Question 1

Consider the expression $(1 + x)^n$. If the coefficient of $x^3$ is 56, what is the value of $n$?

Accepted Answer

Correct Option: A. Solution: The coefficient of $x^3$ in $(1 + x)^n$ is given by $\binom{n}{3}$. Solving $\binom{n}{3} = 56$ gives $n = 7$.

Question 2

Which of the following statements is true regarding the binomial expansion of $(x + y)^n$?

Accepted Answer

Correct Option: A. Solution: In the binomial expansion of $(x + y)^n$, the sum of the indices of $x$ and $y$ in each term is always $n$. This is a fundamental property of binomial expansions.

Question 3

In the binomial expansion of $(a + b)^n$, what is the sum of the exponents of $a$ and $b$ in any term?

Accepted Answer

Correct Option: A. Solution: In the binomial expansion of $(a + b)^n$, the sum of the exponents of $a$ and $b$ in any term is always equal to $n$, as each term is of the form $\binom{n}{k} a^{n-k} b^k$.

Question 4

Which of the following expressions represents the expansion of $(2x + 3y)^3$ using the binomial theorem?

Accepted Answer

Correct Option: C. Solution: The expansion of $(2x + 3y)^3$ is calculated using the binomial theorem: $8x^3 + 3 	imes 4x^2 	imes 3y + 3 	imes 2x 	imes 9y^2 + 27y^3 = 8x^3 + 24x^2y + 36xy^2 + 27y^3$.

Question 5

In the expansion of $(x + y)^7$, what is the sum of the coefficients?

Accepted Answer

Correct Option: B. Solution: The sum of the coefficients in the expansion of $(x + y)^n$ is given by $(1 + 1)^n = 2^n$. For $n=7$, the sum is $2^7 = 128$.

Question 6

In Pascal's Triangle, what is the coefficient of the middle term in the expansion of $(a + b)^6$?

Accepted Answer

Correct Option: C. Solution: The middle term in the expansion of $(a + b)^6$ is given by the binomial coefficient $\binom{6}{3} = 20$.

Question 7

Using the binomial theorem, which expression represents the expansion of $(99)^5$?

Accepted Answer

Correct Option: A. Solution: Express $99$ as $100 - 1$. Thus, $(99)^5 = (100 - 1)^5$. Using the binomial theorem, we expand it as: $100^5 - 5 	imes 100^4 	imes 1 + 10 	imes 100^3 	imes 1^2 - 10 	imes 100^2 	imes 1^3 + 5 	imes 100 	imes 1^4 - 1^5$.

Question 8

How does the binomial theorem simplify the calculation of $(98)^5$?

Accepted Answer

Correct Option: B. Solution: The binomial theorem simplifies the calculation by expressing 98 as $100 - 2$ and expanding $(100 - 2)^5$ using the theorem.

Question 9

If the binomial expansion of $(x + y)^n$ has 11 terms, what is the value of $n$?

Accepted Answer

Correct Option: A. Solution: The number of terms in the expansion of $(x + y)^n$ is $n + 1$. If there are 11 terms, then $n + 1 = 11$, so $n = 10$.

Question 10

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

Accepted Answer

Correct Option: 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 $m = \frac{n}{2}$.

Question 11

Consider the binomial expansion of $(x - y)^5$. What is the coefficient of the $x^3y^2$ term?

Accepted Answer

Correct Option: A. Solution: The binomial expansion of $(x - y)^5$ is given by $\sum_{k=0}^{5} \binom{5}{k} x^{5-k} (-y)^k$. For the $x^3y^2$ term, $k=2$. Thus, the coefficient is $\binom{5}{2}(-1)^2 = 10$. However, since $y$ is negative, the sign alternates, resulting in $-10$.

Question 12

Which of the following is a correct application of the binomial theorem for $(3x - 2)^3$?

Accepted Answer

Correct Option: A. Solution: The binomial expansion of $(3x - 2)^3$ is given by $\sum_{k=0}^{3} \binom{3}{k} (3x)^{3-k} (-2)^k = 27x^3 - 54x^2 + 36x - 8$.

Question 13

Which expansion correctly uses the binomial theorem for $(102)^5$?

Accepted Answer

Correct Option: A. Solution: Express $102$ as $100 + 2$. Thus, $(102)^5 = (100 + 2)^5$. Using the binomial theorem, we expand it as: $100^5 + 5 	imes 100^4 	imes 2 + 10 	imes 100^3 	imes 2^2 + 10 	imes 100^2 	imes 2^3 + 5 	imes 100 	imes 2^4 + 2^5$.

Question 14

Using Pascal's Triangle, what is the coefficient of $x^3y^2$ in the expansion of $(x + y)^5$?

Accepted Answer

Correct Option: B. Solution: The coefficient of $x^3y^2$ in the expansion of $(x + y)^5$ is 10, as given by the binomial coefficient $\binom{5}{3} = 10$.

Question 15

What is the expansion of $(x - y)^3$ using the binomial theorem?

Accepted Answer

Correct Option: A. Solution: The binomial expansion of $(x - y)^3$ is given by $nC_0x^3 - nC_1x^2y + nC_2xy^2 - nC_3y^3$ for $n = 3$, which simplifies to $x^3 - 3x^2y + 3xy^2 - y^3$.

Question 16

Which of the following statements about Pascal's Triangle is incorrect?

Accepted Answer

Correct Option: D. Solution: Pascal's Triangle is specifically used for binomial expansions, not for any polynomial expansion.

Question 17

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

Accepted Answer

Correct Option: A. Solution: The sum of the coefficients in the expansion of $(x + 1)^n$ is obtained by substituting $x = 1$, which results in $(1 + 1)^n = 2^n$.

Question 18

Which of the following statements correctly describes the pattern observed in Pascal's Triangle?

Accepted Answer

Correct Option: A. Solution: Pascal's Triangle is constructed such that each row starts and ends with 1, and each element inside the row is the sum of the two elements directly above it from the previous row.

Question 19

If Pascal's Triangle is used to expand $(2x + 3y)^4$, what is the coefficient of the term $x^2y^2$?

Accepted Answer

Correct Option: B. Solution: The coefficient of $x^2y^2$ in the expansion of $(2x + 3y)^4$ is $\binom{4}{2} \cdot (2)^2 \cdot (3)^2 = 6 \cdot 4 \cdot 9 = 36$.

Question 20

Prove that $6^n - 5n$ always leaves a remainder of 1 when divided by 25.

Accepted Answer

Correct Option: A. Solution: Using the binomial theorem, $6^n - 5n = 25k + 1$, thus leaving a remainder of 1.

Question 21

Evaluate the expression $(98)^5$ using the binomial theorem.

Accepted Answer

Correct Option: A. Solution: Express $98$ as $100 - 2$. Using the binomial theorem, $(98)^5 = (100 - 2)^5 = \sum_{r=0}^{5} \binom{5}{r} (100)^{5-r} (-2)^r$. Calculating each term and summing gives $9039207968$.

Question 22

Which of the following expressions correctly represents the expansion of $(101)^4$ using the binomial theorem?

Accepted Answer

Correct Option: A. Solution: The correct expansion of $(100 + 1)^4$ uses the coefficients from Pascal's triangle: $1, 4, 6, 4, 1$.

Question 23

In the binomial expansion of $(a + b)^n$, the sum of the coefficients is 1024. What is the value of $n$?

Accepted Answer

Correct Option: A. Solution: The sum of the coefficients in $(a + b)^n$ is $(1 + 1)^n = 2^n$. Given $2^n = 1024$, solving gives $n = 10$.

Question 24

Which of the following expressions represents the middle term in the expansion of $(a + b)^6$?

Accepted Answer

Correct Option: A. Solution: The middle term in the expansion of $(a + b)^6$ is the fourth term, which is $\binom{6}{3}a^3b^3$.

Question 25

Using the binomial theorem, evaluate $(102)^5$ by expressing it as $(100 + 2)^5$. What is the coefficient of the $2^3$ term in the expansion?

Accepted Answer

Correct Option: A. Solution: The coefficient of the $2^3$ term is given by $\binom{5}{3} 	imes 100^2 = 10 	imes 100^2$.

Question 26

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

Accepted Answer

Correct Option: 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 $\binom{6}{3} 	imes 2^3 = 20 	imes 8 = 160$.

Question 27

What is the base case used in the proof of the binomial theorem by mathematical induction?

Accepted Answer

Correct Option: B. Solution: The base case for the proof by mathematical induction for the binomial theorem is $n = 1$, where $(a + b)^1 = a + b$.

Question 28

Which statement is true about Pascal's Triangle?

Accepted Answer

Correct Option: A. Solution: Pascal's Triangle is constructed such that each number is the sum of the two numbers directly above it. This property helps generate the coefficients for binomial expansions.

Question 29

What is the value of the binomial coefficient $\binom{5}{3}$ in Pascal's Triangle?

Accepted Answer

Correct Option: B. Solution: The binomial coefficient $\binom{5}{3}$ corresponds to the third position in the row for index 5 in Pascal's Triangle, which is 10.

Question 30

Which of the following represents the middle term in the expansion of $(x + 1)^8$?

Accepted Answer

Correct Option: A. Solution: For $n = 8$, the middle term is the 5th term, which is $8C_4 \cdot x^4$.

Question 31

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

Accepted Answer

Correct Option: 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 $\binom{6}{2} x^4 2^2 = 60x^4$.

Question 32

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

Accepted Answer

Correct Option: B. Solution: The sum of the coefficients in the expansion of $(1 + x)^n$ is $(1 + 1)^n = 2^n$. Given $2^n = 2048$, solving gives $n = 11$.

Question 33

Consider the binomial expansion of $(a + b)^n$. If the coefficient of the middle term in the expansion is the greatest, what can be inferred about the value of $n$?

Accepted Answer

Correct Option: A. Solution: In the binomial expansion of $(a + b)^n$, the coefficient of the middle term is greatest when $n$ is even. This is because the middle term occurs at $\frac{n}{2}$, and the binomial coefficient $\binom{n}{\frac{n}{2}}$ is maximized when $n$ is even.

Question 34

If the middle term in the expansion of $(1 + x)^{10}$ is $\binom{10}{5} x^5$, what is its value?

Accepted Answer

Correct Option: A. Solution: The middle term in the expansion of $(1 + x)^{10}$ is $\binom{10}{5} x^5$. Calculating $\binom{10}{5}$ gives $252$. Therefore, the value is 252.

Question 35

Which of the following expressions represents the expansion of $(101)^4$ using the binomial theorem?

Accepted Answer

Correct Option: A. Solution: Express $101$ as $100 + 1$. Thus, $(101)^4 = (100 + 1)^4$. Using the binomial theorem, we expand it as: $100^4 + 4 	imes 100^3 	imes 1 + 6 	imes 100^2 	imes 1^2 + 4 	imes 100 	imes 1^3 + 1^4$.

Question 36

In the expansion of $(x + y)^5$, what is the coefficient of the middle term?

Accepted Answer

Correct Option: A. Solution: In the expansion of $(x + y)^5$, the middle term is the third term, which is $\binom{5}{2}x^3y^2$. The coefficient is $\binom{5}{2} = 10$.

Question 37

If the expansion of $(a + b)^n$ has 11 terms, what is the value of $n$?

Accepted Answer

Correct Option: B. Solution: The number of terms in the expansion of $(a + b)^n$ is $n + 1$. Therefore, if there are 11 terms, $n = 10$.

Question 38

In the expansion of $(a + b)^n$, if the coefficient of the middle term is 252 and $n = 10$, what is the value of $a$ and $b$ if $a = b$?

Accepted Answer

Correct Option: B. Solution: The middle term for $n = 10$ is the 6th term, which is $10C_5 \cdot a^5 \cdot b^5 = 252$. Given $a = b$, $10C_5 \cdot a^{10} = 252$. Solving gives $a = 3$.

Question 39

Which of the following expressions correctly represents the binomial expansion of $(x + 2)^4$ using Pascal's Triangle?

Accepted Answer

Correct Option: A. Solution: Using Pascal's Triangle, the coefficients for the expansion of $(x + 2)^4$ are 1, 4, 6, 4, 1. Therefore, the expansion is $x^4 + 4x^3 \cdot 2 + 6x^2 \cdot 4 + 4x \cdot 8 + 16$.

Question 40

Which of the following is a correct observation about the binomial expansion $(a + b)^n$?

Accepted Answer

Correct Option: A. Solution: In the binomial expansion $(a + b)^n$, the sum of the indices of $a$ and $b$ in each term is equal to the index $n$.

Question 41

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

Accepted Answer

Correct Option: 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.

Question 42

Which of the following expressions represents the general term in the expansion of $(a + b)^n$?

Accepted Answer

Correct Option: A. Solution: The general term in the expansion of $(a + b)^n$ is given by $\binom{n}{k} a^{n-k} b^k$, where $k$ is the term number starting from 0.

Question 43

Which of the following expressions correctly represents the binomial expansion of $(3x - 2)^4$?

Accepted Answer

Correct Option: B. Solution: Using the binomial theorem, $(3x - 2)^4 = \sum_{k=0}^{4} \binom{4}{k} (3x)^{4-k} (-2)^k$. Calculating each term gives: $81x^4 - 216x^3 + 216x^2 - 144x + 16$.

Question 44

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

Accepted Answer

Correct Option: C. Solution: The coefficient of the $x^4$ term in $(x + 2)^6$ is given by $\binom{6}{4}(2)^2 = 15 	imes 4 = 60$.

Question 45

Which of the following statements about the binomial expansion $(a + b)^n$ is true?

Accepted Answer

Correct Option: A. Solution: In the binomial expansion $(a + b)^n$, the sum of the exponents in each term is $n$.

Question 46

In the expansion of $(a + b)^4$, what is the coefficient of the term $a^2b^2$?

Accepted Answer

Correct Option: B. Solution: The term $a^2b^2$ in the expansion of $(a + b)^4$ corresponds to the binomial coefficient $\binom{4}{2} = 6$.

Question 47

Using the binomial theorem, evaluate $(98)^5$ by expressing 98 as a binomial.

Accepted Answer

Correct Option: A. Solution: Express $98$ as $100 - 2$. Using the binomial theorem: $$(98)^5 = (100 - 2)^5 = 10000000000 - 5 	imes 100000000 	imes 2 + 10 	imes 1000000 	imes 4 - 10 	imes 10000 	imes 8 + 5 	imes 100 	imes 16 - 32 = 9039207968.$$

Question 48

Using the binomial theorem, what is the first term in the expansion of $(98)^5$ when expressed as $(100 - 2)^5$?

Accepted Answer

Correct Option: A. Solution: The first term in the expansion of $(100 - 2)^5$ is $\binom{5}{0} 	imes 100^5 = 100^5$.

Question 49

Using the binomial theorem, what is the expansion of $(1 + 2x)^4$?

Accepted Answer

Correct Option: A. Solution: The binomial expansion of $(1 + 2x)^4$ is given by $\sum_{k=0}^{4} \binom{4}{k} (1)^{4-k} (2x)^k = 1 + 8x + 24x^2 + 32x^3 + 16x^4$.

Question 50

What is the general term $T_{r+1}$ in the binomial expansion of $(a + b)^n$?

Accepted Answer

Correct Option: A. Solution: The general term $T_{r+1}$ in the binomial expansion of $(a + b)^n$ is given by $nC_r a^{n-r} b^r$.

Question 51

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

Accepted Answer

Correct Option: B. Solution: The sum of the coefficients in the expansion of $(a + b)^n$ is $2^n$. For $(a + b)^7$, it is $2^7 = 128$.

Question 52

Using the binomial theorem, what is the coefficient of $x^3$ in the expansion of $(2x + 3)^5$?

Accepted Answer

Correct Option: B. Solution: The expansion of $(2x + 3)^5$ using the binomial theorem is given by: $\sum_{k=0}^{5} \binom{5}{k} (2x)^{5-k} (3)^k$. The coefficient of $x^3$ corresponds to the term where $k=2$, which is $\binom{5}{2} (2x)^3 (3)^2 = 720$.

Question 53

In the expansion of $(a + b)^n$, the general term $T_{r+1}$ is given by $nC_r a^{n-r}b^r$. If $a = 2$, $b = 3$, and $n = 5$, what is the coefficient of the term containing $a^3b^2$?

Accepted Answer

Correct Option: A. Solution: The term containing $a^3b^2$ corresponds to $T_4$ in the expansion, where $r = 2$. The coefficient is $5C_2 \cdot 2^{3} \cdot 3^{2} = 10 \cdot 8 \cdot 9 = 720$.

Question 54

Which of the following is a correct representation of the binomial coefficient $\binom{n}{k}$?

Accepted Answer

Correct Option: A. Solution: The binomial coefficient $\binom{n}{k}$ is defined as $\frac{n!}{k!(n-k)!}$, which represents the number of ways to choose $k$ elements from a set of $n$ elements.

Question 55

What is the expansion of $(a + b)^3$ using the binomial theorem?

Accepted Answer

Correct Option: A. Solution: The binomial expansion of $(a + b)^3$ is $a^3 + 3a^2b + 3ab^2 + b^3$.

Question 56

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

Accepted Answer

Correct Option: A. Solution: The binomial expansion of $(a + b)^3$ is $a^3 + 3a^2b + 3ab^2 + b^3$.

Question 57

Using the binomial theorem, expand the expression $(x + 2)^5$ and find the coefficient of the $x^3$ term.

Accepted Answer

Correct Option: C. Solution: Using the binomial theorem, $(x + 2)^5 = \sum_{k=0}^{5} \binom{5}{k} x^{5-k} (2)^k$. For the $x^3$ term, $k = 2$, so the coefficient is $\binom{5}{2} \cdot 2^2 = 10 \cdot 4 = 40$. Thus, the correct coefficient is 160.

Question 58

Using the binomial theorem, determine the constant term in the expansion of $\left(\frac{2}{x} + x^2\right)^6$.

Accepted Answer

Correct Option: B. Solution: The constant term occurs when the exponents of $x$ in the terms cancel out. For $\left(\frac{2}{x} + x^2\right)^6$, the term is $\binom{6}{k} \left(\frac{2}{x}\right)^{6-k} (x^2)^k$ where $6 - k = 2k$. Solving gives $k = 2$. The constant term is $\binom{6}{2} \cdot 2^4 = 64$.

Question 59

If $(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k$, what is the coefficient of $x^{n-2}y^2$?

Accepted Answer

Correct Option: A. Solution: The coefficient of $x^{n-2}y^2$ in the expansion is given by $\binom{n}{2}$, as it corresponds to the term where $k=2$.

Question 60

If $(2x + 3y)^5$ is expanded using the binomial theorem, what is the term containing $x^3y^2$?

Accepted Answer

Correct Option: A. Solution: The term containing $x^3y^2$ is $10 \cdot (2x)^3 \cdot (3y)^2$ which simplifies to $10 \cdot 8x^3 \cdot 9y^2 = 720x^3y^2$.

Question 61

Which of the following expressions is equal to $\sum_{r=0}^{n} \binom{n}{r} 3^r$?

Accepted Answer

Correct Option: A. Solution: Using the binomial theorem, $\sum_{r=0}^{n} \binom{n}{r} 3^r = (1 + 3)^n = 4^n$.

Question 62

Using Pascal's Triangle, determine the coefficient of $x^3y^2$ in the expansion of $(x + y)^5$.

Accepted Answer

Correct Option: B. Solution: The coefficient of $x^3y^2$ in $(x + y)^5$ is given by the binomial coefficient $\binom{5}{3} = 10$.

Question 63

In the expansion of $(x - 1)^6$, what is the sum of the coefficients?

Accepted Answer

Correct Option: A. Solution: The sum of the coefficients in the expansion of $(x - 1)^6$ is obtained by substituting $x = 1$. Thus, $(1 - 1)^6 = 0^6 = 0$. Therefore, the sum of the coefficients is 0.

Question 64

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

Accepted Answer

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.

Binomial Theorem

Learning Objectives

Learning Objectives

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

Exam Tips & Common Mistakes

Common Mistakes & Exam Tips

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

General Tips

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Multiple Choice Questions

Q1.Consider the expression (1+x)n(1 + x)^n(1+x)n. If the coefficient of x3x^3x3 is 56, what is the value of nnn?

Correct Answer: A

Q2.Which of the following statements is true regarding the binomial expansion of (x+y)n(x + y)^n(x+y)n?

Correct Answer: A

Q3.In the binomial expansion of (a+b)n(a + b)^n(a+b)n, what is the sum of the exponents of aaa and bbb in any term?

Correct Answer: A

Q4.Which of the following expressions represents the expansion of (2x+3y)3(2x + 3y)^3(2x+3y)3 using the binomial theorem?

Correct Answer: C

Q5.In the expansion of (x+y)7(x + y)^7(x+y)7, what is the sum of the coefficients?

Correct Answer: B

Q6.In Pascal's Triangle, what is the coefficient of the middle term in the expansion of (a+b)6(a + b)^6(a+b)6?

Correct Answer: C

Q7.Using the binomial theorem, which expression represents the expansion of (99)5(99)^5(99)5?

Correct Answer: A

Q8.How does the binomial theorem simplify the calculation of (98)5(98)^5(98)5?

Correct Answer: B

Q9.If the binomial expansion of (x+y)n(x + y)^n(x+y)n has 11 terms, what is the value of nnn?

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

Q11.Consider the binomial expansion of (x−y)5(x - y)^5(x−y)5. What is the coefficient of the x3y2x^3y^2x3y2 term?

Correct Answer: A

Q12.Which of the following is a correct application of the binomial theorem for (3x−2)3(3x - 2)^3(3x−2)3?

Correct Answer: A

Q13.Which expansion correctly uses the binomial theorem for (102)5(102)^5(102)5?

Correct Answer: A

Q14.Using Pascal's Triangle, what is the coefficient of x3y2x^3y^2x3y2 in the expansion of (x+y)5(x + y)^5(x+y)5?

Correct Answer: B

Q15.What is the expansion of (x−y)3(x - y)^3(x−y)3 using the binomial theorem?

Correct Answer: A

Q16.Which of the following statements about Pascal's Triangle is incorrect?

Correct Answer: D

Q17.What is the sum of the coefficients in the expansion of (x+1)n(x + 1)^n(x+1)n?

Correct Answer: A

Q18.Which of the following statements correctly describes the pattern observed in Pascal's Triangle?

Correct Answer: A

Q19.If Pascal's Triangle is used to expand (2x+3y)4(2x + 3y)^4(2x+3y)4, what is the coefficient of the term x2y2x^2y^2x2y2?

Correct Answer: B

Q20.Prove that 6n−5n6^n - 5n6n−5n always leaves a remainder of 1 when divided by 25.

Correct Answer: A

Q21.Evaluate the expression (98)5(98)^5(98)5 using the binomial theorem.

Correct Answer: A

Q22.Which of the following expressions correctly represents the expansion of (101)4(101)^4(101)4 using the binomial theorem?

Correct Answer: A

Q23.In the binomial expansion of (a+b)n(a + b)^n(a+b)n, the sum of the coefficients is 1024. What is the value of nnn?

Correct Answer: A

Q24.Which of the following expressions represents the middle term in the expansion of (a+b)6(a + b)^6(a+b)6?