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
5th term from the beginning = 
5th 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 
= 
= 
= 
= 
= 
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