)^{2 n-1}$ are in the ratio $\rightarrow$ NCERT Exemplar
1. [ ] $1: 2$
2. [ ] $1: 3$
3. [ ] $3: 1$
4. [x] $2: 1$
##### The value of $(1.002)^{12}$ upto fourth place of decimal is
1. [x] 1.0242
2. [ ] 1.0245
3. [ ] 1.0004
4. [ ] 1.0254
##### The coefficient of $x^{4}$ in the expansion of $(1+x+x^{2}+x^{3})^{n}$ is
1. [ ] ${ }^{n} C _4$
2. [ ] ${ }^{n} C _4+{ }^{n} C _2$
3. [ ] ${ }^{n} C _4+{ }^{n} C _2+{ }^{n} C _2$
4. [x] ${ }^{n} C _4+{ }^{n} C _2+{ }^{n} C _1 \cdot{ }^{n} C _2$
##### If the middle term of $\Big( \dfrac{1}{x}+x \sin x\Big)^{10}$ is equal to $7 \dfrac{7}{8}$, then the value of $x$ is $\rightarrow$ NCERT Exemplar
1. [ ] $2 n \pi+ \dfrac{\pi}{6}$
2. [ ] $n \pi+ \dfrac{\pi}{6}$
3. [x] $n \pi+(-1)^{n} \dfrac{\pi}{6}$
4. [ ] $n \pi+(-1) \dfrac{\pi}{3}$
##### If the 7 th term in the binomial expansion of $\Big( \dfrac{3}{\sqrt[3]{84}}+\sqrt{3} \ln x\Big)^{9}, x>0$ is equal to 729, then $x$ can be
1. [ ] $e^{2}$
2. [x] $e$
3. [ ] $ \dfrac{e}{2}$
4. [ ] $2 e$
BETA
