.

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.