** According to the CBSE Syllabus 2023-24, this chapter has been renumbered as Chapter 7.*

The **NCERT Solutions Class 11 Chapter 8 Binomial Theorem** can be downloaded at BYJUâ€™S easily. Practising these solutions can help the students clear their doubts as well as solve problems faster. Students can learn new tips and methods to answer a particular question in different ways using NCERT Solutions, which gives them an edge in exam preparation.

The concepts covered in Chapter 8 of the Maths textbook include the study of essential topics such as Positive Integral Indices, Pascalâ€™s Triangle, the Binomial Theorem for any positive integer and some special cases. Students can score high marks in the exams easily by practising the NCERT Class 11 Solutions for all the questions in the textbook. Each solution is solved step-by-step, considering the understanding level of the students. Therefore, it is important to understand the logic set behind each answer and develop a better comprehension of the concepts.

## NCERT Solutions for Class 11 Maths Chapter 8 â€“ Binomial Theorem

### Access Answers to NCERT Class 11 Maths Chapter 8 â€“ Binomial Theorem

### Access Answers to NCERT Class 11 Maths Chapter 8

Exercise 8.1 Page No: 166

**Expand each of the expressions in Exercises 1 to 5.**

**1. (1 â€“ 2x) ^{5}**

**Solution:**

From binomial theorem expansion, we can write as

(1 â€“ 2x)^{5}

= ^{5}C_{o }(1)^{5} â€“ ^{5}C_{1 }(1)^{4} (2x) + ^{5}C_{2 }(1)^{3 }(2x)^{2} â€“ ^{5}C_{3 }(1)^{2 }(2x)^{3} + ^{5}C_{4 }(1)^{1} (2x)^{4} â€“ ^{5}C_{5 }(2x)^{5}

= 1 â€“ 5 (2x) + 10 (4x)^{2} â€“ 10 (8x^{3}) + 5 ( 16 x^{4}) â€“ (32 x^{5})

= 1 â€“ 10x + 40x^{2} â€“ 80x^{3}Â + 80x^{4}â€“ 32x^{5}

**Solution:**

From the binomial theorem, the given equation can be expanded as

**3. (2x â€“ 3) ^{6}**

**Solution:**

From the binomial theorem, the given equation can be expanded as

**Solution:**

From the binomial theorem, the given equation can be expanded as

**Solution:**

From the binomial theorem, the given equation can be expanded as

**6. Using the binomial theorem, find (96) ^{3}**

**.Â Â**

**Solution:**

Given (96)^{3}

96 can be expressed as the sum or difference of two numbers, and then the binomial theorem can be applied.

The given question can be written as 96 = 100 â€“ 4

(96)^{3} = (100 â€“ 4)^{3}

= ^{3}C_{0} (100)^{3} â€“ ^{3}C_{1} (100)^{2} (4) â€“ ^{3}C_{2} (100) (4)^{2}â€“ ^{3}C_{3} (4)^{3}

= (100)^{3} â€“ 3 (100)^{2} (4) + 3 (100) (4)^{2} â€“ (4)^{3}

= 1000000 â€“ 120000 + 4800 â€“ 64

= 884736

**7. Using the binomial theorem, find (102) ^{5}**

**.**

**Solution:**

Given (102)^{5}

102 can be expressed as the sum or difference of two numbers, and then the binomial theorem can be applied.

The given question can be written as 102 = 100 + 2

(102)^{5} = (100 + 2)^{5}

= ^{5}C_{0} (100)^{5} + ^{5}C_{1} (100)^{4} (2) + ^{5}C_{2} (100)^{3} (2)^{2} + ^{5}C_{3} (100)^{2} (2)^{3} + ^{5}C_{4} (100) (2)^{4} + ^{5}C_{5} (2)^{5}

= (100)^{5} + 5 (100)^{4} (2) + 10 (100)^{3} (2)^{2} + 5 (100) (2)^{3} + 5 (100) (2)^{4} + (2)^{5}

= 1000000000 + 1000000000 + 40000000 + 80000 + 8000 + 32

= 11040808032

**8. Using the binomial theorem, find (101) ^{4}**

**.**

**Solution:**

Given (101)^{4}

101 can be expressed as the sum or difference of two numbers, and then the binomial theorem can be applied.

The given question can be written as 101 = 100 + 1

(101)^{4} = (100 + 1)^{4}

= ^{4}C_{0} (100)^{4} + ^{4}C_{1} (100)^{3} (1) + ^{4}C_{2} (100)^{2} (1)^{2} + ^{4}C_{3} (100) (1)^{3} + ^{4}C_{4 }(1)^{4}

= (100)^{4} + 4 (100)^{3} + 6 (100)^{2} + 4 (100) + (1)^{4}

= 100000000 + 4000000 + 60000 + 400 + 1

= 104060401

**9. Using the binomial theorem, find (99) ^{5}**

**m.**

**Solution:**

Given (99)^{5}

99 can be written as the sum or difference of two numbers then the binomial theorem can be applied.

The given question can be written as 99 = 100 -1

(99)^{5} = (100 â€“ 1)^{5}

= ^{5}C_{0} (100)^{5} â€“ ^{5}C_{1} (100)^{4} (1) + ^{5}C_{2} (100)^{3} (1)^{2} â€“ ^{5}C_{3} (100)^{2} (1)^{3} + ^{5}C_{4} (100) (1)^{4} â€“ ^{5}C_{5} (1)^{5}

= (100)^{5} â€“ 5 (100)^{4} + 10 (100)^{3} â€“ 10 (100)^{2} + 5 (100) â€“ 1

= 1000000000 â€“ 5000000000 + 10000000 â€“ 100000 + 500 â€“ 1

= 9509900499

**10. Using Binomial Theorem, indicate which number is larger (1.1) ^{10000}Â or 1000.**

**Solution:**

By splitting the given 1.1 and then applying the binomial theorem, the first few terms of (1.1)^{10000} can be obtained as

(1.1)^{10000} = (1 + 0.1)^{10000}

= (1 + 0.1)^{10000} C_{1 }(1.1) + other positive terms

= 1 + 10000 Ã— 1.1 + other positive terms

= 1 + 11000 + other positive terms

> 1000

(1.1)^{10000} > 1000

**11.** **Find (a + b) ^{4}Â â€“ (a â€“ b)^{4}. Hence, evaluateÂ **

**Solution:**

Using the binomial theorem, the expression (a + b)^{4} and (a â€“ b)^{4}Â can be expanded

(a + b)^{4} = ^{4}C_{0} a^{4} + ^{4}C_{1} a^{3} b + ^{4}C_{2} a^{2} b^{2} + ^{4}C_{3} a b^{3} + ^{4}C_{4} b^{4}

(a â€“ b)^{4 }= ^{4}C_{0} a^{4} â€“ ^{4}C_{1} a^{3} b + ^{4}C_{2} a^{2} b^{2} â€“ ^{4}C_{3} a b^{3} + ^{4}C_{4} b^{4}

Now (a + b)^{4} â€“ (a â€“ b)^{4} = ^{4}C_{0} a^{4} + ^{4}C_{1} a^{3} b + ^{4}C_{2} a^{2} b^{2} + ^{4}C_{3} a b^{3} + ^{4}C_{4} b^{4} â€“ [^{4}C_{0} a^{4} â€“ ^{4}C_{1} a^{3} b + ^{4}C_{2} a^{2} b^{2} â€“ ^{4}C_{3} a b^{3} + ^{4}C_{4} b^{4}]

= 2 (^{4}C_{1} a^{3} b + ^{4}C_{3} a b^{3})

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

= 8ab (a^{2} + b^{2})

Now by substituting a = âˆš3 and b = âˆš2, we get

(âˆš3 + âˆš2)^{4} â€“ (âˆš3 â€“ âˆš2)^{4} = 8 (âˆš3) (âˆš2) {(âˆš3)^{2} + (âˆš2)^{2}}

= 8 (âˆš6) (3 + 2)

= 40 âˆš6

**12.** **Find (x + 1) ^{6}Â + (x â€“ 1)^{6}. Hence or otherwise evaluateÂ **

**Â **

**Solution:**

Using binomial theorem, the expressions (x + 1)^{6} and (x â€“ 1)^{6} can be expressed as

(x + 1)^{6} = ^{6}C_{0} x^{6} + ^{6}C_{1} x^{5} + ^{6}C_{2} x^{4} + ^{6}C_{3} x^{3} + ^{6}C_{4} x^{2} + ^{6}C_{5} x + ^{6}C_{6}

(x â€“ 1)^{6} = ^{6}C_{0} x^{6} â€“ ^{6}C_{1} x^{5} + ^{6}C_{2} x^{4} â€“ ^{6}C_{3} x^{3} + ^{6}C_{4} x^{2} â€“ ^{6}C_{5} x + ^{6}C_{6}

Now, (x + 1)^{6} â€“ (x â€“ 1)^{6} = ^{6}C_{0} x^{6} + ^{6}C_{1} x^{5} + ^{6}C_{2} x^{4} + ^{6}C_{3} x^{3} + ^{6}C_{4} x^{2} + ^{6}C_{5} x + ^{6}C_{6} â€“ [^{6}C_{0} x^{6} â€“ ^{6}C_{1} x^{5} + ^{6}C_{2} x^{4} â€“ ^{6}C_{3} x^{3} + ^{6}C_{4} x^{2} â€“ ^{6}C_{5} x + ^{6}C_{6}]

= 2 [^{6}C_{0} x^{6 }+ ^{6}C_{2} x^{4} + ^{6}C_{4} x^{2} + ^{6}C_{6}]

= 2 [x^{6} + 15x^{4} + 15x^{2} + 1]

Now by substituting x = âˆš2, we get

(âˆš2 + 1)^{6} â€“ (âˆš2 â€“ 1)^{6} = 2 [(âˆš2)^{6} + 15(âˆš2)^{4} + 15(âˆš2)^{2} + 1]

= 2 (8 + 15 Ã— 4 + 15 Ã— 2 + 1)

= 2 (8 + 60 + 30 + 1)

= 2 (99)

= 198

**13. Show that 9 ^{n+1} â€“ 8n â€“ 9 is divisible by 64 whenever n is a positive integer.**

**Solution:**

In order to show that 9^{n+1} â€“ 8n â€“ 9 is divisible by 64, it has to be shown that 9^{n+1} â€“ 8n â€“ 9 = 64 k, where k is some natural number.

Using the binomial theorem,

(1 + a)^{m} = ^{m}C_{0} + ^{m}C_{1} a + ^{m}C_{2} a^{2} + â€¦. + ^{m }C _{m} a^{m}

For a = 8 and m = n + 1 we get

(1 + 8)^{n+1} = ^{n+1}C_{0} + ^{n+1}C_{1} (8) + ^{n+1}C_{2} (8)^{2} + â€¦. + ^{n+1 }C _{n+1 }(8)^{n+1}

9^{n+1} = 1 + (n + 1) 8 + 8^{2} [^{n+1}C_{2} + ^{n+1}C_{3} (8) + â€¦. + ^{n+1 }C _{n+1 }(8)^{n-1}]

9^{n+1} = 9 + 8n + 64 [^{n+1}C_{2} + ^{n+1}C_{3} (8) + â€¦. + ^{n+1 }C _{n+1 }(8)^{n-1}]

9^{n+1} â€“ 8n â€“ 9 = 64 k

Where k = [^{n+1}C_{2} + ^{n+1}C_{3} (8) + â€¦. + ^{n+1 }C _{n+1 }(8)^{n-1}] is a natural number

Thus, 9^{n+1} â€“ 8n â€“ 9 is divisible by 64 whenever n is a positive integer.

Hence proved.

**14. Prove thatÂ **

**Solution:**

Exercise 8.2 Page No: 171

**Find the coefficient of**

**1. x ^{5} in (x + 3)^{8} **

**Solution:**

The general term T_{r+1}Â in the binomial expansion is given by T_{r+1}Â =Â ^{n }C _{r}Â a^{n-r}Â b^{r}

Here x^{5}Â is the T_{r+1}Â term so a= x, b = 3 and n =8

T_{r+1}Â =Â ^{8}C_{r}Â x^{8-r}Â 3^{r}â€¦â€¦â€¦â€¦â€¦ (i)

To find out x^{5}

We have to equate x^{5}= x^{8-r}

â‡’Â r= 3

Putting the value of r in (I), we get

= 1512 x^{5}

Hence the coefficient of x^{5}= 1512.

**2. a ^{5}b^{7}Â in (a â€“ 2b)^{12}**

**Solution:**

The general term T_{r+1}Â in the binomial expansion is given by T_{r+1}Â =Â ^{n }C _{r}Â a^{n-r}Â b^{r}

Here a = a, b = -2b & n =12

Substituting the values, we get

T_{r+1}Â =Â ^{12}C_{r}Â a^{12-r}Â (-2b)^{r}â€¦â€¦â€¦. (i)

To find a^{5}

We equate a^{12-r}Â =a^{5}

r = 7

Putting r = 7 in (i)

T_{8}Â =Â ^{12}C_{7}Â a^{5}Â (-2b)^{7}

= -101376 a^{5}Â b^{7}

Hence, the coefficient of a^{5}b^{7}= -101376.

**Write the general term in the expansion of**

**3. (x ^{2}Â â€“ y)^{6}**

**Solution:**

The general term T_{r+1}Â in the binomial expansion is given by

T_{r+1}Â =Â ^{n }C _{r}Â a^{n-r}Â b^{r}â€¦â€¦.. (i)

Here, a = x^{2} , n = 6 and b = -y

Putting values in (i)

T_{r+1}Â =Â ^{6}C_{r}Â xÂ ^{2(6-r)}Â (-1)^{r}Â y^{r}

= -1^{r 6}c_{r }.x^{12 â€“ 2r}. y^{r}

**4.** **(x ^{2}Â â€“ y x)^{12}, x â‰ 0**

**Solution:**

The general term T_{r+1}Â in the binomial expansion is given by T_{r+1}Â =Â ^{n }C _{r}Â a^{n-r}Â b^{r}

Here n = 12, a= x^{2}Â and b = -y x

Substituting the values, we get

T_{n+1}Â =^{12}C_{r}Â Ã— x^{2(12-r)}Â (-1)^{r}Â y^{r}Â x^{r}

= -1^{r 12}c_{r }.x^{24 â€“2r}. y^{r}

**5. Find the 4th term in the expansion of (x â€“ 2y) ^{12}.**

**Solution:**

_{r+1}Â in the binomial expansion is given by T_{r+1}Â =Â ^{n }C _{r}Â a^{n-r}Â b^{r}

Here, a= x, n =12, r= 3 and b = -2y

By substituting the values, we get

T_{4}Â =Â ^{12}C_{3}Â x^{9}Â (-2y)^{3}

= -1760 x^{9}Â y^{3}

**6.** **Find the 13 ^{th}Â term in the expansion ofÂ **

**Solution:**

**Find the middle terms in the expansions of
**

**Solution:**

**Solution:**

**9. In the expansion of (1 + a) ^{m+n}, prove that coefficients of a^{m}Â and a^{n}Â are equal.**

**Solution:**

We know that the general term T_{r+1}Â in the binomial expansion is given by T_{r+1}Â =Â ^{n}C_{r}Â a^{n-r}Â b^{r}

Here n= m+n, a = 1 and b= a

Substituting the values in the general form

T_{r+1}Â =Â ^{m+n }C_{r}Â 1^{m+n-r}Â a^{r}

=Â ^{m+n }C_{r}Â a^{r}â€¦â€¦â€¦â€¦. (i)

Now, we have that the general term for the expression is,

T_{r+1}Â =Â Â ^{m+n }C_{r}Â a^{r
}

Now, for coefficient of a^{m}

T_{m+1}Â =Â Â ^{m+n }C_{m}Â a^{m
}

Hence, for the coefficient of a^{m}, the value of r = m

So, the coefficient isÂ ^{m+n }C _{m}

Similarly, the coefficient of a^{n}Â isÂ ^{m+n }C _{n
}

**10. The coefficients of the (r â€“ 1) ^{th}, r^{th}Â and (r + 1)^{th}Â terms in the expansion of (x + 1)^{n} are in the ratio 1:3:5. Find n and r.**

**Solution:**

The general term T_{r+1}Â in the binomial expansion is given by T_{r+1}Â =Â ^{n}C_{r}Â a^{n-r}Â b^{r}

Here, the binomial is (1+x)^{n}Â with a = 1 , b = x and n = n

The (r+1)^{th}Â term is given by

T_{(r+1)}Â =Â ^{n}C_{r}Â 1^{n-r}Â x^{r}

T_{(r+1)}Â =Â ^{n}C_{r}Â x^{r}

The coefficient of (r+1)^{th}Â term isÂ ^{n}C_{r}

The r^{th}Â term is given by (r-1)^{th}Â term

T_{(r+1-1)}Â =Â ^{n}C_{r-1}Â x^{r-1}

T_{r}Â =Â ^{n}C_{r-1}Â x^{r-1}

âˆ´Â the coefficient of r^{th}Â term isÂ ^{n}C_{r-1}

For (r-1)^{th} term, we will take (r-2)^{th}Â term

T_{r-2+1}Â =Â ^{n}C_{r-2}Â x^{r-2}

T_{r-1}Â =Â ^{n}C_{r-2}Â x^{r-2}

âˆ´Â the coefficient of (r-1)^{th}Â term isÂ ^{n}C_{r-2}

Given that the coefficient of (r-1)^{th}, r^{th}Â and r+1^{th}Â term are in ratio 1:3:5

Therefore,

â‡’Â 5r = 3n â€“ 3r + 3

â‡’Â 8rÂ â€“Â 3n â€“ 3 =0â€¦â€¦â€¦â€¦.2

We have 1 and 2 as

n â€“ 4rÂ Â±Â 5 =0â€¦â€¦â€¦â€¦1

8r â€“ 3n â€“ 3 =0â€¦â€¦â€¦â€¦â€¦.2

Multiplying equation 1 by number 2

2n -8r +10 =0â€¦â€¦â€¦â€¦â€¦â€¦.3

Adding equations 2 and 3

2n -8r +10 =0

-3n â€“ 8r â€“ 3 =0

â‡’Â -n = -7

n =7Â and r = 3

**11. Prove that the coefficient of x ^{n}Â in the expansion of (1 + x)^{2n}Â is twice the coefficient of x^{n}Â in the expansion of (1 + x)^{2n â€“ 1}.**

**Solution:**

The general term T_{r+1}Â in the binomial expansion is given by T_{r+1}Â =Â ^{n}C_{r}Â a^{n-r}Â b^{r}

The general term for binomial (1+x)^{2n}Â is

T_{r+1}Â =Â ^{2n}C_{r}Â x^{r}Â â€¦â€¦â€¦â€¦â€¦â€¦â€¦..1

To find the coefficient of x^{n}

r = n

T_{n+1}Â =Â ^{2n}C_{n}Â x^{n}

The coefficient of x^{n}Â =Â ^{2n}C_{n}

The general term for binomial (1+x)^{2n-1}Â is

T_{r+1}Â =Â ^{2n-1}C_{r}Â x^{r}

To find the coefficient of x^{n}

Putting n = r

T_{r+1}Â =Â ^{2n-1}C_{r}Â x^{n}

The coefficient of x^{n}Â =Â ^{2n-1}C_{n}

We have to prove

Coefficient of x^{n}Â in (1+x)^{2n}Â = 2 coefficient of x^{n}Â in (1+x)^{2n-1}

Consider LHS = ^{2n}C_{n}

**12. Find a positive value of m for which the coefficient of x ^{2}Â in the expansion (1 + x)^{m}Â is 6.**

**Solution:**

The general term T_{r+1}Â in the binomial expansion is given by T_{r+1}Â =Â ^{n}C_{r}Â a^{n-r}Â b^{r}

Here, a = 1, b = x and n = m

Putting the value

T_{r+1}Â =Â ^{m }C_{r}Â 1^{m-r}Â x^{r}

=Â ^{m }C_{r}Â x^{r}

We need the coefficient of x^{2}

âˆ´Â putting r = 2

T_{2+1}Â =Â ^{m}C_{2}Â x^{2}

The coefficient of x^{2}Â =Â ^{m}C_{2}

Given thatÂ coefficient of x^{2}Â =Â ^{m}C_{2}Â = 6

â‡’Â m (m â€“ 1) = 12

â‡’Â m^{2}â€“ m â€“ 12 =0

â‡’Â m^{2}â€“ 4m + 3m â€“ 12 =0

â‡’Â m (m â€“ 4) + 3 (m â€“ 4) = 0

â‡’Â (m+3) (m â€“ 4) = 0

â‡’Â m = â€“ 3, 4

We need the positive value of m, so m = 4

Miscellaneous Exercise Page No: 175

**1. Find a, b and n in the expansion of (a + b) ^{n}Â if the first three terms of the expansion are 729, 7290 and 30375, respectively.**

**Solution:**

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

T_{r+1} = ^{n}C_{r} a^{n-t} b^{r}

The first three terms of the expansion are given as 729, 7290 and 30375, respectively. Then we have,

T_{1} = ^{n}C_{0} a^{n-0} b^{0} = a^{n} = 729â€¦.. 1

T_{2} = ^{n}C_{1} a^{n-1} b^{1 }= na^{n-1} b = 7290â€¦. 2

T_{3} = ^{n}C_{2} a^{n-2} b^{2} = {n (n -1)/2 }a^{n-2} b^{2} = 30375â€¦â€¦3

Dividing 2 by 1, we get

Dividing 3 by 2, we get

From 4 and 5, we have

n. 5/3 = 10

n = 6

Substituting n = 6 in 1, we get

a^{6} = 729

a = 3

From 5, we have, b/3 = 5/3

b = 5

Thus a = 3, b = 5 and n = 76

**2. Find a if the coefficients of x ^{2}Â and x^{3}Â in the expansion of (3 + a x)^{9}Â are equal.**

**Solution:**

**3. Find the coefficient of x ^{5}Â in the product (1 + 2x)^{6}Â (1 â€“ x)^{7}Â using binomial theorem.**

**Solution:**

(1 + 2x)^{6} = ^{6}C_{0 }+ ^{6}C_{1} (2x) + ^{6}C_{2} (2x)^{2} + ^{6}C_{3} (2x)^{3} + ^{6}C_{4} (2x)^{4} + ^{6}C_{5} (2x)^{5} + ^{6}C_{6} (2x)^{6}

= 1 + 6 (2x) + 15 (2x)^{2} + 20 (2x)^{3} + 15 (2x)^{4} + 6 (2x)^{5} + (2x)^{6}

= 1 + 12 x + 60x^{2} + 160 x^{3} + 240 x^{4} + 192 x^{5} + 64x^{6}

(1 â€“ x)^{7} = ^{7}C_{0} â€“ ^{7}C_{1} (x) + ^{7}C_{2 }(x)^{2} â€“ ^{7}C_{3 }(x)^{3} + ^{7}C_{4 }(x)^{4} â€“ ^{7}C_{5 }(x)^{5} + ^{7}C_{6 }(x)^{6 }â€“ ^{7}C_{7 }(x)^{7}

= 1 â€“ 7x + 21x^{2} â€“ 35x^{3} + 35x^{4} â€“ 21x^{5} + 7x^{6} â€“ x^{7}

(1 + 2x)^{6} (1 â€“ x)^{7} = (1 + 12 x + 60x^{2} + 160 x^{3} + 240 x^{4} + 192 x^{5} + 64x^{6}) (1 â€“ 7x + 21x^{2} â€“ 35x^{3} + 35x^{4} â€“ 21x^{5} + 7x^{6} â€“ x^{7})

192 â€“ 21 = 171

Thus, the coefficient of x^{5}Â in the expression (1+2x)^{6}(1-x)7 is 171.

**4. If a and b are distinct integers, prove that a â€“ b is a factor of a ^{n}Â â€“ b^{n}, whenever n is a positive integer. [Hint write a^{n}Â = (a â€“ b + b)^{n}Â and expand]**

**Solution:**

In order to prove that (a â€“ b) is a factor of (a^{n} â€“ b^{n}), it has to be proved that

a^{n} â€“ b^{n} = k (a â€“ b) where k is some natural number.

a can be written as a = a â€“ b + b

a^{n} = (a â€“ b + b)^{n } = [(a â€“ b) + b]^{n}

= ^{n}C_{0} (a â€“ b)^{n} + ^{n}C_{1} (a â€“ b)^{n-1 } b + â€¦â€¦ + ^{n }C _{n} b^{n}

a^{n} â€“ b^{n} = (a â€“ b) [(a â€“b)^{n-1} + ^{n}C_{1} (a â€“ b)^{n-1 } b + â€¦â€¦ + ^{n }C _{n} b^{n}]

a^{n} â€“ b^{n} = (a â€“ b) k

Where k = [(a â€“b)^{n-1} + ^{n}C_{1} (a â€“ b)^{n-1 } b + â€¦â€¦ + ^{n }C _{n} b^{n}] is a natural number

This shows that (a â€“ b) is a factor of (a^{n} â€“ b^{n}), where n is a positive integer.

**5. EvaluateÂ **

**Solution:**

Using the binomial theorem, the expression (a + b)^{6} and (a â€“ b)^{6}Â can be expanded

(a + b)^{6} = ^{6}C_{0} a^{6} + ^{6}C_{1} a^{5} b + ^{6}C_{2} a^{4} b^{2} + ^{6}C_{3} a^{3} b^{3} + ^{6}C_{4} a^{2} b^{4} + ^{6}C_{5} a b^{5 }+ ^{6}C_{6} b^{6}

(a â€“ b)^{6 }= ^{6}C_{0} a^{6} â€“ ^{6}C_{1} a^{5} b + ^{6}C_{2} a^{4} b^{2} â€“ ^{6}C_{3} a^{3} b^{3} + ^{6}C_{4} a^{2} b^{4} â€“ ^{6}C_{5} a b^{5 }+ ^{6}C_{6} b^{6}

Now (a + b)^{6} â€“ (a â€“ b)^{6} =^{6}C_{0} a^{6} + ^{6}C_{1} a^{5} b + ^{6}C_{2} a^{4} b^{2} + ^{6}C_{3} a^{3} b^{3} + ^{6}C_{4} a^{2} b^{4} + ^{6}C_{5} a b^{5 }+ ^{6}C_{6} b^{6} â€“ [^{6}C_{0} a^{6} â€“ ^{6}C_{1} a^{5} b + ^{6}C_{2} a^{4} b^{2} â€“ ^{6}C_{3} a^{3} b^{3} + ^{6}C_{4} a^{2} b^{4} â€“ ^{6}C_{5} a b^{5 }+ ^{6}C_{6} b^{6}]

Now by substituting a = âˆš3 and b = âˆš2, we get

(âˆš3 + âˆš2)^{6} â€“ (âˆš3 â€“ âˆš2)^{6} = 2 [6 (âˆš3)^{5} (âˆš2) + 20 (âˆš3)^{3} (âˆš2)^{3} + 6 (âˆš3) (âˆš2)^{5}]

= 2 [54(âˆš6) + 120 (âˆš6) + 24 âˆš6]

= 2 (âˆš6) (198)

= 396 âˆš6

**6. Find the value ofÂ **

**Solution:**

**7. Find an approximation of (0.99) ^{5}Â using the first three terms of its expansion.**

**Solution:**

0.99 can be written as

0.99 = 1 â€“ 0.01

Now by applying the binomial theorem, we get

(o. 99)^{5} = (1 â€“ 0.01)^{5}

= ^{5}C_{0 }(1)^{5} â€“ ^{5}C_{1 }(1)^{4} (0.01) + ^{5}C_{2 }(1)^{3} (0.01)^{2 }

= 1 â€“ 5 (0.01) + 10 (0.01)^{2}

= 1 â€“ 0.05 + 0.001

= 0.951

**8. Find n, if the ratio of the fifth term from the beginning to the fifth term from the end, in the expansion of , is âˆš6: 1**

**Solution:**

**9. Expand using the Binomial TheoremÂ **

**Solution:**

Using the binomial theorem, the given expression can be expanded as

Again by using the binomial theorem to expand the above terms, we get

From equations 1, 2 and 3, we get

**10. Find the expansion of (3x ^{2}Â â€“ 2ax + 3a^{2})^{3}Â using binomial theorem.**

**Solution:**

We know that (a + b)^{3}Â = a^{3}Â + 3a^{2}b + 3ab^{2}Â + b^{3}

Putting a = 3x^{2}Â & b = -a (2x-3a), we get

^{2}Â + (-a (2x-3a))]

^{3}

= (3x^{2})^{3}+3(3x^{2})^{2}(-a (2x-3a)) + 3(3x^{2}) (-a (2x-3a))^{2}Â + (-a (2x-3a))^{3}

= 27x^{6}Â â€“ 27ax^{4 }(2x-3a) + 9a^{2}x^{2 }(2x-3a)^{2}Â â€“ a^{3}(2x-3a)^{3}

=Â 27x^{6}Â â€“ 54ax^{5}Â + 81a^{2}x^{4}Â +Â 9a^{2}x^{2 }(4x^{2}-12ax+9a^{2}) â€“ a^{3 }[(2x)^{3}Â â€“ (3a)^{3}Â â€“ 3(2x)^{2}(3a) + 3(2x)(3a)^{2}]

=Â 27x^{6}Â â€“ 54ax^{5}Â + 81a^{2}x^{4}Â +Â 36a^{2}x^{4}Â â€“ 108a^{3}x^{3}Â + 81a^{4}x^{2} â€“ 8a^{3}x^{3}Â + 27a^{6}Â + 36a^{4}x^{2}Â â€“ 54a^{5}x

=Â 27x^{6}Â â€“ 54ax^{5}+Â 117a^{2}x^{4}Â â€“ 116a^{3}x^{3}Â + 117a^{4}x^{2}Â â€“ 54a^{5}x + 27a^{6}

Thus,Â (3x^{2}Â â€“ 2ax + 3a^{2})^{3}

=Â 27x^{6}Â â€“ 54ax^{5}+Â 117a^{2}x^{4}Â â€“ 116a^{3}x^{3}Â + 117a^{4}x^{2}Â â€“ 54a^{5}x + 27a^{6}

Also Access |

NCERT Exemplar for Class 11 Maths Chapter 8 |

CBSE Notes for Class 11 Maths Chapter 8 |

## NCERT Solutions for Class 11 Maths Chapter 8 â€“ Binomial Theorem

The Chapter 8 Binomial Theorem of NCERT Solutions for Class 11 covers the topics given below.

8.1 Introduction to Binomial TheoremÂ

8.2 Binomial Theorem for Positive Integral IndicesÂ

Â Â Â Â Pascalâ€™s TriangleÂ

8.2.1 Binomial theorem for any positive integer nÂ

8.2.2 Some special cases

8.3 General and Middle TermsÂ

Exercise 8.1 Solutions 14 Questions

Exercise 8.2 Solutions 12 Questions

Miscellaneous Exercise On Chapter 8 Solutions 10 Questions

## NCERT Solutions for Class 11 Maths Chapter 8 â€“ Binomial Theorem

The unit Algebra houses the chapter Binomial Theorem, adding up to 30 marks of the total 80 marks. A total of 3 exercises, including the miscellaneous exercise, are present in this chapter. Chapter 8 of NCERT Solutions for Class 11 Maths discusses the concepts provided underneath:

- The expansion of a binomial for any positive integral n is given by the Binomial Theorem, which is (a+b)
^{n}=^{n}C_{0}a^{n}+^{n}C_{1}a^{n â€“ 1}b +^{n}C_{2}a^{n â€“ 2}b^{2}+ â€¦+^{n}C_{n â€“ 1}a.b^{n â€“ 1}+^{n}C_{n}b^{n}. - The coefficients of the expansions are arranged in an array. This array is called Pascalâ€™s triangle.
- The general term of an expansion (a + b)
^{n}is T_{r + 1}=^{n}C_{r}a^{n â€“ r}. b^{r}

Therefore, it is ensured that a student who is thorough with Chapter 8 of Class 11, the Binomial Theorem, will be well-versed in the history of the Binomial Theorem, statement and proof of the binomial theorem for positive integral indices, Pascalâ€™s triangle, general and middle term in binomial expansion as well as simple applications of Binomial theorem.

**Disclaimer â€“Â **

**Dropped Topics â€“Â **

8.3 General Middle Terms

Example 17 and Ques. 1â€“3, and 8 (Miscellaneous Exercise)

Last two points in the Summary

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