Binomial Theorem Class 11 Ncert Solutions Miscellaneous
NCERT Solutions for Class 11 Maths Chapter 8 – Binomial Theorem Miscellaneous Exercise
Page No 175:
Question 1:
Finda,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, (T r +1), in the binomial expansion of (a+b) n is given by
.
The first three terms of the expansion are given as 729, 7290, and 30375 respectively.
Therefore, we obtain
Dividing (2) by (1), we obtain
Dividing (3) by (2), we obtain
From (4) and (5), we obtain
Substitutingn = 6 in equation (1), we obtain
a 6 = 729
From (5), we obtain
Thus,a = 3,b = 5, andn = 6.
Question 2:
Finda if the coefficients ofx 2 andx 3 in the expansion of (3 +ax)9 are equal.
Answer:
It is known that (r+ 1)th term, (T r +1), in the binomial expansion of (a+b) n is given by
.
Assuming thatx 2 occurs in the (r + 1)th term in the expansion of (3 +ax)9, we obtain
Comparing the indices ofx inx 2 and inT r + 1, we obtain
r = 2
Thus, the coefficient ofx 2 is
Assuming thatx 3 occurs in the (k + 1)th term in the expansion of (3 +ax)9, we obtain
Comparing the indices ofx inx 3 and inT k + 1, we obtain
k= 3
Thus, the coefficient ofx 3 is
It is given that the coefficients ofx 2 andx 3 are the same.
Thus, the required value ofa is
.
Question 3:
Find the coefficient ofx 5 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
The complete multiplication of the two brackets is not required to be carried out. Only those terms, which involvex 5, are required.
The terms containingx 5 are
Thus, the coefficient ofx 5 in the given product is 171.
Question 4:
If a andb are distinct integers, prove thata –b is a factor ofa n –b n , whenevern is a positive integer.
[Hint: writea n = (a – b+b) n and expand]
Answer:
In order to prove that (a –b) is a factor of (a n –b n ), it has to be proved that
a n –b n =k (a –b), wherek is some natural number
It can be written that,a =a –b +b
This shows that (a –b) is a factor of (a n –b n ), wheren is a positive integer.
Question 5:
Evaluate
.
Answer:
Firstly, the expression (a +b)6 – (a –b)6 is simplified by using Binomial Theorem.
This can be done as
Question 6:
Find the value of
.
Answer:
Firstly, the expression (x +y)4 + (x –y)4 is simplified by using Binomial Theorem.
This can be done as
Question 7:
Find an approximation of (0.99)5 using the first three terms of its expansion.
Answer:
0.99 = 1 – 0.01
Thus, the value of (0.99)5 is approximately 0.951.
Question 8:
Findn, if the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of
Answer:
In the expansion,
,
Fifth term from the beginning
Fifth term from the end
Therefore, it is evident that in the expansion of
, the fifth term from the beginning is
and the fifth term from the end is
.
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
Thus, the value ofn is 10.
Page No 176:
Question 9:
Expand using Binomial Theorem
.
Answer:
Using Binomial Theorem, the given expression
can be expanded as
Again by using Binomial Theorem, we obtain
From (1), (2), and (3), we obtain
Question 10:
Find the expansion of
using binomial theorem.
Answer:
Using Binomial Theorem, the given expression can be expanded as
Again by using Binomial Theorem, we obtain
From (1) and (2), we obtain
Binomial Theorem Class 11 Ncert Solutions Miscellaneous
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