RD Sharma Class 11 Solutions Chapter - Binomial Theorem

 Page 1


18. Binomial Theorem
Very Short Answer
1. Question
Write the number of terms in the expansion of .
Answer
Given:
 (1)
 (2)
Add both equations;
The even terms; i.e. k=1,3,5,7 & 9 cancel each other
So, we are left with only terms with k=0,2,4,6,8 &10
So total number of terms = 6
2. Question
Write the sum of the coefficients in the expansion .
Answer
Given:
Page 2


18. Binomial Theorem
Very Short Answer
1. Question
Write the number of terms in the expansion of .
Answer
Given:
 (1)
 (2)
Add both equations;
The even terms; i.e. k=1,3,5,7 & 9 cancel each other
So, we are left with only terms with k=0,2,4,6,8 &10
So total number of terms = 6
2. Question
Write the sum of the coefficients in the expansion .
Answer
Given:
(1-3x+x
2
 )
111
For sum of coefficients; put x=1
We have;
(1-3+1)
111
=(-1)
111
= -1
3. Question
Write the number of terms in the expansion of .
Answer
Given:
Highest power is 
And lowest power is 
So the expansion contains all the terms ranging from 0 to 24
Therefore, total number of terms = 25
4. Question
Write the middle term in the expansion of .
Answer
Given:
Total number of terms = n+1 = 11
So middle term = 6
th
 term, i.e. k=5
=252
5. Question
Which term is independent of x, in the expansion of ?
Answer
Given:
Page 3


18. Binomial Theorem
Very Short Answer
1. Question
Write the number of terms in the expansion of .
Answer
Given:
 (1)
 (2)
Add both equations;
The even terms; i.e. k=1,3,5,7 & 9 cancel each other
So, we are left with only terms with k=0,2,4,6,8 &10
So total number of terms = 6
2. Question
Write the sum of the coefficients in the expansion .
Answer
Given:
(1-3x+x
2
 )
111
For sum of coefficients; put x=1
We have;
(1-3+1)
111
=(-1)
111
= -1
3. Question
Write the number of terms in the expansion of .
Answer
Given:
Highest power is 
And lowest power is 
So the expansion contains all the terms ranging from 0 to 24
Therefore, total number of terms = 25
4. Question
Write the middle term in the expansion of .
Answer
Given:
Total number of terms = n+1 = 11
So middle term = 6
th
 term, i.e. k=5
=252
5. Question
Which term is independent of x, in the expansion of ?
Answer
Given:
? x
(9-k-2k)
=x
0
? 9-3k=0
? k=3
So 4
th
 term is independent of x.
6. Question
If a and b denote respectively the coefficients of and  in the expansion of , then write the
relation between a and b.
Answer
Given:
Coefficient of ;k=m
 …. (1)
Coefficient of ;k=n
 …. (2)
Divide both equations;
We get;
a=b
7. Question
If a and b are coefficients of in the expansion of  and  respectively, then write the
relation between a and b.
Answer
Given:
Page 4


18. Binomial Theorem
Very Short Answer
1. Question
Write the number of terms in the expansion of .
Answer
Given:
 (1)
 (2)
Add both equations;
The even terms; i.e. k=1,3,5,7 & 9 cancel each other
So, we are left with only terms with k=0,2,4,6,8 &10
So total number of terms = 6
2. Question
Write the sum of the coefficients in the expansion .
Answer
Given:
(1-3x+x
2
 )
111
For sum of coefficients; put x=1
We have;
(1-3+1)
111
=(-1)
111
= -1
3. Question
Write the number of terms in the expansion of .
Answer
Given:
Highest power is 
And lowest power is 
So the expansion contains all the terms ranging from 0 to 24
Therefore, total number of terms = 25
4. Question
Write the middle term in the expansion of .
Answer
Given:
Total number of terms = n+1 = 11
So middle term = 6
th
 term, i.e. k=5
=252
5. Question
Which term is independent of x, in the expansion of ?
Answer
Given:
? x
(9-k-2k)
=x
0
? 9-3k=0
? k=3
So 4
th
 term is independent of x.
6. Question
If a and b denote respectively the coefficients of and  in the expansion of , then write the
relation between a and b.
Answer
Given:
Coefficient of ;k=m
 …. (1)
Coefficient of ;k=n
 …. (2)
Divide both equations;
We get;
a=b
7. Question
If a and b are coefficients of in the expansion of  and  respectively, then write the
relation between a and b.
Answer
Given:
Coefficient of ;k=n
 ….. (1)
Coefficient of x
n
;k=n
 ….. (2)
Divide both equations;
a=2b
8. Question
Write the middle term in the expansion of .
Answer
Given:
Total terms = n+1 =11
So middle term= 6
th
 term ; i.e. k=5
For k=5;
= 
10
C
5
9. Question
If a and b denote the sum of the coefficients in the expansions of and respectively,
then write the relation between a and b.
Page 5


18. Binomial Theorem
Very Short Answer
1. Question
Write the number of terms in the expansion of .
Answer
Given:
 (1)
 (2)
Add both equations;
The even terms; i.e. k=1,3,5,7 & 9 cancel each other
So, we are left with only terms with k=0,2,4,6,8 &10
So total number of terms = 6
2. Question
Write the sum of the coefficients in the expansion .
Answer
Given:
(1-3x+x
2
 )
111
For sum of coefficients; put x=1
We have;
(1-3+1)
111
=(-1)
111
= -1
3. Question
Write the number of terms in the expansion of .
Answer
Given:
Highest power is 
And lowest power is 
So the expansion contains all the terms ranging from 0 to 24
Therefore, total number of terms = 25
4. Question
Write the middle term in the expansion of .
Answer
Given:
Total number of terms = n+1 = 11
So middle term = 6
th
 term, i.e. k=5
=252
5. Question
Which term is independent of x, in the expansion of ?
Answer
Given:
? x
(9-k-2k)
=x
0
? 9-3k=0
? k=3
So 4
th
 term is independent of x.
6. Question
If a and b denote respectively the coefficients of and  in the expansion of , then write the
relation between a and b.
Answer
Given:
Coefficient of ;k=m
 …. (1)
Coefficient of ;k=n
 …. (2)
Divide both equations;
We get;
a=b
7. Question
If a and b are coefficients of in the expansion of  and  respectively, then write the
relation between a and b.
Answer
Given:
Coefficient of ;k=n
 ….. (1)
Coefficient of x
n
;k=n
 ….. (2)
Divide both equations;
a=2b
8. Question
Write the middle term in the expansion of .
Answer
Given:
Total terms = n+1 =11
So middle term= 6
th
 term ; i.e. k=5
For k=5;
= 
10
C
5
9. Question
If a and b denote the sum of the coefficients in the expansions of and respectively,
then write the relation between a and b.
Answer
Given:
(1-3x+10x
2
)
n
Sum of coefficients = a
a 
=(2
3
)
n
=(2
n
)
3
(1+x
2
)
n
Sum of coefficients = b
b= (1+1)
n
=2
n
Put value of b in a; we get:
a=b
3
10. Question
Write the coefficient of the middle term in the expansion of (1 + x)
2n
Answer
Given:
(1 + x)
2n
Total terms = 2n+1
Middle term = (2n+1)/2
i.e. (n+1)th term
so k=n
= 
2n
C
n
11. Question
Write the number of terms in the expansion of {(2x + y
3
)
4
}
7
Answer
Given:
{(2x + y
3
)
4
}
7
 = (2x + y
3
)
28