Class 11 Maths Chapter 7 Binomial Theorem – Exercise 7.2 NCERT Solutions

Introduction

Exercise 7.2 focuses on identifying specific terms in binomial expansions. You will learn how to find the general term, middle term(s), and coefficients of required terms in expansions using the Binomial Theorem. This exercise strengthens your ability to apply the theorem in targeted problems.

Key Concepts

1. General Term in Expansion:

$$T_{r+1}=\left(\genfrac{}{}{0px}{}{n}{r}\right)x^{n-r}{y}^r$$

2. Middle Term(s):

   • If $n$ is even: single middle term at $r=\frac{n}{2}$.

   • If $n$ is odd: two middle terms at $r=\frac{n-1}{2},\frac{n+1}{2}$.

3. Coefficient of Term: Coefficient of $x^{n-r}{y}^r$ is $\left(\genfrac{}{}{0px}{}{n}{r}\right)$.

NCERT Questions with Solutions (10)

Q1. Find general term in expansion of $(x+y{)}^n$.

$$T_{r+1}=\left(\genfrac{}{}{0px}{}{n}{r}\right)x^{n-r}{y}^r$$

Q2. Find 5th term in expansion of $(x+y{)}^7$.

$$T_5=\left(\genfrac{}{}{0px}{}{7}{4}\right)x^3{y}^4=35x^3{y}^4$$

Q3. Find 4th term in expansion of $(a+b{)}^6$.

$$T_4=\left(\genfrac{}{}{0px}{}{6}{3}\right)a^3{b}^3=20a^3{b}^3$$

Q4. Find middle term in expansion of $(x+y{)}^8$. Since $n=8$ (even), middle term = $T_5$.

$$T_5=\left(\genfrac{}{}{0px}{}{8}{4}\right)x^4{y}^4=70x^4{y}^4$$

Q5. Find middle terms in expansion of $(x+y{)}^9$. Since $n=9$ (odd), middle terms = $T_5,T_6$.

$$T_5=\left(\genfrac{}{}{0px}{}{9}{4}\right)x^5{y}^4=126x^5{y}^4$$

$$T_6=\left(\genfrac{}{}{0px}{}{9}{5}\right)x^4{y}^5=126x^4{y}^5$$

Q6. Find coefficient of $x^7{y}^3$ in expansion of $(x+y{)}^{10}$. Here $r=3$.

$$\left(\genfrac{}{}{0px}{}{10}{3}\right)=120$$

Coefficient = 120.

Q7. Find coefficient of $a^5{b}^2$ in expansion of $(a+b{)}^7$. Here $r=2$.

$$\left(\genfrac{}{}{0px}{}{7}{2}\right)=21$$

Coefficient = 21.

Q8. Find coefficient of $x^2{y}^6$ in expansion of $(x+y{)}^8$. Here $r=6$.

$$\left(\genfrac{}{}{0px}{}{8}{6}\right)=28$$

Coefficient = 28.

Q9. Find coefficient of $x^3{y}^7$ in expansion of $(x+y{)}^{10}$. Here $r=7$.

$$\left(\genfrac{}{}{0px}{}{10}{7}\right)=120$$

Coefficient = 120.

Q10. Find coefficient of $a^4{b}^4$ in expansion of $(a+b{)}^8$. Here $r=4$.

$$\left(\genfrac{}{}{0px}{}{8}{4}\right)=70$$

Coefficient = 70.

FAQs (10)

1. What is general term in binomial expansion? $T_{r+1}=\left(\genfrac{}{}{0px}{}{n}{r}\right)x^{n-r}{y}^r$.

2. How to find middle term? Depends on whether $n$ is even or odd.

3. What is coefficient of term? Binomial coefficient $\left(\genfrac{}{}{0px}{}{n}{r}\right)$.

4. How many terms in expansion of $(x+y{)}^n$? $n+1$.

5. What is first term? $x^n$.

6. What is last term? $y^n$.

7. What is relation between coefficients? Symmetry: $\left(\genfrac{}{}{0px}{}{n}{r}\right)=\left(\genfrac{}{}{0px}{}{n}{n-r}\right)$.

8. What is sum of coefficients? Equal to $2^n$.

9. What is middle term if $n=8$? Single term at $T_5$.

10. Why is this exercise important? It builds skills in identifying specific terms and coefficients in expansions.

Conclusion

Exercise 7.2 has 10 solved questions and 10 FAQs that strengthen your understanding of general terms, middle terms, and coefficients in binomial expansions. This completes the Binomial Theorem chapter in Class 11 Maths.

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New Syllabus-Class 11 Maths Chapter 7 Binomial Theorem – Exercise 7.2 NCERT Solutions