NCERT Solutions for Exercise 8.2 Class 11 Maths Chapter 8

NCERT Solutions for Exercise 8.2 Class 11 Maths Chapter 8 - Binomial Theorem

Edited By Ravindra Pindel | Updated on Jul 13, 2022 03:42 PM IST

In the previous exercise, you have already learned about the binomial expansion for positive integer and pascal triangle. In the NCERT solutions for Class 11 Maths chapter 8 exercise 8.2, you will learn about finding the general and middle terms of the binomial expansion. If you have understood the previous exercise of this chapter, you won't find any difficulty the get Class 11 Maths chapter 8 exercise 8.2. This exercise is very important for probability and statistics.

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Binomial Theorem Class 11 Chapter 8 Exercise: 8.1
Question:1 Find the coefficient of
More About NCERT Solutions for Class 11 Maths Chapter 8 Exercise 8.2:-
NCERT Solutions of Class 11 Subject Wise
Subject Wise NCERT Exampler Solutions

You must try to solve all the NCERT book Class 11 Maths chapter 8 exercise 8.2 problems by yourself. If you have any doubt about this exercise, Class 11 Maths chapter 8 exercise 8.2 solutions are here to help you. These solutions are designed in a step-by-step manner so you can understand them very easily. Here you will get NCERT Solutions from Classes 6 to 12 for Science and Maths.

Also, see

Binomial Theorem Class 11 Chapter 8 Exercise: 8.1

Question:1 Find the coefficient of

$x^5$ in $(x + 3)^8$

Answer:

As we know that the $(r+1)^{th}$ term $T_{r+1}$ in the binomial expansion of $(a+b)^n$ is given by

$T_{r+1}=^nC_ra^{n-r}b^r$

Now let's assume $x^5$ happens in the $(r+1)^{th}$ term of the binomial expansion of $(x + 3)^8$

So,

$T_{r+1}=^8C_rx^{8-r}3^r$

On comparing the indices of x we get,

$r=3$

Hence the coefficient of the $x^5$ in $(x + 3)^8$ is

$^8C_3\times3^3=\frac{8!}{5!3!}\times 9=\frac{8\times7\times6}{3\times2}\times9=1512$

Question:2 Find the coefficient of $a^5b^7$ in $(a- 2b)^{12}$

Answer:

As we know that the $(r+1)^{th}$ term $T_{r+1}$ in the binomial expansion of $(a+b)^n$ is given by

$T_{r+1}=^nC_ra^{n-r}b^r$

Now let's assume $a^5b^7$ happens in the $(r+1)^{th}$ term of the binomial expansion of $(a- 2b)^{12}$

So,

$T_{r+1}=^{12}C_rx^{12-r}(-2b)^r$

On comparing the indices of x we get,

$r=7$

Hence the coefficient of the $a^5b^7$ in $(a- 2b)^{12}$ is

$\\ \Rightarrow ^{12}C_7\times(-2)^7=\frac{12!}{5!7!}\times (-128)\\=\frac{12\times11\times10\times 9\times8}{5\times4\times3\times2}\times(-128) \\=-(729)(128) \\=-101376$

Question:3 Write the general term in the expansion of

$(x^2 - y)^6$

Answer:

As we know that the general $(r+1)^{th}$ term $T_{r+1}$ in the binomial expansion of $(a+b)^n$ is given by

$T_{r+1}=^nC_ra^{n-r}b^r$

So the general term of the expansion of $(x^2 - y)^6$ :

$T_{r+1}=^6C_r(x^2)^{6-r}(-y)^r=(-1)^r\times^6C_rx^{12-2r}y^r$ .

Question:4 Write the general term in the expansion of

$(x^2 - xy)^{12}, \ x\neq 0$

Answer:

As we know that the general $(r+1)^{th}$ term $T_{r+1}$ in the binomial expansion of $(a+b)^n$ is given by

$T_{r+1}=^nC_ra^{n-r}b^r$

So the general term of the expansion of $(x^2 - xy)^{12},$ is

$\\\Rightarrow T_{r+1}\\=^{12}C_r(x^2)^{12-r}(-xy)^r\\=(-1)^r\times^{12}C_rx^{24-2r+r}y^r\\=(-1)^r\times^{12}C_rx^{24-r}y^r$ .

Question:5 Find the 4^th term in the expansion of $(x-2y)^{12}$ .

Answer:

As we know that the general $(r+1)^{th}$ term $T_{r+1}$ in the binomial expansion of $(a+b)^n$ is given by

$T_{r+1}=^nC_ra^{n-r}b^r$

So the $4^{th}$ term of the expansion of $(x-2y)^{12}$ is

$\\\Rightarrow T_4= T_{3+1}\\=^{12}C_3(x)^{12-3}(-2y)^3\\=(-2)^3\times^{12}C_3x^{9}y^3\\=-8\times\frac{12!}{3!9!}x^{9}y^3$

$\\=-8\times \frac{12\times11\times10}{3\times2}\times x^9y^3\\=-8\times220\times x^9y^3 \\=-1760x^9y^3$ .

Question:6 Find the 13^th term in the expansion of $\left(9x - \frac{1}{3\sqrt x} \right )^{18},\ x\neq 0$

Answer:

As we know that the general $(r+1)^{th}$ term $T_{r+1}$ in the binomial expansion of $(a+b)^n$ is given by

$T_{r+1}=^nC_ra^{n-r}b^r$

So the $13^{th}$ term of the expansion of $\left(9x - \frac{1}{3\sqrt x} \right )^{18}$ is

$\\\Rightarrow T_{13}= T_{12+1}\\=^{18}C_{12}(9x)^{18-12}\left(\frac{1}{3\sqrt{x}}\right)^{12}\\ \\ \\=\frac{18!}{12!6!}\times9^{6}\left ( \frac{1}{3} \right )^{12}\times x^{6-6}$

$\\=\frac{18\times17\times16\times15\times14\times13}{6\times5\times4\times3\times2}\times9^{6}\left ( \frac{1}{3^{12}} \right )$

$=18564$

Question:7 Find the middle terms in the expansion of $\left(3 - \frac{x^3}{6} \right )^7$

Answer:

As we know that the middle terms in the expansion of $(a+b)^n$ when n is odd are,

$\left ( \frac{n+1}{2} \right )^{th}\:term\: \:and\:\:\left ( \frac{n+1}{2}+1 \right )^{th}\:term$

Hence the middle term of the expansion $\left(3 - \frac{x^3}{6} \right )^7$ are

$\left ( \frac{7+1}{2} \right )^{th}\:term\: \:and\:\:\left ( \frac{7+1}{2}+1 \right )^{th}\:term$

Which are $4^{th}\:term\:and\:\:5^{th} \:term$

Now,

As we know that the general $(r+1)^{th}$ term $T_{r+1}$ in the binomial expansion of $(a+b)^n$ is given by

$T_{r+1}=^nC_ra^{n-r}b^r$

So the $4^{th}$ term of the expansion of $\left(3 - \frac{x^3}{6} \right )^7$ is

$\\\Rightarrow T_4= T_{3+1}\\=^{7}C_3(3)^{7-3}\left (- \frac{x^3}{6} \right )^3\\=\left ( -\frac{1}{6} \right )^3\times 3^4\times^{7}C_3\times x^{9}\\=\left ( -\frac{1}{6} \right )^3\times 3^4\times\frac{7!}{3!4!}\times x^{9} \\=-\frac{3\times3\times3\times3}{6\times6\times6}\times\frac{7\times 6\times5}{3\times2}\times x^9$

$=-\frac{105}{8}x^9$

And the $5^{th}$ Term of the expansion of $\left(3 - \frac{x^3}{6} \right )^7$ is,

$\\\Rightarrow T_5= T_{4+1}\\=^{7}C_4(3)^{7-4}\left (- \frac{x^3}{6} \right )^4\\=\left ( -\frac{1}{6} \right )^4\times 3^3\times^{7}C_4\times x^{12}\\=\left ( -\frac{1}{6} \right )^4\times 3^3\times\frac{7!}{3!4!}\times x^{12} \\=\frac{3\times3\times3}{6\times6\times6\times6}\times\frac{7\times 6\times5}{3\times2}\times x^{12}$

$=\frac{35}{48}x^{12}$

Hence the middle terms of the expansion of given expression are

$-\frac{105}{8}x^9\:and\:\frac{35}{48}x^{12}.$

Question:8 Find the middle terms in the expansion of $\left(\frac{x}{3} + 9y \right )^{10}$

Answer:

As we know that the middle term in the expansion of $(a+b)^n$ when n is even is,

$\left ( \frac{n}{2}+1 \right )^{th}\:term\:$ ,

Hence the middle term of the expansion $\left(\frac{x}{3} + 9y \right )^{10}$ is,

$\left ( \frac{10}{2}+1 \right )^{th}\:term\:$

Which is $6^{th}\:term$

Now,

As we know that the general $(r+1)^{th}$ term $T_{r+1}$ in the binomial expansion of $(a+b)^n$ is given by

$T_{r+1}=^nC_ra^{n-r}b^r$

So the $6^{th}$ term of the expansion of $\left(\frac{x}{3} + 9y \right )^{10}$ is

$\\\Rightarrow T_6= T_{5+1}\\=^{10}C_5\left ( \frac{x}{3} \right )^{10-5}\left ( 9y \right )^5\\$

$=\left ( \frac{1}{3} \right )^5\times9^5\times^{10}C_5\times x^5y^5$

$=\left ( \frac{1}{3} \right )^5\times9^5\times\left ( \frac{10!}{5!5!} \right )\times x^5y^5$

$=\left ( \frac{1}{3^5} \right )\times9^5\times\left ( \frac{10\times9\times8\times7\times6}{5\times4\times3\times2} \right )\times x^5y^5$

$=61236x^5y^5$

Hence the middle term of the expansion of $\left(\frac{x}{3} + 9y \right )^{10}$ is nbsp; $61236x^5y^5$ .

Question:9 In the expansion of $(1 + a)^{m+n}$ , prove that coefficients of $a^m$ and $a^n$ are equal

Answer:

As we know that the general $(r+1)^{th}$ term $T_{r+1}$ in the binomial expansion of $(a+b)^n$ is given by

$T_{r+1}=^nC_ra^{n-r}b^r$

So, the general $(r+1)^{th}$ term $T_{r+1}$ in the binomial expansion of $(1 + a)^{m+n}$ is given by

$T_{r+1}=^{m+n}C_r1^{m+n-r}a^r=^{m+n}C_ra^r$

Now, as we can see $a^m$ will come when $r=m$ and $a^n$ will come when $r=n$

So,

Coefficient of $a^m$ :

$K_{a^m}=^{m+n}C_m=\frac{(m+n)!}{m!n!}$

CoeficientCoefficient of $a^n$ :

$K_{a^n}=^{m+n}C_n=\frac{(m+n)!}{m!n!}$

As we can see $K_{a^m}=K_{a^n}$ .

Hence it is proved that the coefficients of $a^m$ and $a^n$ are equal.

Question:10 The coefficients of the $(r-1)$ ^th , r^th and $(r + 1)$ ^th terms in the expansion of $(x+1)^{n}$ are in the ratio 1 : 3 : 5. Find n and r.

Answer:

As we know that the general $(r+1)^{th}$ term $T_{r+1}$ in the binomial expansion of $(a+b)^n$ is given by

$T_{r+1}=^nC_ra^{n-r}b^r$

So,

$(r+1)^{th}$ Term in the expansion of $(x+1)^{n}$ :

$T_{r+1}=^nC_rx^{n-r}1^r=^nC_rx^{n-r}$

$r^{th}$ Term in the expansion of $(x+1)^{n}$ :

$T_{r}=^nC_{r-1}x^{n-r+1}1^{r-1}=^nC_{r-1}x^{n-r+1}$

$(r-1)^{th}$ Term in the expansion of $(x+1)^{n}$ :

$T_{r-1}=^nC_{r-2}x^{n-r+2}1^{r-2}=^nC_{r-2}x^{n-r+2}$

Now, As given in the question,

$T_{r-1}:T_r:T_{r+1}=1:3:5$

$^nC_{r-2}:^nC_{r-1}:^nC_{r}=1:3:5$

$\frac{n!}{(r-2)!(n-r+2)!}:\frac{n!}{(r-1)!(n-r+1)!}:\frac{n!}{r!(n-r)!}=1:3:5$

From here, we get ,

$\frac{r-1}{n-r+2}=\frac{1}{3}\:\:and\:\:\frac{r}{n-r+1}=\frac{3}{5}$

Which can be written as

$n-4r+5=0\:\:and\:\:3n-8r+3=0$

From these equations we get,

$n=7\:\:and\:\:r=3$

Question:11 Prove that the coefficient of $x^n$ in the expansion of $(1+x)^{2n}$ is twice the coefficient of $x^n$ in the expansion of $(1+x)^{2n-1}$ .

Answer:

As we know that the general $(r+1)^{th}$ term $T_{r+1}$ in the binomial expansion of $(a+b)^n$ is given by

$T_{r+1}=^nC_ra^{n-r}b^r$

So, general $(r+1)^{th}$ term $T_{r+1}$ in the binomial expansion of $(1+x)^{2n}$ is,

$T_{r+1}=^{2n}C_r1^{2n-r}x^r$

$x^n$ will come when $r=n$ ,

So, Coefficient of $x^n$ in the binomial expansion of $(1+x)^{2n}$ is,

$K_{1x^n}=^{2n}C_n$

Now,

the general $(r+1)^{th}$ term $T_{r+1}$ in the binomial expansion of $(1+x)^{2n-1}$ is,

$T_{r+1}=^{2n-1}C_r1^{2n-1-r}x^r$

Here also $x^n$ will come when $r=n$ ,

So, Coefficient of $x^n$ in the binomial expansion of $(1+x)^{2n-1}$ is,

$K_{2x^n}=^{2n-1}C_n$

Now, As we can see

$^{2n-1}C_n=\frac{(2n-1)!}{n!(2n-1-n)!}=\frac{(2n-1)!}{n!(n-1)!}=\frac{(2n)!}{2n(n!)(n-1)!}=\frac{(2n)!}{2(n!)(n!)}$

$^{2n-1}C_n=\frac{1}{2}\times^{2n}C_n$

$2\times^{2n-1}C_n=^{2n}C_n$

$2\times K_{2x^n}=K_{1x^n}$

Hence, the coefficient of $x^n$ in the expansion of $(1+x)^{2n}$ is twice the coefficient of $x^n$ in the expansion of $(1+x)^{2n-1}$ .

Question:12 Find a positive value of m for which the coefficient of $x^2$ in the expansion $(1 + x)^ m$ is 6.

Answer:

As we know that the general $(r+1)^{th}$ term $T_{r+1}$ in the binomial expansion of $(a+b)^n$ is given by

$T_{r+1}=^nC_ra^{n-r}b^r$

So, the general $(r+1)^{th}$ term $T_{r+1}$ in the binomial expansion of $(1 + x)^ m$ is

$T_{r+1}=^mC_r1^{m-r}x^r=^mC_rx^r$

$x^2$ will come when $r=2$ . So,

The coeficient of $x^2$ in the binomial expansion of $(1 + x)^ m$ = 6

$\Rightarrow ^mC_2=6$

$\Rightarrow \frac{m!}{2!(m-2)!}=6$

$\Rightarrow \frac{m(m-1)}{2}=6$

$\Rightarrow m(m-1)=12$

$\Rightarrow m^2-m-12=0$

$\Rightarrow (m+3)(m-4)=0$

$\Rightarrow m=4\:or\:-3$

Hence the positive value of m for which the coefficient of $x^2$ in the expansion $(1 + x)^ m$ is 6, is 4.

NCERT Solutions of Class 11 Subject Wise

Subject Wise NCERT Exampler Solutions

Happy learning!!!

Frequently Asked Questions (FAQs)

1. In the expansion of (a+b)^5 which term is the middle term ?

The 3rd and fourth terms of this expansion are the middle terms of this expansion.

2. The number of terms in the expansion of (a+b)^5 ?

The number of terms in the expansion = 5+1=6

3. What is the weightage of the binomial theorem in the CBSE Class 11 Maths ?

The weightage of the Algebra part in the CBSE Class 11 Maths is 30 marks. The chapter-wise weightage is not provided by CBSE.

4. Are these useful for revision before the CBSE exam ?

Yes, NCERT solutions are useful for quick revision before the CBSE exam.

5. In the expansion of (a+b)^6 which term is the middle term ?

Here the value of n is 6 so the number of terms will be 7 and the middle term would be fourth.

6. Do I need to buy CBSE class 11 solution book ?

No, you don't need to buy any solutions book for Class 11 Maths. You can find all the NCERT solutions for Class 11 online.

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