NCERT Solutions for Class 11 Maths Chapter 8 – Binomial Theorem Miscellaneous Exercise
Page No 175:
Question 1:
Find a, b and n in the expansion of (a + b)n if the first three terms of the expansion are 729, 7290 and 30375, respectively.
Answer:
It is known that (r + 1)th term, (Tr+1), in the binomial expansion of (a + b)n is given by
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/168/5587/Chapter%208_html_m71136a02.gif)
The first three terms of the expansion are given as 729, 7290, and 30375 respectively.
Therefore, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/168/5587/Chapter%208_html_m3fa5b83b.gif)
Dividing (2) by (1), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/168/5587/Chapter%208_html_mc5c2725.gif)
Dividing (3) by (2), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/168/5587/Chapter%208_html_716c74d9.gif)
From (4) and (5), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/168/5587/Chapter%208_html_5ea130da.gif)
Substituting n = 6 in equation (1), we obtain
a6 = 729
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/168/5587/Chapter%208_html_m2f2d5bbf.gif)
From (5), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/168/5587/Chapter%208_html_2e433e0d.gif)
Thus, a = 3, b = 5, and n = 6.
Question 2:
Find a if the coefficients of x2 and x3 in the expansion of (3 + ax)9 are equal.
Answer:
It is known that (r + 1)th term, (Tr+1), in the binomial expansion of (a + b)n is given by
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/168/5588/Chapter%208_html_m71136a02.gif)
Assuming that x2 occurs in the (r + 1)th term in the expansion of (3 + ax)9, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/168/5588/Chapter%208_html_d4145cc.gif)
Comparing the indices of x in x2 and in Tr + 1, we obtain
r = 2
Thus, the coefficient of x2 is
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/168/5588/Chapter%208_html_m51af93b2.gif)
Assuming that x3 occurs in the (k + 1)th term in the expansion of (3 + ax)9, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/168/5588/Chapter%208_html_2e8a4ee.gif)
Comparing the indices of x in x3 and in Tk+ 1, we obtain
k = 3
Thus, the coefficient of x3 is
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/168/5588/Chapter%208_html_4e1d4da6.gif)
It is given that the coefficients of x2 and x3 are the same.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/168/5588/Chapter%208_html_m387ab972.gif)
Thus, the required value of a is
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/168/5588/Chapter%208_html_m177f6162.gif)
Question 3:
Find the coefficient of x5 in the product (1 + 2x)6 (1 – x)7 using binomial theorem.
Answer:
Using Binomial Theorem, the expressions, (1 + 2x)6 and (1 – x)7, can be expanded as
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/168/5217/NS_14-10-08_Smita_11_Math_Miscellaneous%20Exe_Chapter%208_10_SU_SS_html_4724063d.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/168/5217/NS_14-10-08_Smita_11_Math_Miscellaneous%20Exe_Chapter%208_10_SU_SS_html_595ae41c.gif)
The complete multiplication of the two brackets is not required to be carried out. Only those terms, which involve x5, are required.
The terms containing x5 are
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/168/5217/NS_14-10-08_Smita_11_Math_Miscellaneous%20Exe_Chapter%208_10_SU_SS_html_m63e9e10c.gif)
Thus, the coefficient of x5 in the given product is 171.
Question 4:
If a and b are distinct integers, prove that a – b is a factor of an – bn, whenever n is a positive integer.
[Hint: write an = (a – b + b)n and expand]
Answer:
In order to prove that (a – b) is a factor of (an – bn), it has to be proved that
an – bn = k (a – b), where k is some natural number
It can be written that, a = a – b + b
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/168/5218/NS_14-10-08_Smita_11_Math_Miscellaneous%20Exe_Chapter%208_10_SU_SS_html_m681aceaf.gif)
This shows that (a – b) is a factor of (an – bn), where n is a positive integer.
Question 5:
Evaluate
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/168/5589/Chapter%208_html_56514e41.gif)
Answer:
Firstly, the expression (a + b)6 – (a – b)6 is simplified by using Binomial Theorem.
This can be done as
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/168/5589/Chapter%208_html_7e88dc0a.gif)
Question 6:
Find the value of
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/168/5590/Chapter%208_html_m720bf876.gif)
Answer:
Firstly, the expression (x + y)4 + (x – y)4 is simplified by using Binomial Theorem.
This can be done as
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/168/5590/Chapter%208_html_m348a1bf.gif)
Question 7:
Find an approximation of (0.99)5 using the first three terms of its expansion.
Answer:
0.99 = 1 – 0.01
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/168/5221/NS_14-10-08_Smita_11_Math_Miscellaneous%20Exe_Chapter%208_10_SU_SS_html_5670c523.gif)
Thus, the value of (0.99)5 is approximately 0.951.
Question 8:
Find n, if the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of ![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/168/5591/Chapter%208_html_m3edad97c.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/168/5591/Chapter%208_html_m3edad97c.gif)
Answer:
In the expansion,
,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/168/5591/Chapter%208_html_53d5b14b.gif)
Fifth term from the beginning ![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/168/5591/Chapter%208_html_m53e4e1cd.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/168/5591/Chapter%208_html_m53e4e1cd.gif)
Fifth term from the end ![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/168/5591/Chapter%208_html_m35a17a21.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/168/5591/Chapter%208_html_m35a17a21.gif)
Therefore, it is evident that in the expansion of
, the fifth term from the beginning is
and the fifth term from the end is
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/168/5591/Chapter%208_html_m52204e8e.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/168/5591/Chapter%208_html_2c3e0675.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/168/5591/Chapter%208_html_68211c8f.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/168/5591/Chapter%208_html_m464d9448.gif)
It is given that the ratio of the fifth term from the beginning to the fifth term from the end is
. Therefore, from (1) and (2), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/168/5591/Chapter%208_html_479957b.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/168/5591/Chapter%208_html_m1d463ce2.gif)
Thus, the value of n is 10.
Page No 176:
Question 9:
Expand using Binomial Theorem
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/168/5592/Chapter%208_html_m7f596d86.gif)
Answer:
Using Binomial Theorem, the given expression
can be expanded as
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/168/5592/Chapter%208_html_16cda96.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/168/5592/Chapter%208_html_5fc34bfc.gif)
Again by using Binomial Theorem, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/168/5592/Chapter%208_html_78af9023.gif)
From (1), (2), and (3), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/168/5592/Chapter%208_html_54800fb3.gif)
Question 10:
Find the expansion of
using binomial theorem.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/168/5593/Chapter%208_html_736f9e8f.gif)
Answer:
Using Binomial Theorem, the given expression
can be expanded as
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/168/5593/Chapter%208_html_736f9e8f.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/168/5593/Chapter%208_html_75825dd7.gif)
Again by using Binomial Theorem, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/168/5593/Chapter%208_html_264e91a9.gif)
From (1) and (2), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/168/5593/Chapter%208_html_m40b6a80c.gif)