.
Answer:
term from the end would be
i.e.
term from the beginning for expression
.
Answer:
.
Answer:
term from the end would be
i.e.
term from the beginning for expression
Question 4: Find the term in the expansion of
.
Answer:
.
.
Answer:
.
Answer:
term from the end would be
i.e.
term from the beginning for expression
.
Answer:
term from the end would be
i.e.
term from the beginning for expression
.
Answer:
term from the end would be
i.e.
term from the beginning for expression
Question 9: Find the coefficient of:
Answer:
.
.
.
.
Hence the coefficient of the term with
Answer:
Therefore the required term term.
.
Answer:
Suppose occurs in the given expression at the
term.
,
can only be an integer, this is not possible. Hence there is no term in the expansion of
that contains
.
Answer:
Suppose occurs in the given expression at the
term.
,
can only be an integer, this is not possible. Hence there is no term in the expansion of
.
Question 13: Find the middle term in the expansion of:
Answer:
Question 14: Find the middle term in the expansion of:
Answer:
terms are the middle term of the given expression.
terms are the middle term of the given expression.
terms are the middle term of the given expression.
terms are the middle term of the given expression.
Question 15: Find the middle term in the expansion of:
Answer:
terms are the middle term of the given expression.
terms are the middle term of the given expression.
terms are the middle term of the given expression.
terms are the middle term of the given expression.
Question 16: Find the term independent of in the expansion of the following expressions:
Answer:
term be independent of
term be independent of
term be independent of
term be independent of
term be independent of
term be independent of
,
i.e
term be independent of
,
term be independent of
.
Coefficient of the term
term be independent of
. This is not possible. Hence there is no term with
in the expansion.
.
Coefficient of the term
Therefore the coefficient of the term independent of
term be independent of
term.
term be independent of
,
.
term.
Question 17: If the coefficients of and
terms in the expansion of
are equal, find
.
Answer:
Question 18: If the coefficient of and
term in the expansion of
are equal, find
.
Answer:
Question 19: Prove that the coefficient of term in the expansion of
is equal to the sum of the coefficients of the
and
terms in the expansion of
.
Answer:
For expression:
For expression:
Sum of the coefficients
Hence proven.
Question 20: Prove that the term independent of in the expansion of
is
.
Answer:
If the term is independent of
Therefore the term independent of is
Question 21: The coefficient of the and
terms in the expansion of
are in A.P. , find
.
Answer:
Coefficient of
Coefficient of
Coefficient of
Since they are in AP
Question 22: If the coefficient of and
terms in the expansion of
are in A.P., show that
.
Answer:
Hence proved.
Question 23: If the coefficient of and
terms in the expansion of
are in A.P., then find the value on
.
Answer:
is not possible as then
in the
term.
Hence .
Question 24: If in the expansion of , the coefficients of
and
terms are equal, prove the
where
.
Answer:
. Hence proved.
Question 25: Find if the coefficients of
and
are equal.
Answer:
Given expression:
and coefficient of
Question 26: Find the coefficient of in the product
using binomial theorem.
Answer:
Given expression:
coefficient of
Question 27: In the expansion of the binomial coefficients of three consecutive terms are respectively
and
find the value of
.
Answer:
Let the consecutive terms be
The binomial coefficient for these terms will be respectively.
It is given, and
… … … … … i)
… … … … … ii)
From i) and ii) we get
Question 28: In the expansion of the coefficients of three consecutive terms are respectively
and
then find
and the position of the terms of these coefficients.
Answer:
Let the consecutive terms be
The binomial coefficient for these terms will be respectively.
We have
Now, and
Dividing
Therefore the required terms are and
.
Question 29: If and
terms in the expansion
be respectively
and
prove that
.
Answer:
Given:
We have to prove:
… … … … … i)
Substituting the values in i) we get
We k
Therefore LHS RHS. Hence proved.
Question 30: If and
in any binomial expansion be the
and
terms respectively, then prove that
.
Answer:
Let the expression be
We have to prove:
… … … … … i)
We know:
Substituting in i) we get
We k
Therefore LHS RHS. Hence proved.
Question 31: If the coefficient of three consecutive terms in the expansion be
and
find
.
Answer:
Let the three consecutive terms be
Coefficient of
Coefficient of
Coefficient of
Now,
… … … … … i)
… … … … … ii)
Subtracting ii) from i) we get
Question 32: If the and
are respectively
and
, find
.
Answer:
and
Now,
… … … … … i)
… … … … … ii)
From i) and ii)
Substituting we get,
Substituting in i) we get
Question 33: If the and
term in the expansion of
are
and
respectively, find
.
Answer:
and
Now,
… … … … … i)
… … … … … ii)
From i) and ii) we get
Substituting in i) we get
… … … … … iii)
From
Question 34: Find and
in the expansion
if the first three terms in the expansion are
and
Answer:
… … … … … i)
… … … … … ii)
Dividing ii) by i) we get
.
Answer:
Question 36: Find the term in the expansion
if the binomial coefficient of the third term from the end is
.
Answer:
Third term from the end for is third term from the beginning for the expression
(not possible as
cannot ne negative)
term from the beginning is
Question 37: If is a real number and if the middle term in the expansion of
Answer:
i.e.
term
.
Answer:
term from the end
term from the beginning.
Question 39: If the term from the beginning and from the end in the
.
Answer:
term from the end
term from the beginning.