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**Miscellaneous Exercise**

**1. Find **** and **** in the expansion of **** if the first three terms of the expansion are 729, 7290 and 30375 respectively.**

**Ans. **Given:

…..(i)

…..(ii)

And

…..(iii)

Multiplying eq. (i) and eq. (iii),

…..(iv)

Squaring both sides in eq. (ii),

…..(v)

Dividing eq. (iv) by eq. (v),

From eq. (i),

And From eq. (ii),

Therefore, , and

**2. Find **** if the coefficient of **** and **** in the expansion of **** are equal.**

**Ans. **Here

Coefficient of = and Coefficient of =

According to question,

=

**3. Find the coefficient of **** in the product **** using binomial theorem.**

**Ans. **Using Binomial Theorem,

=

Coefficient of in the product

=

=

**4. If **** and **** are distinct integers, prove that **** is a factor of **** whenever **** is a positive integer.**

**Ans. **We have,

which shows that is a factor of

**5. Evaluate: **

**Ans. **Given:

Using Binomial Theorem,

=

=

=

=

**6. Find the value of **

**Ans. **Putting and we have

=

=

= =

= =

= =

**7. Find an approximation of **** using the first three terms of its expansion.**

**Ans. **Here =

= = = 0.951

**8. Find **** if the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of **** is **

**Ans. **Here

5^{th} term from the beginning =

5^{th} term from the end = term from beginning = term from the beginning

=

Now,

**9. Expand using binomial theorem **

**Ans. **We have

=

=

=

=

**10. Find the expansion of **** using binomial theorem.**

**Ans. **Here

=

=

=

=

=

Thanks for help our study

Thknuu

Thanks…..this really help a lot.