Question 1: Find the term from the beginning and the
term from the end in the expansion of
.
Answer:
Given expression:
Therefore there are terms in the expansion.
term from the end would be
i.e.
term from the beginning for expression
We know that for
Therefore
Also
Question 2: Find the term in the expansion
.
Answer:
Given expression:
We know that for
Question 3: Find the terms from the end in the expansion of
.
Answer:
Given expression:
Therefore there are terms in the expansion.
term from the end would be
i.e.
term from the beginning for expression
Question 4: Find the term in the expansion of
.
Answer:
Given expression: .
Question 5: Find the term in the expansion of
.
Answer:
Given expression:
Question 6: Find the term in the beginning and the
term from the end in the expansion
.
Answer:
Given expression:
Therefore there are terms in the expansion.
term from the end would be
i.e.
term from the beginning for expression
Question 7: Find the term from the end in the expansion of
.
Answer:
Given expression:
Therefore there are terms in the expansion.
term from the end would be
i.e.
term from the beginning for expression
Question 8: Find the term from the end in the expansion of
.
Answer:
Given expression:
Therefore there are terms in the expansion.
term from the end would be
i.e.
term from the beginning for expression
Question 9: Find the coefficient of:
i) in the expansion of
ii) in the expansion of
iii) in the expansion of
iv) in the expansion of
v) in the expansion of
vi) in the expansion of
vii) in the expansion of
viii) in the expansion of
Answer:
i) The given expression:
.
Let term has
Therefore
Therefore coefficient of
ii) The given expression:
.
Let term has
Therefore
Therefore coefficient of
iii) The given expression:
Let term has
Therefore
Therefore coefficient of
iv) The given expression:
.
Let term has
Therefore
Therefore coefficient of
v) The given expression:
.
Let term has
Therefore
Therefore coefficient of
vi) Given expression:
Therefore terms with
Hence the coefficient of the term with
vii) Given expression:
Therefore coefficient of
viii) Given expression:
Therefore coefficient of
Question 10: Which term in the expansion of contains
and
to one and the same power?
Answer:
Given expression:
Therefore
Therefore the required term term.
Question 11: Does the expansion of
contain any term involving
.
Answer:
General expression:
Suppose occurs in the given expression at the
term.
Fr the term to contain ,
Since can only be an integer, this is not possible. Hence there is no term in the expansion of
that contains
.
Question 12: Show that the expansion of
does not contain any term involving
.
Answer:
Given expression:
Suppose occurs in the given expression at the
term.
For the term to contain ,
Since 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:
i)
ii)
iii)
iv)
Answer:
i)
Here is even. Therefore
i.e.
term is the middle term.
ii)
Here is even. Therefore
i.e.
term is the middle term.
iii)
Here is even. Therefore
i.e.
term is the middle term.
iv)
Here is even. Therefore
i.e.
term is the middle term.
Question 14: Find the middle term in the expansion of:
i)
ii)
iii)
iv)
Answer:
i) Given expression:
Here is odd. Therefore
and
i.e.
and
terms are the middle term of the given expression.
ii) Given expression:
Here is odd. Therefore
and
i.e.
and
terms are the middle term of the given expression.
iii) Given expression:
Here is odd. Therefore
and
i.e.
and
terms are the middle term of the given expression.
iv) Given expression:
Here is odd. Therefore
and
i.e.
and
terms are the middle term of the given expression.
Question 15: Find the middle term in the expansion of:
i)
ii)
iii)
iv)
v)
vi)
vii)
viii)
ix)
x)
Answer:
i) Given expression:
Here is even. Therefore
i.e.
term is the middle term.
ii) Given expression:
Here is even. Therefore
i.e.
term is the middle term.
iii) Given expression:
Here is even. Therefore
i.e.
term is the middle term.
iv) Given expression:
Here is odd. Therefore
and
i.e.
and
terms are the middle term of the given expression.
v) Given expression:
Here is odd. Therefore
and
i.e.
and
terms are the middle term of the given expression.
vi) Given expression:
Here is even. Therefore
i.e.
term is the middle term.
vii) Given expression:
Here is odd. Therefore
and
i.e.
and
terms are the middle term of the given expression.
viii) Given expression:
Here is even. Therefore
i.e.
term is the middle term.
ix) Given expression:
Here is odd. Therefore
and
i.e.
and
terms are the middle term of the given expression.
x) Given expression:
Here is even. Therefore
i.e.
term is the middle term.
Question 16: Find the term independent of in the expansion of the following expressions:
i)
ii)
iii)
iv)
v)
vi)
vii)
viii)
ix)
x)
Answer:
i) Given expression:
Let term be independent of
Therefore
If is to be independent of
, then
Therefore the
term.
Therefore the required term
ii) Given expression:
Let term be independent of
Therefore
If is to be independent of
, then
Therefore the
term.
Therefore the required term
iii) Given expression:
Let term be independent of
Therefore
If is to be independent of
, then
Therefore the
term.
Therefore the required term
iv) Given expression:
Let term be independent of
Therefore
If is to be independent of
, then
Therefore the
term.
Therefore the required term
v) Given expression:
Let term be independent of
Therefore
If is to be independent of
, then
Therefore the
term.
Therefore the required term
vi) Given expression:
Let term be independent of
If is to be independent of
, then,
i.e
term
vii)
Let term be independent of
If is to be independent of
, then,
Therefore the
term.
viii) Given expression:
Now
Let term be independent of
. Therefore it is the
term.
Coefficient of the term
Let term be independent of
. This is not possible. Hence there is no term with
in the expansion.
If is to be independent of
, then,
. Therefore it is the
term.
Coefficient of the term
Therefore the coefficient of the term independent of
ix) Given expression:
Let term be independent of
If is to be independent of
, then,
Therefore it is the
term.
x) Given expression:
Let term be independent of
If is to be independent of
, then,
. Therefore it is the
term.
Question 17: If the coefficients of and
terms in the expansion of
are equal, find
.
Answer:
Given expression:
Given
Question 18: If the coefficient of and
term in the expansion of
are equal, find
.
Answer:
Given expression:
Given
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:
Given expression:
If the term is independent of , then
Therefore the term independent of is
Question 21: The coefficient of the and
terms in the expansion of
are in A.P. , find
.
Answer:
Given expression:
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:
Given expression:
Therefore