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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
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
=
=
=
=
=
Thanks for help our study
Thknuu
Thanks…..this really help a lot.