Binomial Theorem
Class 11th Mathematics RD Sharma Solution
- Write the number of terms in the expansion of ( 2 + root {3}x ) ^{10} + ( 2 -…
- Write the sum of the coefficients in the expansion ( 1-3x+x^{2} ) ^{111} .…
- Write the number of terms in the expansion of ( 1-3x+3x^{2} - x^{3} ) ^{8} .…
- Write the middle term in the expansion of ( { 2x^{2} }/{3} + frac {3}/{ 2x^{2} } )…
- Which term is independent of x, in the expansion of ( x - {1}/{ 3x^{2} } ) ^{9} ?…
- If a and b denote respectively the coefficients of x^{m} and x^{n} in the…
- If a and b are coefficients of x^{n} in the expansion of (1+x)^{2n} and…
- Write the middle term in the expansion of ( x + {1}/{x} ) ^{10} .…
- If a and b denote the sum of the coefficients in the expansions of ( 1-3x+10x^{2} )…
- Write the coefficient of the middle term in the expansion of (1 + x)2n…
- Write the number of terms in the expansion of {(2x + y3)4}7
- Find the sum of coefficients of two middle terms in the binomial expansion of (1 +…
- Find the ratio of the coefficients of x^{p} and x^{q} in the expansion of…
- Write last two digits of the number 3400.
- Find the number of terms in the expansion of (a + b + c)n.
- If a and b are the coefficients of x^{n} in the expansions (1+x)^{2n} and…
- Write the total number of terms in the expansion of (x+a)^{100} + (x-a)^{100} .…
- If (1 – x + x2)n = a0 + a1x + a2x2 + … + a2nx2n, find the value of a0 + a2 + a4 + … +…
- If in the expansion of (1+x)^{20} , the coefficient of rth and (r +4) th terms are…
- The term without x in the expansion of ( 2x - {1}/{ 2x^{2} } ) ^{12} is Mark the…
- If rth term in the expansion of ( 2x^{2} - {1}/{x} ) ^{12} is without x, then r is…
- If in the expansion of (a+b)^{n} and (a+b)^{n+3} , the ratio of the coefficients…
- If A and B are the sums of odd and even terms respectively in the expansion of…
- The number of irrational terms in the expansion of (4^{1/5}+7^{1/10})^{45} is Mark…
- The coefficient of x-17 in the expansion of ( x^{4} - {1}/{ x^{3} } ) ^{15} is…
- In the expansion of ( x^{2} - {1}/{3x} ) ^{9} , the term without x is equal to…
- If in the expansion of (1+x)^{15} , the coefficients of (21+3)^ and (1-1)^…
- The middle term in the expansion of ( { 2x^{2} }/{3} + frac {3}/{ 2x^{2} } ) ^{10}…
- If in the expansion of ( x^{4} - {1}/{3} ) ^{15} , x^{-17} occurs in rth term,…
- In the expansion of ( x - {1}/{ 3x^{2} } ) ^{9} , the term independent of x is…
- If in the expansion of (1 + y)n, the coefficients of 5th, 6th and 7th terms are in…
- In the expansion of ( {1}/{2} x^{1/3}+x^{-1/5} ) ^{8} , the term independent of x…
- If the sum of odd numbered terms and the sum of even numbered terms in the expansion…
- If the coefficient of x in ( x^{2} + { lambda }/{x} ) ^{5} is 270, then lambda…
- The coefficient of x^{4} in ( {x}/{2} - frac {3}/{2} ) ^{10} is. Mark the…
- The total number of terms in the expansion of (x+a)^{100} + (x-a)^{100} after…
- If t_{2}/t_{3} in the expansion of (a+b)^{n} and t_{3}/t_{4} in the expansion of…
- The coefficient of {1}/{x} in the expansion of (1+x)^{n} ( 1 + {1}/{x} ) ^{n}…
- If the sum of the binomial coefficients of the expansion ( 2x + {1}/{x} ) ^{n} is…
- If the fifth term of the expansion ( a^{2/3}+a^{-1} ) ^{n} does not contain ‘a’.…
- The coefficient of x^{-3} in the expansion of ( x - {m}/{x} ) ^{11} is Mark the…
- The coefficient of the term independent of x in the expansion of ( ax + {b}/{x} )…
- The coefficient of x^{5} in the expansion of (1+x)^{21} + (1+x)^{22} + l. s +…
- The coefficient of x^{8}y^{10} in the expansion (x+y)^{18} is. Mark the correct…
- If the coefficients of the (n+1)^{th} term and the (n+3)^{th} term in the…
- If the coefficients of 2nd, 3rd and 4th terms in the expansion of (1+x)^{n} , n inn…
- The middle term in the expansion of ( {2x}/{3} - frac {3}/{ 2x^{2} } ) ^{2n} is.…
- If rth term is the middle term in the expansion of ( x^{2} - {1}/{2x} ) ^{20} ,…
- The number of terms with integral coefficients in the expansion of…
- Constant term in the expansion of ( x - {1}/{x} ) ^{10} is Mark the correct…
- If the coefficients of x2 and x3 in the expansion of (3+ax)^{9} are the same, then…
Very Short Answer
Question 1.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
Question 2.
Write the sum of the coefficients in the expansion
.
Answer:
Given:
(1-3x+x2 )111
For sum of coefficients; put x=1
We have;
(1-3+1)111=(-1)111
= -1
Question 3.
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
Question 4.
Write the middle term in the expansion of 
.
Answer:
Given:
Total number of terms = n+1 = 11
So middle term = 6th term, i.e. k=5
=252
Question 5.
Which term is independent of x, in the expansion of 
?
Answer:
Given:
⇒ x(9-k-2k)=x0
⇒ 9-3k=0
⇒ k=3
So 4th term is independent of x.
Question 6.
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
Question 7.
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 xn;k=n
….. (2)
Divide both equations;
a=2b
Question 8.
Write the middle term in the expansion of 
.
Answer:
Given:
Total terms = n+1 =11
So middle term= 6th term ; i.e. k=5
For k=5;
= 10C5
Question 9.
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+10x2)n
Sum of coefficients = a
a 
=(23)n
=(2n)3
(1+x2)n
Sum of coefficients = b
b= (1+1)n
=2n
Put value of b in a; we get:
a=b3
Question 10.
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
= 2nCn
Question 11.
Write the number of terms in the expansion of {(2x + y3)4}7
Answer:
Given:
{(2x + y3)4}7 = (2x + y3)28
So total number of terms = n+1
= 28+1
= 29
Question 12.
Find the sum of coefficients of two middle terms in the binomial expansion of (1 + x)2n-1
Answer:
Given:
Total terms after expansion = 2n-1+1=2n
Middle term = 2n/2 = nth term
So two required middle terms are : nth & (n+1)th term
k= (n-1) & n for both terms respectively.
(1 + x)2n-1
Coefficient of nth term;
=2n-1Cn-1
Coefficient of (n+1)th term ;
= 2n-1Cn
Sum of coefficients = 2n-1Cn-1 + 2n-1Cn
= 2n-1+1Cn
=2nCn
Question 13.
Find the ratio of the coefficients of 
and 
in the expansion of 
.
Answer:
Given:
For xp; k=p
Coefficient = p+qCp (1)
For xq; k=q
Coefficient = p+qCq (2)
Divide both equations;
Question 14.
Write last two digits of the number 3400.
Answer:
Given:
3400
By binomial expansion, 
+
=1-2000+ 102 {I}
=1+100(I-20)
So, the last two digits would be 01.
Question 15.
Find the number of terms in the expansion of (a + b + c)n.
Answer:
Given:
Tn = 
;
Where p + q + r = n
Since number of ways in which we can divide n different things into r different things is : n+r-1Cr-1
Here, n=n & r=3
So, n+3-1C3-1 = n+2C2
so, the number of terms 
Question 16.
If a and b are the coefficients of in the expansions and respectively, find 
.
Answer:
Given:
Coefficient of ;k=n
(1)
Coefficient of ;k=n
(2)
Divide both equations;
a=2b
Question 17.
Write the total number of terms in the expansion of 
.
Answer:
Given:
So odd powers of x cancel each other, we are left with even powers of x or say odd terms of expansion.
So total number of terms are T1,T3,…T99,T101
=51
Question 18.
If (1 – x + x2)n = a0 + a1x + a2x2 + … + a2nx2n, find the value of a0 + a2 + a4 + … + a2n.
Answer:
(1 – x + x2)n = a0 + a1x + a2x2 + … + a2nx2n
At x = 1
(1 – 1 + 12)n = a0 + a1(1) + a2(1)2 + … + a2n(1)2n
a0 + a1 + a2 + … + a2n = 1 …(1)
At x = -1
(1 – (-1) + (-1)2)n = a0 + a1(-1) + a2(-1)2 + … + a2n(-1)2n
a0 - a1 + a2 - … + a2n = 3n …(2)
On adding eq.1 and eq.2
(a0 + a1 + a2 + … + a2n) + (a0 - a1 + a2 - … + a2n) = 1 + 3n
2(a0 + a2 + a4 + … + a2n) = 1 + 3n
a0 + a2 + a4 + … + a2n = 
Mcq
Question 1.Mark the correct alternative in the following :
If in the expansion of 
, the coefficient of rth and (r +4) th terms are equal, then r is equal to
A.7
B. 8
C. 9
D. 10
Answer:
Given:
In rth term; k=r-1
& in (r+4)th term ; k=r+3
So, the terms are;
& 
Coefficients of both terms are equal:
]
So, r= (21-r);
(r+1)= (20-r);
(r+2)= (19-r);
(r+3)= (18-r)
We get;
r=9
Question 2.
Mark the correct alternative in the following :
The term without x in the expansion of 
is
A.495
B. -495
C. -7920
D. 7920
Answer:
Given:
The term without x is where :
12-3k=0
k=4
for k=4; the term is :
Question 3.
Mark the correct alternative in the following :
If rth term in the expansion of 
is without x, then r is equal to.
A.8
B. 7
C. 9
D. 10
Answer:
Given:
For term without x:
24-2k-k=0
24-3k=0
k=8
for k= 8;
term = 8+1=9th term
Question 4.
Mark the correct alternative in the following :
If in the expansion of 
and 
, the ratio of the coefficients of second and third terms, and third and fourth terms respectively are equal, then n is
A.3
B. 4
C. 5
D. 6
Answer:
Given:
T2
; T3
(1)
T3
; T4
(2)
Equating both equations:
2(n+1)=3(n-1)
2n+2 = 3n-3
n=5
Question 5.
Mark the correct alternative in the following :
If A and B are the sums of odd and even terms respectively in the expansion of 
, then 
is equal to
A.4 (A+B)
B. 4 (A – B)
C. AB
D. 4 AB
Answer:
Given:
So, 
= 2A
So, 
=2B
=2A
2B
=4AB
Question 6.
Mark the correct alternative in the following :
The number of irrational terms in the expansion of 
is
A. 40
B. 5
C. 41
D. None of these
Answer:
Given:
Total number of terms in expansion =n+1
=45+1
=46
irrational terms = total terms – rational terms
For rational terms; the power of each term should be integer.
Therefore, k must be divisible by 5 and (45-k) by 10.
i.e. the terms having power as multiples of 5.
i.e. 0,5,10,15,20,25,30,35,40 & 45
for k= 5,15,25,35 & 45;
(45-k) do not give an integral power, so these powers have to be rejected.
Now, we have k= 0,10,20,30 & 40 which give us rational terms.
Hence, irrational terms = 46-5 = 41
Question 7.
Mark the correct alternative in the following :
The coefficient of x-17 in the expansion of 
is
A.1365
B. -1365
C. 3003
D. -3003
Answer:
Given:
60-4k-3k = -17
-7k = -77
k= 11
Coefficient = -1365
Question 8.
Mark the correct alternative in the following :
In the expansion of 
, the term without x is equal to
A.
B. 
C. 
D. None of these
Answer:
Given:
⇒ x2(9 – k)-k = x0
⇒18-2k-k = 0
⇒18-3k = 0
⇒k = 6
Question 9.
Mark the correct alternative in the following :
If in the expansion of 
, the coefficients of 
and 
terms are equal, then the value of r is
A.5
B. 6
C. 4
D. 3
Answer:
Given:
For (2r+3)th term; k=(2r+2)
For (r-1)th term; k=r-2
Coefficients of both terms are equal;
Question 10.
Mark the correct alternative in the following :
The middle term in the expansion of 
is
A.251
B. 252
C. 250
D. None of these
Answer:
Given:
n= 10
Total number of terms on expansion = n+1 = 11
So middle term is 6th term; i.e. k=5
= 252
Question 11.
Mark the correct alternative in the following :
If in the expansion of 
occurs in rth term, then
A.r = 10
B. r = 11
C. r = 12
D. r = 13
Answer:
Given:
⇒ x4(15-k)-3k = x-17
⇒ 60-4k-3k = -17
⇒ -7k = -77
⇒ k= 11
So, the term is 12th term.
Question 12.
Mark the correct alternative in the following :
In the expansion of 
, the term independent of x is
A.T3
B. T4
C. T5
D. None of these
Answer:
Given:
⇒ x9-k-2k = x0
⇒ 9-3k = 0
⇒ k = 3
So, the term is 4th term.
Question 13.
Mark the correct alternative in the following :
If in the expansion of (1 + y)n, the coefficients of 5th, 6th and 7th terms are in A.P., then n is equal to
A.7, 11
B. 7, 14
C. 8, 16
D. None of these
Answer:
Given:
T5
; T6
& T7
Since T5 , T6 & T7 are in AP
Then; 2(T6) = T5 + T7
i.e. 
⇒ 30+(n-4) (n-5)-12(n-4)=0
⇒30+n2-9n+20-12n+48=0
⇒n2-21n+98=0
⇒ (n-7) (n-14)=0
⇒n=7,14
Question 14.
Mark the correct alternative in the following :
In the expansion of 
, the term independent of x is
A.T5
B. T6
C. T7
D. T8
Answer:
Given:
⇒40-5k-3k = 0
⇒40-8k =0
⇒ k = 5
So, the term is 6th term.
Question 15.
Mark the correct alternative in the following :
If the sum of odd numbered terms and the sum of even numbered terms in the expansion of are A and B respectively, then the value of 
is.
A.
B. 
C. 4 AB
D. None of these
Answer:
Given:
= A+B
= A-B
(x2-a2 )n= [(x+a)(x-a)]n
=(x+a)n (x-a)n
= (A+B) (A-B)
=A2-B2
Question 16.
Mark the correct alternative in the following :
If the coefficient of x in 
is 270, then 
=
A.3
B. 4
C. 5
D. None of these
Answer:
Given:
⇒ x2(5-k)-k = x1
⇒10-2k-k=1
⇒9-3k=0
⇒ k=3
for k=3;
⇒λ3=27
⇒λ=3
Question 17.
Mark the correct alternative in the following :
The coefficient of 
in 
is.
A.
B. 
C. 
D. None of these
Answer:
Given:
⇒ x10-k = x4
⇒ 10-k=4
⇒ k=6
for k=6;
So, the coefficient of 
Question 18.
Mark the correct alternative in the following :
The total number of terms in the expansion of 
after simplification is
A.202
B. 51
C. 50
D. None of these
Answer:
Given:
So odd powers of x cancel each other, we are left with even powers of x or say odd terms of expansion.
So total number of terms are T1,T3,…T99,T101
=51
Question 19.
Mark the correct alternative in the following :
If 
in the expansion of 
and 
in the expansion of are equal, then n =
A.3
B. 4
C. 5
D. 6
Answer:
Given:
T2 ; T3
(1)
T3 ; T4
(2)
Equating both equations:
⇒2(n+1)=3(n-1)
⇒ 2n+2 = 3n-3
⇒n=5
Question 20.
Mark the correct alternative in the following :
The coefficient of 
in the expansion of 
is.
A.
B. 
C. 
D. None of these
Answer:
Given:
For x-1;
⇒k-n=-1
⇒k=n-1
So, coefficient 
Question 21.
Mark the correct alternative in the following :
If the sum of the binomial coefficients of the expansion 
is equal to 256, then the term independent of x is
A.1120
B. 1020
C. 512
D. None of these
Answer:
Given:
Sum of binomial coefficients = 2n
=256
⇒2n=28
⇒ n=8
so total terms = n+1
=9
Middle term = 5th term; i.e. k=4
So, term independent of 
= 1120
Question 22.
Mark the correct alternative in the following :
If the fifth term of the expansion 
does not contain ‘a’. Then n is equal to
A.2
B. 5
C. 10
D. None of these
Answer:
Given:
Term 5 ; i.e. k=4:
⇒2n-8-12=0
⇒ n=10
Question 23.
Mark the correct alternative in the following :
The coefficient of 
in the expansion of 
is
A.
B. 
C. 
D. 
Answer:
Given:
⇒11-2k=-3
⇒14-2k=0
⇒ k =7
for k=7; coefficient is:
=-330m7
Question 24.
Mark the correct alternative in the following :
The coefficient of the term independent of x in the expansion of 
is
A.
B. 
C. 
D. 
Answer:
Given:
14-2k=0
k = 7
So, the coefficient is:
Question 25.
Mark the correct alternative in the following :
The coefficient of 
in the expansion of 
is.
A.
B. 
C. 
D. 
Answer:
Given:
(1+x)21+(1+x)22+...+(1+x)30
Coefficient of x5in any expansion = 
; i.e. nC5
So, coefficient of x5 in above expansion = 21C5 + 22C5 + 23C5 +…+ 30C5
Question 26.
Mark the correct alternative in the following :
The coefficient of 
in the expansion 
is.
A.
B. 
C. 
D. None of these
Answer:
Given:
For x8 y10; k=10
So coefficient is 18C10
Also 18C10 = 18C8
So coefficient = 18C8
Question 27.
Mark the correct alternative in the following :
If the coefficients of the 
term and the 
term in the expansion of 
are equal, then the value of n is
A.10
B. 8
C. 9
D. None of these
Answer:
Given:
For nth term ; k=n-1
So for (n+1)th term ; k= n
& for (n+3)th term ; k =n+2
Coefficients for the above terms are equal;
(20-n)(19-n) = (n+2)(n+1)
380-39n+n2 = n2+3n+2
42n-378=0
n=9
Question 28.
Mark the correct alternative in the following :
If the coefficients of 2nd, 3rd and 4th terms in the expansion of 
are in A.P., then n =
A.7
B. 14
C. 2
D. None of these
Answer:
Given:
T2
; T3
& T4
Since T2 , T3 & T4 are in AP
Then; 2(T3) = T2 + T4
i.e. 
(n-1)(n-2)-6(n-1)+6=0
n2-3n+2-6n+6+6=0
n2-9n+14=0
(n-2)(n-7)=0
n= 2,7
n=2 rejected for term 3rd
So n=7
Question 29.
Mark the correct alternative in the following :
The middle term in the expansion of 
is.
A.
B. 
C.
D. None of these
Answer:
Given:
For middle term,
Tn
=(-1)n 2nCn x-n
Question 30.
Mark the correct alternative in the following :
If rth term is the middle term in the expansion of 
, then 
term is
A.
B. 
C. 
D. None of these
Answer:
Given:
Total terms = n+1 = 21
Mid term = 21/2 = 11th term
For k= 10,it is rth term.
So (r+3)th term = 11th term
k=13
T14
= -20C13 x. 2 -13
= -20C7 x. 2 -13
Question 31.
Mark the correct alternative in the following :
The number of terms with integral coefficients in the expansion of 
is
A.2n
B. 50
C. 150
D. 101
Answer:
Given:
For integral coefficients; (600-k) should be divisible by 3 and k should be disable bye 2.
It indicates that k should be multiple of 6.
So, the values of k would be = 6,12,18…,594,600
Question 32.
Mark the correct alternative in the following :
Constant term in the expansion of 
is
A.152
B. -152
C. -252
D. 252
Answer:
Given:
For constant term,
10-2k = 0
k = 5
Term = 
= -252
Question 33.
Mark the correct alternative in the following :
If the coefficients of x2 and x3 in the expansion of 
are the same, then the value of a is.
A.
B. 
C. 
D. 
Answer:
Given:
Coefficient of x2 ; k=2
(1)
Coefficient of x3 ; k=3
(2)
Equate both equations;
