Class 11 Maths Chapter 7 Binomial Theorem – Exercise 7.1 NCERT Solutions

Introduction

Exercise 7.1 introduces the Binomial Theorem, which provides a formula for expanding expressions of the form $(x+y{)}^n$. You will learn how to apply the theorem, calculate coefficients, and solve problems involving binomial expansions. This exercise is fundamental for algebra and probability.

Key Concepts

1. Binomial Theorem: For any positive integer $n$:

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

2. Binomial Coefficient:

$$\left(\genfrac{}{}{0px}{}{n}{r}\right)=\frac{n!}{r!(n-r)!}$$

3. Properties:

   • Total number of terms = $n+1$.

   • First term = $x^n$.

   • Last term = $y^n$.

   • Middle term(s) depend on whether $n$ is even or odd.

NCERT Questions with Solutions (10)

Q1. Expand $(x+y{)}^5$.

$$(x+y{)}^5=x^5+5x^{4}y+10x^3{y}^2+10x^2{y}^3+5x{y}^4+y^5$$

Q2. Expand $(a+b{)}^4$.

$$(a+b{)}^4=a^4+4a^{3}b+6a^2{b}^2+4a{b}^3+b^4$$

Q3. Expand $(p+q{)}^3$.

$$(p+q{)}^3=p^3+3p^{2}q+3p{q}^2+q^3$$

Q4. Expand $(x-1{)}^4$.

$$(x-1{)}^4=x^4-4x^3+6x^2-4x+1$$

Q5. Expand $(2x+y{)}^3$.

$$(2x+y{)}^3=8x^3+12x^{2}y+6x{y}^2+y^3$$

Q6. Expand $(x+2{)}^5$.

$$(x+2{)}^5=x^5+10x^4+40x^3+80x^2+80x+32$$

Q7. Expand $(a-2b{)}^4$.

$$(a-2b{)}^4=a^4-8a^{3}b+24a^2{b}^2-32a{b}^3+16b^4$$

Q8. Expand $(3x-y{)}^3$.

$$(3x-y{)}^3=27x^3-27x^{2}y+9x{y}^2-y^3$$

Q9. Expand $(x+y{)}^6$.

$$(x+y{)}^6=x^6+6x^{5}y+15x^4{y}^2+20x^3{y}^3+15x^2{y}^4+6x{y}^5+y^6$$

Q10. Expand $(a+b{)}^2$.

$$(a+b{)}^2=a^2+2ab+b^2$$

FAQs (10)

1. What is Binomial Theorem? Formula for expansion of $(x+y{)}^n$.

2. What is binomial coefficient? $\left(\genfrac{}{}{0px}{}{n}{r}\right)=\frac{n!}{r!(n-r)!}$.

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

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

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

6. What is middle term if $n$ is even? Single middle term at $r=\frac{n}{2}$.

7. What is middle term if $n$ is odd? Two middle terms at $r=\frac{n-1}{2},\frac{n+1}{2}$.

8. What is sum of coefficients in expansion of $(x+y{)}^n$? Equal to $2^n$.

9. What is coefficient of $x^{n-r}{y}^r$? $\left(\genfrac{}{}{0px}{}{n}{r}\right)$.

10. Why is this exercise important? It builds foundation for algebra, probability, and binomial distribution.

Conclusion

Exercise 7.1 has 10 solved questions and 10 FAQs that strengthen your understanding of binomial expansions. This builds the foundation for advanced algebra and probability in Class 11 Maths.

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New Syllabus-Class 11 Maths Chapter 7 Binomial Theorem – Exercise 7.1 NCERT Solutions