Skip to content
← Mathematics

Mathematics · Class 11 Science

Ch 7Binomial Theorem — Class 11 Mathematics, concept-first.

In earlier classes you learned to expand squares and cubes of binomials such as , , , and . These ready-made expansions double as a numerical shortcut: if you can write a number as a sum or difference of two convenient numbers, you can find its square or cube without a long multiplication.

64

Q&A

5

Concepts

~5m

Unit weightage

Start learning — read this chapter →

Key concepts

Hover a concept to preview it and jump to its most relevant Q&A.

Binomial Theorem Expansion

Imagine you have to expand . That's easy: . Now try : . Still manageable.

Most relevant Q&A

Start with this concept →

Chapter contents

The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.

7.1

Introduction

In earlier classes you learned to expand squares and cubes of binomials such as , , , and . These ready-made expansions double as a numerical shortcut: if you can write a number as a sum or difference…

7.2

Binomial Theorem for Positive Integral Indices

We begin by recalling expansions we already know for small powers of :

7.2.1

Binomial Theorem for Any Positive Integer n

The binomial theorem gives us a formula for expanding when is a positive integer — without having to multiply the bracket by itself times.

7.2.2

Some Special Cases

18 Q

The binomial theorem gives us a general expansion for , but in practice we rarely work with generic and .

+Worked Examplesi4 questions
  1. Example 1Expand $\left(x^2 + \dfrac{3}{x}\right)^4$, $x \neq 0$.Free
  2. Example 2Compute $(98)^5$.Free
  3. Example 3Which is larger $(1.01)^{1000000}$ or $10,000$?Preview
  4. Example 4Using binomial theorem, prove that $6^n - 5n$ always leaves remainder 1 when divided by 25.Preview
+Exercise 7.1i14 questions
  1. Q1Expand the expression $(1 - 2x)^5$.Free
  2. Q2Expand the expression $\left(\dfrac{2}{x} - \dfrac{x}{2}\right)^5$.Free
  3. Q3Expand the expression $(2x - 3)^6$.Free
  4. Q4Expand the expression $\left(\dfrac{x}{3} + \dfrac{1}{x}\right)^5$.Preview
  5. Q5Expand the expression $\left(x + \dfrac{1}{x}\right)^6$.Preview
  6. Q6Using binomial theorem, evaluate $(96)^3$.Preview
  7. Q7Using binomial theorem, evaluate $(102)^5$.Preview
  8. Q8Using binomial theorem, evaluate $(101)^4$.Preview
  9. Q9Using binomial theorem, evaluate $(99)^5$.Preview
  10. Q10Using Binomial Theorem, indicate which number is larger $(1.1)^{10000}$ or $1000$.Preview
  11. Q11Find $(a + b)^4 - (a - b)^4$. Hence, evaluate $\left(\sqrt{3} + \sqrt{2}\right)^4 - \left(\sqrt{3} - \sqrt{2}\right)^4$.Preview
  12. Q12Find $(x + 1)^6 + (x - 1)^6$. Hence or otherwise evaluate $\left(\sqrt{2} + 1\right)^6 + \left(\sqrt{2} - 1\right)^6$.Preview
  13. Q13Show that $9^{n+1} - 8n - 9$ is divisible by 64, whenever $n$ is a positive integer.Preview
  14. Q14Prove that $\displaystyle\sum_{r=0}^{n} 3^r \, {}^nC_r = 4^n$.Preview

Miscellaneous

The exercises above practised the binomial theorem, Pascal's triangle, and the special-case expansions one idea at a time.

+Miscellaneous Exercisei6 questions
  1. Q1If $a$ and $b$ are distinct integers, prove that $a - b$ is a factor of $a^n - b^n$, whenever $n$ is a positive integer. [Hint: write $a^n =…Free
  2. Q2Evaluate $\left(\sqrt{3} + \sqrt{2}\right)^6 - \left(\sqrt{3} - \sqrt{2}\right)^6$.Free
  3. Q3Find the value of $\left(a^2 + \sqrt{a^2 - 1}\right)^4 + \left(a^2 - \sqrt{a^2 - 1}\right)^4$.Preview
  4. Q4Find an approximation of $(0.99)^5$ using the first three terms of its expansion.Preview
  5. Q5Expand using Binomial Theorem $\left(1 + \dfrac{x}{2} - \dfrac{2}{x}\right)^4$, $x \neq 0$.Preview
  6. Q6Find the expansion of $(3x^2 - 2ax + 3a^2)^3$ using binomial theorem.Preview

Summary

- The Binomial Theorem gives the expansion of for any positive integer :

Exemplar Problems

Higher-order thinking / exemplar-style practice problems.

+Show 40 questions40 questions
  1. Q1Find the term independent of $x$, $x \neq 0$, in the expansion of $\left(\dfrac{3x^2}{2} - \dfrac{1}{3x}\right)^{15}$.Free
  2. Q2If the term free from $x$ in the expansion of $\left(\sqrt{x} - \dfrac{k}{x^2}\right)^{10}$ is $405$, find the value of $k$.Free
  3. Q3Find the coefficient of $x$ in the expansion of $(1 - 3x + 7x^2)(1 - x)^{16}$.Free
  4. Q4Find the term independent of $x$ in the expansion of $\left(3x - \dfrac{2}{x^2}\right)^{15}$.Preview
  5. Q5Find the middle term (terms) in the expansion of (i) $\left(\dfrac{x}{a} - \dfrac{a}{x}\right)^{10}$ (ii) $\left(3x - \dfrac{x^3}{6}\right)^…Preview
  6. Q6Find the coefficient of $x^{15}$ in the expansion of $(x - x^2)^{10}$.Preview
  7. Q7Find the coefficient of $\dfrac{1}{x^{17}}$ in the expansion of $\left(x^4 - \dfrac{1}{x^3}\right)^{15}$.Preview
  8. Q8Find the sixth term of the expansion $\left(y^{1/2} + x^{1/3}\right)^{n}$, if the binomial coefficient of the third term from the end is $45…Preview
  9. Q9Find the value of $r$, if the coefficients of $(2r + 4)^{\text{th}}$ and $(r - 2)^{\text{th}}$ terms in the expansion of $(1 + x)^{18}$ are…Preview
  10. Q10If the coefficient of second, third and fourth terms in the expansion of $(1 + x)^{2n}$ are in A.P., show that $2n^2 - 9n + 7 = 0$.Preview
  11. Q11Find the coefficient of $x^4$ in the expansion of $(1 + x + x^2 + x^3)^{11}$.Preview
  12. Q12If $p$ is a real number and if the middle term in the expansion of $\left(\dfrac{p}{2} + 2\right)^{8}$ is $1120$, find $p$.Preview
  13. Q13Show that the middle term in the expansion of $\left(x - \dfrac{1}{x}\right)^{2n}$ is $\dfrac{1 \times 3 \times 5 \times \dots (2n - 1)}{n!}…Preview
  14. Q14Find $n$ in the binomial $\left(\sqrt[3]{2} + \dfrac{1}{\sqrt[3]{3}}\right)^{n}$ if the ratio of $7^{\text{th}}$ term from the beginning to…Preview
  15. Q15In the expansion of $(x + a)^n$ if the sum of odd terms is denoted by $O$ and the sum of even terms by $E$, then prove that (i) $O^2 - E^2 =…Preview
  16. Q16If $x^p$ occurs in the expansion of $\left(x^2 + \dfrac{1}{x}\right)^{2n}$, prove that its coefficient is $\dfrac{(2n)!}{\left(\dfrac{4n - p…Preview
  17. Q17Find the term independent of $x$ in the expansion of $(1 + x + 2x^3)\left(\dfrac{3}{2}x^2 - \dfrac{1}{3x}\right)^{9}$.Preview
  18. Q18The total number of terms in the expansion of $(x + a)^{100} + (x - a)^{100}$ after simplification is (A) $50$ (B) $202$ (C) $51$ (D) none o…Preview
  19. Q19Given the integers $r > 1$, $n > 2$, and coefficients of $(3r)^{\text{th}}$ and $(r + 2)^{\text{nd}}$ terms in the binomial expansion of $(1…Preview
  20. Q20The two successive terms in the expansion of $(1 + x)^{24}$ whose coefficients are in the ratio $1 : 4$ are (A) $3^{\text{rd}}$ and $4^{\tex…Preview
  21. Q21The coefficient of $x^n$ in the expansion of $(1 + x)^{2n}$ and $(1 + x)^{2n - 1}$ are in the ratio (A) $1 : 2$ (B) $1 : 3$ (C) $3 : 1$ (D)…Preview
  22. Q22If the coefficients of $2^{\text{nd}}$, $3^{\text{rd}}$ and the $4^{\text{th}}$ terms in the expansion of $(1 + x)^n$ are in A.P., then valu…Preview
  23. Q23If $A$ and $B$ are coefficient of $x^n$ in the expansions of $(1 + x)^{2n}$ and $(1 + x)^{2n - 1}$ respectively, then $\dfrac{A}{B}$ equals…Preview
  24. Q24If the middle term of $\left(\dfrac{1}{x} + x \sin x\right)^{10}$ is equal to $7\dfrac{7}{8}$, then value of $x$ is (A) $2n\pi + \dfrac{\pi}…Preview
  25. Q25The largest coefficient in the expansion of $(1 + x)^{30}$ is ______ .Preview
  26. Q26The number of terms in the expansion of $(x + y + z)^n$ ______ .Preview
  27. Q27In the expansion of $\left(x^2 - \dfrac{1}{x^2}\right)^{16}$, the value of constant term is ______ .Preview
  28. Q28If the seventh terms from the beginning and the end in the expansion of $\left(\sqrt[3]{2} + \dfrac{1}{\sqrt[3]{3}}\right)^{n}$ are equal, t…Preview
  29. Q29The coefficient of $a^{-6} b^4$ in the expansion of $\left(\dfrac{1}{a} - \dfrac{2b}{3}\right)^{10}$ is ______ .Preview
  30. Q30Middle term in the expansion of $(a^3 + ba)^{28}$ is ______ .Preview
  31. Q31The ratio of the coefficients of $x^p$ and $x^q$ in the expansion of $(1 + x)^{p + q}$ is ______ .Preview
  32. Q32The position of the term independent of $x$ in the expansion of $\left(\sqrt{\dfrac{x}{3}} + \dfrac{3}{2x^2}\right)^{10}$ is ______ .Preview
  33. Q33If $25^{15}$ is divided by $13$, the remainder is ______ .Preview
  34. Q34The sum of the series $\displaystyle\sum_{r=0}^{10} {}^{20}C_r$ is $2^{19} + \dfrac{{}^{20}C_{10}}{2}$.Preview
  35. Q35The expression $7^9 + 9^7$ is divisible by $64$.Preview
  36. Q36The number of terms in the expansion of $[(2x + y^3)^4]^7$ is $8$.Preview
  37. Q37The sum of coefficients of the two middle terms in the expansion of $(1 + x)^{2n - 1}$ is equal to ${}^{2n - 1}C_n$.Preview
  38. Q38The last two digits of the numbers $3^{400}$ are $01$.Preview
  39. Q39If the expansion of $\left(x - \dfrac{1}{x^2}\right)^{2n}$ contains a term independent of $x$, then $n$ is a multiple of $2$.Preview
  40. Q40Number of terms in the expansion of $(a + b)^n$ where $n \in \mathbb{N}$ is one less than the power $n$.Preview