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

Hence proved.

Question 23: If the coefficient of and terms in the expansion of are in A.P., then find the value on .

Answer:

Given expression:

Therefore

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:

Given expression:

. Hence proved.

Question 25: Find , if the coefficients of and in the expansion of are equal.

Answer:

Given expression:

Therefore coefficient of

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)

Similarly,

… … … … … 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 expression:

Given:

We have to prove:

… … … … … i)

Substituting the values in i) we get

We know

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

Given:

We have to prove:

… … … … … i)

We know:

Substituting in i) we get

We know

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)

Similarly,

… … … … … ii)

Subtracting ii) from i) we get

Question 32: If the and in the expansion of are respectively and , find .

Answer:

Given expression:

Given and

Now,

… … … … … i)

Similarly,

… … … … … ii)

From i) and ii)

Therefore

Substituting we get,

Substituting in i) we get

Question 33: If the and term in the expansion of are and respectively, find .

Answer:

Given expression:

Given and

Now,

… … … … … i)

Similarly,

… … … … … ii)

From i) and ii) we get

Substituting in i) we get

… … … … … iii)

Now

From iii)

Question 34: Find and in the expansion , if the first three terms in the expansion are and respectively.

Answer:

Given expression:

Given:

Similarly,

… … … … … i)

Similarly,

… … … … … ii)

Dividing ii) by i) we get

Since

Question 35: If the term free from in the expansion is , find the value of .

Answer:

Given expression:

Let term is independent of

Therefore

If is independent of , then

. Therefore it is the term.

Question 36: Find the term in the expansion , if the binomial coefficient of the third term from the end is .

Answer:

Given expression:

Third term from the end for is third term from the beginning for the expression

Given

or (not possible as cannot ne negative)

Therefore

Therefore the term from the beginning is

Question 37: If is a real number and if the middle term in the expansion of is , find .

Answer:

Given expression:

Middle term i.e. term

Question 38: Find in the binomial , if the ratio of term from the beginning to the term from the end is .

Answer:

Given expression:

term from the end term from the beginning.

Given

Question 39: If the term from the beginning and from the end in the binomial expansion are equal, find .

Answer:

Given expression:

term from the end term from the beginning.

Since,