Binomial Theorem

Action Plan Notes Questions Mock-Test

Study Action Plan (Sorted by Difficulty)

General Term and Middle Term

Accuracy: 40%

Total Attempted10

Total Correct4

Total Incorrect6

Difficulty Level:

High

Binomial Identities and Divisibility Applications

Accuracy: 44%

Total Attempted9

Total Correct4

Total Incorrect5

Difficulty Level:

High

Binomial Coefficients

Accuracy: 50%

Total Attempted14

Total Correct7

Total Incorrect7

Difficulty Level:

Medium

Proof by Mathematical Induction for Binomial Theorem

Accuracy: 50%

Total Attempted8

Total Correct4

Total Incorrect4

Difficulty Level:

Medium

Pascal's Triangle

Accuracy: 58%

Total Attempted19

Total Correct11

Total Incorrect8

Difficulty Level:

Medium

Binomial Theorem for Positive Integral Indices

Accuracy: 58%

Total Attempted12

Total Correct7

Total Incorrect5

Difficulty Level:

Medium

Numerical Evaluation Using Binomial Theorem

Accuracy: 84%

Total Attempted19

Total Correct16

Total Incorrect3

Difficulty Level:

Low

Special Cases of Binomial Expansion

Accuracy: 100%

Total Attempted5

Total Correct5

Total Incorrect0

Difficulty Level:

Low

Learning Objectives

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Learning Objectives

Understand the Binomial Theorem for Positive Integral Indices and its application in expanding expressions of the form $(a + b)^n$ .
Analyze the structure and use of Pascal's Triangle to determine binomial coefficients for expansions.
Prove the binomial theorem using Mathematical Induction, establishing the base case and proving the inductive step.
Calculate Binomial Coefficients using combinations and apply them in expansions.
Explore Special Cases of Binomial Expansion, such as $(x - y)^n$ , and apply simplifications for specific values of $a$ and $b$ .
Determine the General Term and Middle Term in binomial expansions, using the formula $T_{r+1} = \binom{n}{r} a^{n-r} b^r$ .
Perform Numerical Evaluation Using Binomial Theorem for expressions like $(98)^5$ and $(101)^4$ .
Prove Binomial Identities and Divisibility Applications, such as showing that $6^n - 5n$ leaves a remainder of 1 when divided by 25.

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Revision Notes & Summary

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Chapter Notes

Binomial Theorem for Positive Integral Indices

Definition: The binomial theorem provides a formula for expanding expressions of the form $(a + b)^n$ , where $n$ is a positive integer.
Expansion: $(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$
Properties:
- The total number of terms in the expansion is $n+1$ .
- Powers of $a$ decrease from $n$ to $0$ , while powers of $b$ increase from $0$ to $n$ .
- In each term, the sum of the powers of $a$ and $b$ equals $n$ .

Pascal's Triangle

Description: A triangular array of numbers where each number is the sum of the two directly above it.
Use: Provides the binomial coefficients for the expansion of $(a + b)^n$ .
Example:
- Row for $n=4$ : $1, 4, 6, 4, 1$

Proof by Mathematical Induction for Binomial Theorem

Base Case: For $n=1$ , $(a + b)^1 = a + b$ .
Inductive Step: Assume true for $n=k$ , prove for $n=k+1$ .
Conclusion: By induction, the theorem holds for all positive integers $n$ .

Binomial Coefficients

Notation: $inom{n}{k}$ , also known as combinations.
Calculation: $inom{n}{k} = \frac{n!}{k!(n-k)!}$

Special Cases of Binomial Expansion

Negative Terms: $(x - y)^n = \sum_{k=0}^{n} (-1)^k \binom{n}{k} x^{n-k} y^k$
Example: $(x - 2y)^5 = x^5 - 10x^4y + 40x^3y^2 - 80x^2y^3 + 80xy^4 - 32y^5$

General Term and Middle Term

General Term: $T_{r+1} = \binom{n}{r} a^{n-r} b^r$
Middle Term: If $n$ is even, middle term is $T_{\frac{n}{2} + 1}$ ; if odd, terms are $T_{\frac{n}{2}}$ and $T_{\frac{n}{2} + 1}$ .

Numerical Evaluation Using Binomial Theorem

Example: $(98)^5 = (100 - 2)^5$ $(98)^{5} = (100 - 2)^{5}$
- Expansion: $= 10000000000 - 5 \times 100000000 \times 2 + 10 \times 1000000 \times 4 - 10 \times 10000 \times 8 + 5 \times 100 \times 16 - 32$
- Result: $= 9039207968$

Binomial Identities and Divisibility Applications

Identity: $inom{n}{0} + inom{n}{1} + inom{n}{2} + ext{...} + inom{n}{n} = 2^n$
Divisibility: Prove $6^n - 5n$ $6^{n} - 5 n$ leaves remainder 1 when divided by 25.
- Formula: $6^n - 5n = 25k + 1$
- Conclusion: Remainder is 1.

Exercises

Expand: $(1 - 2x)^5$ , $(2x + 3)^6$
Evaluate: $(96)^3$ , $(102)^5$ , $(101)^4$ , $(99)^5$
Comparison: Determine which is larger, $(1.1)^{10000}$ or $1000$ .
Divisibility Proof: Show $9^{n+1} - 8n - 9$ is divisible by 64 for positive integers $n$ .

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Practice Test – MCQs, True/False

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Multiple Choice Questions

160

240

320

180

Correct Answer: A

Solution:

The binomial expansion of

(x + 2)^6

is given by:

\sum_{k=0}^{6} \binom{6}{k} x^{6-k} (2)^k

. The coefficient of the

x^3

term is

\binom{6}{3} \times 2^3 = 20 \times 8 = 160

Chapter Concept:

Numerical Evaluation Using Binomial Theorem

a^3 + 3a^2b + 3ab^2 + b^3

a^3 + 2a^2b + ab^2 + b^3

a^3 + 3a^2b + 2ab^2 + b^3

a^3 + 3a^2b + 3ab^2

Correct Answer: A

Solution:

The binomial expansion of

(a + b)^3

a^3 + 3a^2b + 3ab^2 + b^3

Chapter Concept:

Binomial Theorem for Positive Integral Indices

Correct Answer: B

Solution:

The sum of the coefficients in the expansion of

(1 + x)^n

(1 + 1)^n = 2^n

. Given

2^n = 2048

, solving gives

n = 11

Chapter Concept:

General Term and Middle Term

120

Correct Answer: C

Solution:

The coefficient of the

x^4

term in

(x + 2)^6

is given by

\binom{6}{4}(2)^2 = 15 \times 4 = 60

Chapter Concept:

Binomial Coefficients

128

256

512

Correct Answer: B

Solution:

The sum of the coefficients in the expansion of

(a + b)^n

2^n

. For

(a + b)^7

, it is

2^7 = 128

Chapter Concept:

Pascal's Triangle

2^n

n

n+1

0

Correct Answer: A

Solution:

The sum of the coefficients in the expansion of

(x + 1)^n

is obtained by substituting

x = 1

, which results in

(1 + 1)^n = 2^n

Chapter Concept:

Binomial Identities and Divisibility Applications

True

False

Depends on

n

None of the above

Correct Answer: A

Solution:

Using the binomial theorem,

6^n = (1+5)^n = \sum_{r=0}^{n} \binom{n}{r} 1^{n-r} 5^r

. Subtracting

5n

and rearranging terms, we get

6^n - 5n = 1 + 25k

, where

k

is an integer, thus leaving a remainder of 1 when divided by 25.

Chapter Concept:

Binomial Identities and Divisibility Applications

True, because

6^n - 5n = 25k + 1

for some integer

k

False, because

6^n - 5n

is not divisible by 25.

True, because

6^n - 5n = 25k

for some integer

k

False, because

6^n - 5n

leaves a remainder of 0.

Correct Answer: A

Solution:

Using the binomial theorem,

6^n - 5n = 25k + 1

, thus leaving a remainder of 1.

Chapter Concept:

Binomial Identities and Divisibility Applications

60x^4

240x^4

160x^4

192x^4

Correct Answer: A

Solution:

The expansion of

(x + 2)^6

is given by

\sum_{k=0}^{6} \binom{6}{k} x^{6-k} 2^k

. The term containing

x^4

corresponds to

k=2

, which is

\binom{6}{2} x^4 2^2 = 60x^4

Chapter Concept:

Proof by Mathematical Induction for Binomial Theorem

m = \frac{n}{2}

m = \frac{n+1}{2}

m = \frac{n-1}{2}

m = \frac{n+2}{2}

Correct Answer: A

Solution:

When

n

is even, the middle term in the expansion of

(x + y)^n

is the

\left(\frac{n}{2} + 1\right)

-th term, hence

m = \frac{n}{2}

Chapter Concept:

General Term and Middle Term

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Binomial Theorem

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True or False

Correct Answer: True

Solution:

For even

n

, the expansion of

(a + b)^n

has a middle term at position

\frac{n}{2}+1

. This term is

T_{\frac{n}{2}+1} = \binom{n}{\frac{n}{2}} a^{\frac{n}{2}} b^{\frac{n}{2}}

Chapter Concept :

General Term and Middle Term

Correct Answer: True

Solution:

In the expansion of

(x - y)^n

, the binomial coefficients alternate in sign due to the presence of

(-y)

raised to successive powers.

Chapter Concept :

Special Cases of Binomial Expansion

Correct Answer: True

Solution:

In the expansion of

(a + b)^n

, the sum of the indices of

a

and

b

in each term is equal to

n

, as observed in the properties of the binomial expansion.

Chapter Concept :

Binomial Theorem for Positive Integral Indices

Correct Answer: True

Solution:

The general term in the binomial expansion of

(a + b)^n

is indeed given by

T_{r+1} = \binom{n}{r} a^{n-r} b^r

, where

\binom{n}{r}

is the binomial coefficient.

Chapter Concept :

General Term and Middle Term

Correct Answer: True

Solution:

The proof of the binomial theorem for positive integers is obtained by applying the principle of mathematical induction.

Chapter Concept :

Proof by Mathematical Induction for Binomial Theorem

Correct Answer: True

Solution:

Pascal's Triangle is a triangular array of numbers where each number is the sum of the two directly above it. It is used to find binomial coefficients for the expansion of

(a + b)^n

Chapter Concept :

Pascal's Triangle

Correct Answer: True

Solution:

The binomial coefficients

nC_k

are the numbers in Pascal's triangle and can be used to write the row for any index without writing the earlier rows.

Chapter Concept :

Binomial Coefficients

Correct Answer: False

Solution:

In the expansion of

(a + b)^n

, the sum of the indices of

a

and

b

n

in every term of the expansion.

Chapter Concept :

Pascal's Triangle

Correct Answer: False

Solution:

The middle term in the expansion of

(a + b)^n

is not necessarily the term with the highest coefficient. The middle term is determined by the position in the sequence, not by the coefficient size.

Chapter Concept :

General Term and Middle Term

Correct Answer: False

Solution:

While Pascal's triangle provides a convenient way to visualize the coefficients, they are mathematically derived using combinations, represented as

nCk

. The triangle itself is constructed based on these combinations.

Chapter Concept :

Binomial Coefficients

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Binomial Theorem

Study Action Plan (Sorted by Difficulty)

General Term and Middle Term

High

Binomial Identities and Divisibility Applications

High

Binomial Coefficients

Medium

Proof by Mathematical Induction for Binomial Theorem

Medium

Pascal's Triangle

Medium

Binomial Theorem for Positive Integral Indices

Medium

Numerical Evaluation Using Binomial Theorem

Low

Special Cases of Binomial Expansion

Low

Learning Objectives

Learning Objectives

Practice, Analyze & Improve

Revision Notes & Summary

Chapter Notes

Binomial Theorem for Positive Integral Indices

Pascal's Triangle

Proof by Mathematical Induction for Binomial Theorem

Binomial Coefficients

Special Cases of Binomial Expansion

General Term and Middle Term

Numerical Evaluation Using Binomial Theorem

Binomial Identities and Divisibility Applications

Exercises

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Practice Test – MCQs, True/False

Multiple Choice Questions

Q1.Using the binomial theorem, what is the coefficient of the x3x^3x3 term in the expansion of (x+2)6(x + 2)^6(x+2)6?

Correct Answer: A

Q2.What is the binomial expansion of (a+b)3(a + b)^3(a+b)3?

Correct Answer: A

Q3.In the binomial expansion of (1+x)n(1 + x)^n(1+x)n, if the sum of the coefficients is 2048, what is the value of nnn?

Correct Answer: B

Q4.If (x+2)6(x + 2)^6(x+2)6 is expanded using the binomial theorem, what is the coefficient of the x4x^4x4 term?

Correct Answer: C

Q5.What is the value of the sum of the coefficients in the expansion of (a+b)7(a + b)^7(a+b)7?

Correct Answer: B

Q6.What is the sum of the coefficients in the expansion of (x+1)n(x + 1)^n(x+1)n?

Correct Answer: A

Q7.Using the binomial theorem, prove that 6n−5n6^n - 5n6n−5n always leaves a remainder of 1 when divided by 25.

Correct Answer: A

Q8.Prove that 6n−5n6^n - 5n6n−5n always leaves a remainder of 1 when divided by 25.

Correct Answer: A

Q9.If (x+2)6(x + 2)^6(x+2)6 is expanded using the binomial theorem, what is the term containing x4x^4x4?

Correct Answer: A

Q10.For the binomial expansion of (x+y)n(x + y)^n(x+y)n, if the middle term is given by Tm+1T_{m+1}Tm+1​, which of the following is true when nnn is even?

Correct Answer: A

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Q1.The middle term in the expansion of (a+b)n(a + b)^n(a+b)n for even nnn is given by Tn2+1T_{\frac{n}{2}+1}T2n​+1​.

Correct Answer: True

Q2.In the expansion of (x−y)n(x - y)^n(x−y)n, the coefficients alternate in sign.

Correct Answer: True

Q3.In the binomial expansion of (a+b)n(a + b)^n(a+b)n, the sum of the indices of aaa and bbb in each term is always equal to nnn.

Correct Answer: True

Q4.In the binomial expansion of (a+b)n(a + b)^n(a+b)n, the general term Tr+1T_{r+1}Tr+1​ is given by Tr+1=(nr)an−rbrT_{r+1} = \binom{n}{r} a^{n-r} b^rTr+1​=(rn​)an−rbr.

Correct Answer: True

Q5.The binomial theorem for positive integers is proven using mathematical induction.

Correct Answer: True

Q6.Pascal's Triangle is used to find binomial coefficients for the expansion of (a+b)n(a + b)^n(a+b)n.

Correct Answer: True

Q7.The binomial coefficient nCknC_knCk​ can be derived from the rows of Pascal's triangle.

Correct Answer: True

Q8.The sum of the indices of aaa and bbb in each term of the binomial expansion (a+b)n(a + b)^n(a+b)n is different and does not equal nnn.

Correct Answer: False

Q9.In the expansion of (a+b)n(a + b)^n(a+b)n, the middle term is always the term with the highest coefficient.

Correct Answer: False

Q10.The coefficients in the expansion of (a+b)n(a + b)^n(a+b)n can be directly obtained from Pascal's triangle without using combinations.

Correct Answer: False