Please wait...

Solution
Q.1 Correct
Q.1 In-correct
Q.1 Unattempt

[27-Jan-2024 Shift 1]

[27-Jan-2024 Shift 1]
Q.2 Correct
Q.2 In-correct
Q.2 Unattempt

If A denotes the sum of all the coefficients in the expansion of (1 − 3x + 10x2)n and B denotes the sum of all the coefficients in the expansion of (1 + x2)n, then :

[27-Jan-2024 Shift 1]

If A denotes the sum of all the coefficients in the expansion of (1 − 3x + 10x2)n and B denotes the sum of all the coefficients in the expansion of (1 + x2)n, then :

[27-Jan-2024 Shift 1]
Q.3 Correct
Q.3 In-correct
Q.3 Unattempt

The coefficient of x2012 in the expansion of (1 − x)2008 (1 + x + x2)2007 is equal to

[27-Jan-2024 Shift 2]

The coefficient of x2012 in the expansion of (1 − x)2008 (1 + x + x2)2007 is equal to

[27-Jan-2024 Shift 2]
Q.4 Correct
Q.4 In-correct
Q.4 Unattempt

[29-Jan-2024 Shift 1]

[29-Jan-2024 Shift 1]
Q.5 Correct
Q.5 In-correct
Q.5 Unattempt

[29-Jan-2024 Shift 2]

[29-Jan-2024 Shift 2]
Q.6 Correct
Q.6 In-correct
Q.6 Unattempt

Number of integral terms in the expansion of  is equal to

[30-Jan-2024 Shift 1]

Number of integral terms in the expansion of  is equal to

[30-Jan-2024 Shift 1]
Q.7 Correct
Q.7 In-correct
Q.7 Unattempt

Suppose 2 − p, p, 2 − α, α are the coefficient of four consecutive terms in the expansion of (1 + x)n. Then the value of p2 − α2 + 6α + 2p equals

[30-Jan-2024 Shift 2]

Suppose 2 − p, p, 2 − α, α are the coefficient of four consecutive terms in the expansion of (1 + x)n. Then the value of p2 − α2 + 6α + 2p equals

[30-Jan-2024 Shift 2]
Q.8 Correct
Q.8 In-correct
Q.8 Unattempt

[30-Jan-2024 Shift 2]

[30-Jan-2024 Shift 2]
Q.9 Correct
Q.9 In-correct
Q.9 Unattempt

In the expansion of (1 + x)(1 − x2)  the sum of the coefficient of x3 and x−13 is equal to

[31-Jan-2024 Shift 1]

In the expansion of (1 + x)(1 − x2)  the sum of the coefficient of x3 and x−13 is equal to

[31-Jan-2024 Shift 1]
Q.10 Correct
Q.10 In-correct
Q.10 Unattempt

[31-Jan-2024 Shift 2]

[31-Jan-2024 Shift 2]
Q.11 Correct
Q.11 In-correct
Q.11 Unattempt

[31-Jan-2024 Shift 2]

[31-Jan-2024 Shift 2]
Q.12 Correct
Q.12 In-correct
Q.12 Unattempt

If the Coefficient of x30 in the expansion of (1 + 1/x)6(1 + x2)7(1 − x3)8; x ≠ 0 is α, then |α| equals______

[1-Feb-2024 Shift 1]

If the Coefficient of x30 in the expansion of (1 + 1/x)6(1 + x2)7(1 − x3)8; x ≠ 0 is α, then |α| equals______

[1-Feb-2024 Shift 1]
Q.13 Correct
Q.13 In-correct
Q.13 Unattempt

Let m and n be the coefficients of seventh and thirteenth terms respectively in the expansion of  (n/m)1/3 is:

[1-Feb-2024 Shift 2]

Let m and n be the coefficients of seventh and thirteenth terms respectively in the expansion of  (n/m)1/3 is:

[1-Feb-2024 Shift 2]
Q.14 Correct
Q.14 In-correct
Q.14 Unattempt
The value
22
∑
r=0
22
‌
Cr‌23Cr is
[24-Jan-2023 Shift 1]
The value
22
∑
r=0
22
‌
Cr‌23Cr is
[24-Jan-2023 Shift 1]
Q.15 Correct
Q.15 In-correct
Q.15 Unattempt
Suppose
2023
∑
r=0
r2‌‌2023Cr=2023×α×22022. Then the value of α is
[24-Jan-2023 Shift 1]
Suppose
2023
∑
r=0
r2‌‌2023Cr=2023×α×22022. Then the value of α is
[24-Jan-2023 Shift 1]
Q.16 Correct
Q.16 In-correct
Q.16 Unattempt
If (‌30C1)2+2(‌30C2)2+3(‌30C3)2+.......+30(‌30C30)2 =‌
α60!
(30!)2
, then α is equal to
[24-Jan-2023 Shift 2]
If (‌30C1)2+2(‌30C2)2+3(‌30C3)2+.......+30(‌30C30)2 =‌
α60!
(30!)2
, then α is equal to
[24-Jan-2023 Shift 2]
Q.17 Correct
Q.17 In-correct
Q.17 Unattempt
Let the sum of the coefficients of the first three terms in the expansion of (x−‌
3
x2
)n,x≠0,n∈N, be 376 . Then the coefficient of x4 is___
[24-Jan-2023 Shift 2]
Let the sum of the coefficients of the first three terms in the expansion of (x−‌
3
x2
)n,x≠0,n∈N, be 376 . Then the coefficient of x4 is___
[24-Jan-2023 Shift 2]
Q.18 Correct
Q.18 In-correct
Q.18 Unattempt
If ar is the coefficient of x10−r in the Binomial expansion of (1+x)10, then
10
∑
r=1
r3(‌
ar
ar−1
)2 is equal to
[25-Jan-2023 Shift 1]
If ar is the coefficient of x10−r in the Binomial expansion of (1+x)10, then
10
∑
r=1
r3(‌
ar
ar−1
)2 is equal to
[25-Jan-2023 Shift 1]
Q.19 Correct
Q.19 In-correct
Q.19 Unattempt
The constant term in the expansion of
(2x+‌
1
x7
+3x2)5‌ is ‌ ______.
[25-Jan-2023 Shift 1]
The constant term in the expansion of
(2x+‌
1
x7
+3x2)5‌ is ‌ ______.
[25-Jan-2023 Shift 1]
Q.20 Correct
Q.20 In-correct
Q.20 Unattempt
6
∑
k=0
‌‌‌51−kC3 is equal to
[25-Jan-2023 Shift 2]
6
∑
k=0
‌‌‌51−kC3 is equal to
[25-Jan-2023 Shift 2]
Q.21 Correct
Q.21 In-correct
Q.21 Unattempt
The remainder when (2023)2023 is divided by 35 is ________.
[25-Jan-2023 Shift 2]
The remainder when (2023)2023 is divided by 35 is ________.
[25-Jan-2023 Shift 2]
Q.22 Correct
Q.22 In-correct
Q.22 Unattempt
If the co-efficient of x9 in (αx3+‌
1
βx
)11 and the co-efficient of x−9 in (αx−‌
1
βx3
)11 are equal, then (αβ)2 is equal to _______.
[29-Jan-2023 Shift 1]
If the co-efficient of x9 in (αx3+‌
1
βx
)11 and the co-efficient of x−9 in (αx−‌
1
βx3
)11 are equal, then (αβ)2 is equal to _______.
[29-Jan-2023 Shift 1]
Q.23 Correct
Q.23 In-correct
Q.23 Unattempt
Let the coefficients of three consecutive terms in the binomial expansion of (1+2x)n be in the ratio 2:5:8. Then the coefficient of the term, which is in the middle of these three terms, is _______.
[29-Jan-2023 Shift 1]
Let the coefficients of three consecutive terms in the binomial expansion of (1+2x)n be in the ratio 2:5:8. Then the coefficient of the term, which is in the middle of these three terms, is _______.
[29-Jan-2023 Shift 1]
Q.24 Correct
Q.24 In-correct
Q.24 Unattempt
Let K be the sum of the coefficients of the odd powers of x in the expansion of (1+x)99. Let a be the middle term in the expansion of (2+‌
1
√2
)200. If ‌
‌200C99K
a
=‌
2lm
n
, where m and n are odd numbers, then the ordered pair (ℓ,n) is equal to :
[29-Jan-2023 Shift 2]
Let K be the sum of the coefficients of the odd powers of x in the expansion of (1+x)99. Let a be the middle term in the expansion of (2+‌
1
√2
)200. If ‌
‌200C99K
a
=‌
2lm
n
, where m and n are odd numbers, then the ordered pair (ℓ,n) is equal to :
[29-Jan-2023 Shift 2]
Q.25 Correct
Q.25 In-correct
Q.25 Unattempt
If the coefficient of x15 in the expansion of (ax3+‌
1
bx‌
1
3
)15 is equal to the coefficient of x−15 in the expansion of (ax‌
1
3
−‌
1
bx3
)15, where a and b are positive real numbers, then for each such ordered pair (a, b) :
[30-Jan-2023 Shift 1]
If the coefficient of x15 in the expansion of (ax3+‌
1
bx‌
1
3
)15 is equal to the coefficient of x−15 in the expansion of (ax‌
1
3
−‌
1
bx3
)15, where a and b are positive real numbers, then for each such ordered pair (a, b) :
[30-Jan-2023 Shift 1]
Q.26 Correct
Q.26 In-correct
Q.26 Unattempt
The coefficient of x301 in (1+x)500+x(1+x)499+x2(1+x)498+.....+x500 is:
[30-Jan-2023 Shift 1]
The coefficient of x301 in (1+x)500+x(1+x)499+x2(1+x)498+.....+x500 is:
[30-Jan-2023 Shift 1]
Q.27 Correct
Q.27 In-correct
Q.27 Unattempt
Let x=(8√3+13)13 and y=(7√2+9)9. If [t] denotes the greatest integer ≤t, then
[30-Jan-2023 Shift 2]
Let x=(8√3+13)13 and y=(7√2+9)9. If [t] denotes the greatest integer ≤t, then
[30-Jan-2023 Shift 2]
Q.28 Correct
Q.28 In-correct
Q.28 Unattempt
50‌th ‌ root of a number x is 12 and 50‌th ‌ root of another number y is 18 . Then the remainder obtained on dividing (x+y) by 25 is _______.
[30-Jan-2023 Shift 2]
50‌th ‌ root of a number x is 12 and 50‌th ‌ root of another number y is 18 . Then the remainder obtained on dividing (x+y) by 25 is _______.
[30-Jan-2023 Shift 2]
Q.29 Correct
Q.29 In-correct
Q.29 Unattempt
Let α>0, be the smallest number such that the expansion of (x‌
2
3
+‌
2
x3
)30 has a term βx−α,β∈N.
Then α is equal to _______.
[31-Jan-2023 Shift 1]
Let α>0, be the smallest number such that the expansion of (x‌
2
3
+‌
2
x3
)30 has a term βx−α,β∈N.
Then α is equal to _______.
[31-Jan-2023 Shift 1]
Q.30 Correct
Q.30 In-correct
Q.30 Unattempt
The remainder on dividing 599 by 11 is ________.
[31-Jan-2023 Shift 1]
The remainder on dividing 599 by 11 is ________.
[31-Jan-2023 Shift 1]
Q.31 Correct
Q.31 In-correct
Q.31 Unattempt
The Coefficient of x−6, in the expansion of (‌
4x
5
+‌
5
2x2
)9, is _______
[31-Jan-2023 Shift 2]
The Coefficient of x−6, in the expansion of (‌
4x
5
+‌
5
2x2
)9, is _______
[31-Jan-2023 Shift 2]
Q.32 Correct
Q.32 In-correct
Q.32 Unattempt
If the constant term in the binomial expansion of (‌
x
5
2
2
−‌
4
xℓ
)9 is −84 and the Coefficient of x−3ℓ is 2αβ, where β<0 is an odd number, Then |αℓ−β| is equal to _________
[31-Jan-2023 Shift 2]
If the constant term in the binomial expansion of (‌
x
5
2
2
−‌
4
xℓ
)9 is −84 and the Coefficient of x−3ℓ is 2αβ, where β<0 is an odd number, Then |αℓ−β| is equal to _________
[31-Jan-2023 Shift 2]
Q.33 Correct
Q.33 In-correct
Q.33 Unattempt
The value of ‌
1
1!50!
+‌
1
3!48!
+‌
1
5!46!
+....+‌
1
49!2!
+‌
1
51!1!
 is
[1-Feb-2023 Shift 1]
The value of ‌
1
1!50!
+‌
1
3!48!
+‌
1
5!46!
+....+‌
1
49!2!
+‌
1
51!1!
 is
[1-Feb-2023 Shift 1]
Q.34 Correct
Q.34 In-correct
Q.34 Unattempt
The remainder when 19200+23200 is divided by 49 , is _______.
[1-Feb-2023 Shift 1]
The remainder when 19200+23200 is divided by 49 , is _______.
[1-Feb-2023 Shift 1]
Q.35 Correct
Q.35 In-correct
Q.35 Unattempt
Let the sixth term in the binomial expansion of (√2log2(10−3x)+‌5√2(x−2)log23)m, in the increasing powers of 2(x−2)log23, be 21 . If the binomial coefficients of the second, third and fourth terms in the expansion are respectively the first, third and fifth terms of an A.P., then the sum of the squares of all possible values of x is ________.
[1-Feb-2023 Shift 2]
Let the sixth term in the binomial expansion of (√2log2(10−3x)+‌5√2(x−2)log23)m, in the increasing powers of 2(x−2)log23, be 21 . If the binomial coefficients of the second, third and fourth terms in the expansion are respectively the first, third and fifth terms of an A.P., then the sum of the squares of all possible values of x is ________.
[1-Feb-2023 Shift 2]
Q.36 Correct
Q.36 In-correct
Q.36 Unattempt
If the term without x in the expansion of (x‌
2
3
+‌
α
x3
)22 is 7315 , then |α| is equal to _______.
[1-Feb-2023 Shift 2]
If the term without x in the expansion of (x‌
2
3
+‌
α
x3
)22 is 7315 , then |α| is equal to _______.
[1-Feb-2023 Shift 2]
Q.37 Correct
Q.37 In-correct
Q.37 Unattempt
If the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of (‌4√2+‌
1
‌4√3
)a is √6:1, then the third term from the beginning is :
[6-Apr-2023 shift 1]
If the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of (‌4√2+‌
1
‌4√3
)a is √6:1, then the third term from the beginning is :
[6-Apr-2023 shift 1]
Q.38 Correct
Q.38 In-correct
Q.38 Unattempt
If ‌2nC3:‌nC3:10:1, then the ratio (n2+3n):(n2−3n+4) is :
[6-Apr-2023 shift 1]
If ‌2nC3:‌nC3:10:1, then the ratio (n2+3n):(n2−3n+4) is :
[6-Apr-2023 shift 1]
Q.39 Correct
Q.39 In-correct
Q.39 Unattempt
The coefficient of x18 in the expansion of (x4−‌
1
x3
)15 is ______.
[6-Apr-2023 shift 1]
The coefficient of x18 in the expansion of (x4−‌
1
x3
)15 is ______.
[6-Apr-2023 shift 1]
Q.40 Correct
Q.40 In-correct
Q.40 Unattempt
If the coefficients of x7 in (ax2+‌
1
2bx
)11 and x−7 in (ax−‌
1
3bx2
)11 are equal, then :
[6-Apr-2023 shift 2]
If the coefficients of x7 in (ax2+‌
1
2bx
)11 and x−7 in (ax−‌
1
3bx2
)11 are equal, then :
[6-Apr-2023 shift 2]
Q.41 Correct
Q.41 In-correct
Q.41 Unattempt
Among the statements :
(S1) : 20232022−19992022 is divisible by 8
(S2) : 13(13)n−11n−13 is divisible by 144 for infinitely many n∈ℕ
[6-Apr-2023 shift 2]
Among the statements :
(S1) : 20232022−19992022 is divisible by 8
(S2) : 13(13)n−11n−13 is divisible by 144 for infinitely many n∈ℕ
[6-Apr-2023 shift 2]
Q.42 Correct
Q.42 In-correct
Q.42 Unattempt
Let (t) denote the greatest integer ≤t, If the constant term in the expansion of (3x2−‌
1
2x5
)7 is α, then [α] is equal to _______.
[8-Apr-2023 shift 1]
Let (t) denote the greatest integer ≤t, If the constant term in the expansion of (3x2−‌
1
2x5
)7 is α, then [α] is equal to _______.
[8-Apr-2023 shift 1]
Q.43 Correct
Q.43 In-correct
Q.43 Unattempt
If aa is the greatest term in the sequence an=‌
n3
n4+147
,n=1,2,3,..., then a is equal to ______.
[8-Apr-2023 shift 1]
If aa is the greatest term in the sequence an=‌
n3
n4+147
,n=1,2,3,..., then a is equal to ______.
[8-Apr-2023 shift 1]
Q.44 Correct
Q.44 In-correct
Q.44 Unattempt
The largest natural number n such that 3n divides 66 ! is _______.
[8-Apr-2023 shift 1]
The largest natural number n such that 3n divides 66 ! is _______.
[8-Apr-2023 shift 1]
Q.45 Correct
Q.45 In-correct
Q.45 Unattempt
The absolute difference of the coefficients of x10 and x7 in the expansion of (2x2+‌
1
2x
)11 is equal to
[8-Apr-2023 shift 2]
The absolute difference of the coefficients of x10 and x7 in the expansion of (2x2+‌
1
2x
)11 is equal to
[8-Apr-2023 shift 2]
Q.46 Correct
Q.46 In-correct
Q.46 Unattempt
25190−19190−8190+2190 is divisible by
[8-Apr-2023 shift 2]
25190−19190−8190+2190 is divisible by
[8-Apr-2023 shift 2]
Q.47 Correct
Q.47 In-correct
Q.47 Unattempt
If the coefficient of x7 in (ax−‌
1
bx2
)13 and the coefficient of x−5 in (ax+‌
1
bx2
)13 are equal, then a4b4 is equal to :
[10-Apr-2023 shift 1]
If the coefficient of x7 in (ax−‌
1
bx2
)13 and the coefficient of x−5 in (ax+‌
1
bx2
)13 are equal, then a4b4 is equal to :
[10-Apr-2023 shift 1]
Q.48 Correct
Q.48 In-correct
Q.48 Unattempt
The coefficient of x7 in (1−x+2x3)10 is _______.
[10-Apr-2023 shift 1]
The coefficient of x7 in (1−x+2x3)10 is _______.
[10-Apr-2023 shift 1]
Q.49 Correct
Q.49 In-correct
Q.49 Unattempt
Let the number (22)2022+(2022)22 leave the remainder α when divided by 3 and β when divided by 7 . Then (α2. +β2) is equal to
[10-Apr-2023 shift 2]
Let the number (22)2022+(2022)22 leave the remainder α when divided by 3 and β when divided by 7 . Then (α2. +β2) is equal to
[10-Apr-2023 shift 2]
Q.50 Correct
Q.50 In-correct
Q.50 Unattempt
If the coefficients of x and x2 in (1+x)p(1−x)q are 4 and -5 respectively, then 2p+3q is equal to
[10-Apr-2023 shift 2]
If the coefficients of x and x2 in (1+x)p(1−x)q are 4 and -5 respectively, then 2p+3q is equal to
[10-Apr-2023 shift 2]
Q.51 Correct
Q.51 In-correct
Q.51 Unattempt
The number of integral terms in the expansion of (3‌
1
2
+5‌
1
4
)680 is equal to :
[11-Apr-2023 shift 1]
The number of integral terms in the expansion of (3‌
1
2
+5‌
1
4
)680 is equal to :
[11-Apr-2023 shift 1]
Q.52 Correct
Q.52 In-correct
Q.52 Unattempt
The mean of the coefficients of x,x2,...x7 in the binomial expansion of (2+x)9 is ________.
[11-Apr-2023 shift 1]
The mean of the coefficients of x,x2,...x7 in the binomial expansion of (2+x)9 is ________.
[11-Apr-2023 shift 1]
Q.53 Correct
Q.53 In-correct
Q.53 Unattempt
If the 1011‌th ‌ term from the end in the binominal expansion of (‌
4x
5
−‌
5
2x
)2022 is 1024 times 1011‌th ‌ term from the beginning, the |x| is equal to
[11-Apr-2023 shift 2]
If the 1011‌th ‌ term from the end in the binominal expansion of (‌
4x
5
−‌
5
2x
)2022 is 1024 times 1011‌th ‌ term from the beginning, the |x| is equal to
[11-Apr-2023 shift 2]
Q.54 Correct
Q.54 In-correct
Q.54 Unattempt
The sum of the coefficients of three consecutive terms in the binomial expansion of (1+x)n+2, which are in the ratio 1:3:5, is equal to
[11-Apr-2023 shift 2]
The sum of the coefficients of three consecutive terms in the binomial expansion of (1+x)n+2, which are in the ratio 1:3:5, is equal to
[11-Apr-2023 shift 2]
Q.55 Correct
Q.55 In-correct
Q.55 Unattempt
If ‌
1
n+1
‌nCn+‌
1
n
‌nCn−1+...+‌
1
2
‌nC1+‌nC0=‌
1023
10
 then n is equal to
[12-Apr-2023 shift 1]
If ‌
1
n+1
‌nCn+‌
1
n
‌nCn−1+...+‌
1
2
‌nC1+‌nC0=‌
1023
10
 then n is equal to
[12-Apr-2023 shift 1]
Q.56 Correct
Q.56 In-correct
Q.56 Unattempt
The sum, of the coefficients of the first 50 terms in the binomial expansion of (1−x)100, is equal to
[12-Apr-2023 shift 1]
The sum, of the coefficients of the first 50 terms in the binomial expansion of (1−x)100, is equal to
[12-Apr-2023 shift 1]
Q.57 Correct
Q.57 In-correct
Q.57 Unattempt
Fractional part of the number is ‌
42022
15
 equal to
[13-Apr-2023 shift 1]
Fractional part of the number is ‌
42022
15
 equal to
[13-Apr-2023 shift 1]
Q.58 Correct
Q.58 In-correct
Q.58 Unattempt
Let α be the constant term in the binomial expansion of (√x−‌
6
x‌
3
2
)n,n≤15. If the sum of the coefficients of the remaining terms in the expansion is 649 and the coefficient of x−n is λα, then λ is equal to _______.
[13-Apr-2023 shift 1]
Let α be the constant term in the binomial expansion of (√x−‌
6
x‌
3
2
)n,n≤15. If the sum of the coefficients of the remaining terms in the expansion is 649 and the coefficient of x−n is λα, then λ is equal to _______.
[13-Apr-2023 shift 1]
Q.59 Correct
Q.59 In-correct
Q.59 Unattempt
Let for x∈R,S0(x)=x,Sk(x)=Ckx+k‌
x
∫
0
Sk−1(t)‌dt where C0=1,Ck=1−
1
∫
0
Sk−1(x)‌dx,k=1,2,3,... Then S2(3)+6C3 is equal to _______.
[13-Apr-2023 shift 1]
Let for x∈R,S0(x)=x,Sk(x)=Ckx+k‌
x
∫
0
Sk−1(t)‌dt where C0=1,Ck=1−
1
∫
0
Sk−1(x)‌dx,k=1,2,3,... Then S2(3)+6C3 is equal to _______.
[13-Apr-2023 shift 1]
Q.60 Correct
Q.60 In-correct
Q.60 Unattempt
The coefficient of x5 in the expansion of (2x3−‌
1
3x2
)5 is
[13-Apr-2023 shift 2]
The coefficient of x5 in the expansion of (2x3−‌
1
3x2
)5 is
[13-Apr-2023 shift 2]
Q.61 Correct
Q.61 In-correct
Q.61 Unattempt
The remainder, when 7103 is divided by 17 , is ______.
[13-Apr-2023 shift 2]
The remainder, when 7103 is divided by 17 , is ______.
[13-Apr-2023 shift 2]
Q.62 Correct
Q.62 In-correct
Q.62 Unattempt
Let (a+bx+cx2)10=
20
∑
i=0
pixi,a,b,c∈N. If p1=20 and p2=210, then 2(a+b+c) is equal to
[15-Apr-2023 shift 1]
Let (a+bx+cx2)10=
20
∑
i=0
pixi,a,b,c∈N. If p1=20 and p2=210, then 2(a+b+c) is equal to
[15-Apr-2023 shift 1]
Q.63 Correct
Q.63 In-correct
Q.63 Unattempt
The remainder when 32022 is divided by 5 is :
[24-Jun-2022-Shift-1]
The remainder when 32022 is divided by 5 is :
[24-Jun-2022-Shift-1]
Q.64 Correct
Q.64 In-correct
Q.64 Unattempt
The remainder on dividing 1+3+32+33+.....+32021 by 50 is__
[24-Jun-2022-Shift-2]
The remainder on dividing 1+3+32+33+.....+32021 by 50 is__
[24-Jun-2022-Shift-2]
Q.65 Correct
Q.65 In-correct
Q.65 Unattempt
Let Cr denote the binomial coefficient of xr in the expansion of (1+x)10. If for α,β∈R,C1+3.2C2+5.3C3+...... upto 10 terms =‌
α×211
2β−1
(C0+‌
C1
2
+‌
C2
3
+.... upto 10 terms ) then the value of α+β is equal to
[25-Jun-2022-Shift-1]
Let Cr denote the binomial coefficient of xr in the expansion of (1+x)10. If for α,β∈R,C1+3.2C2+5.3C3+...... upto 10 terms =‌
α×211
2β−1
(C0+‌
C1
2
+‌
C2
3
+.... upto 10 terms ) then the value of α+β is equal to
[25-Jun-2022-Shift-1]
Q.66 Correct
Q.66 In-correct
Q.66 Unattempt
The coefficient of x101 in the expression (5+x)500+x(5+x)499+x2(5+x)498+...+x500,x>0, is
[25-Jun-2022-Shift-2]
The coefficient of x101 in the expression (5+x)500+x(5+x)499+x2(5+x)498+...+x500,x>0, is
[25-Jun-2022-Shift-2]
Q.67 Correct
Q.67 In-correct
Q.67 Unattempt
If the sum of the co-efficient of all the positive even powers of x in the binomial expansion of (2x3+‌
3
x
)10 is 510−β.39, then β is equal to____
[25-Jun-2022-Shift-2]
If the sum of the co-efficient of all the positive even powers of x in the binomial expansion of (2x3+‌
3
x
)10 is 510−β.39, then β is equal to____
[25-Jun-2022-Shift-2]
Q.68 Correct
Q.68 In-correct
Q.68 Unattempt
The remainder when (2021)2023 is divided by 7 is :
[26-Jun-2022-Shift-1]
The remainder when (2021)2023 is divided by 7 is :
[26-Jun-2022-Shift-1]
Q.69 Correct
Q.69 In-correct
Q.69 Unattempt
If (‌40C0)+(‌41C1)+(‌42C2)+......+(‌60C20)=‌
m
n
‌60C20, m and n are coprime, then m+n is equal to____
[26-Jun-2022-Shift-2]
If (‌40C0)+(‌41C1)+(‌42C2)+......+(‌60C20)=‌
m
n
‌60C20, m and n are coprime, then m+n is equal to____
[26-Jun-2022-Shift-2]
Q.70 Correct
Q.70 In-correct
Q.70 Unattempt
If the coefficient of x10 in the binomial expansion of (‌
√x
5‌
1
4
+‌
√5
x‌
1
3
)60 is 5k.l, where I, k∈N and I is co-prime to 5 , then k is equal to
[27-Jun-2022-Shift-1]
If the coefficient of x10 in the binomial expansion of (‌
√x
5‌
1
4
+‌
√5
x‌
1
3
)60 is 5k.l, where I, k∈N and I is co-prime to 5 , then k is equal to
[27-Jun-2022-Shift-1]
Q.71 Correct
Q.71 In-correct
Q.71 Unattempt
If the sum of the coefficients of all the positive powers of x, in the Binomial expansion of (xn+‌
2
x5
)7 is 939 , then the sum of all the possible integral values of n is____
[27-Jun-2022-Shift-2]
If the sum of the coefficients of all the positive powers of x, in the Binomial expansion of (xn+‌
2
x5
)7 is 939 , then the sum of all the possible integral values of n is____
[27-Jun-2022-Shift-2]
Q.72 Correct
Q.72 In-correct
Q.72 Unattempt
If
31
∑
k=1
(‌31Ck)(‌31Ck−1)−
30
∑
k=1
(‌31Ck)(‌31Ck−1)=‌
α(60!)
(30!)(31!)

where α∈R, then the value of 16α is equal to
[28-Jun-2022-Shift-1]
If
31
∑
k=1
(‌31Ck)(‌31Ck−1)−
30
∑
k=1
(‌31Ck)(‌31Ck−1)=‌
α(60!)
(30!)(31!)

where α∈R, then the value of 16α is equal to
[28-Jun-2022-Shift-1]
Q.73 Correct
Q.73 In-correct
Q.73 Unattempt
The number of positive integers k such that the constant term in the binomial expansion of (2x3+‌
3
xk
)12,x≠0 is 28.I, where I is an odd integer, is___
[28-Jun-2022-Shift-1]
The number of positive integers k such that the constant term in the binomial expansion of (2x3+‌
3
xk
)12,x≠0 is 28.I, where I is an odd integer, is___
[28-Jun-2022-Shift-1]
Q.74 Correct
Q.74 In-correct
Q.74 Unattempt
The term independent of x in the expansion of (1−x2+3x3)(‌
5
2
x3−‌
1
5x2
)11,x≠0 is :
[28-Jun-2022-Shift-2]
The term independent of x in the expansion of (1−x2+3x3)(‌
5
2
x3−‌
1
5x2
)11,x≠0 is :
[28-Jun-2022-Shift-2]
Q.75 Correct
Q.75 In-correct
Q.75 Unattempt
If the constant term in the expansion of
(3x3−2x2+‌
5
x5
)10 is 2k.I, where I is an odd integer, then the value of k is equal to:
[29-Jun-2022-Shift-1]
If the constant term in the expansion of
(3x3−2x2+‌
5
x5
)10 is 2k.I, where I is an odd integer, then the value of k is equal to:
[29-Jun-2022-Shift-1]
Q.76 Correct
Q.76 In-correct
Q.76 Unattempt
Let n≥5 be an integer. If 9n−8n−1=64α and 6n−5n−1=25β, then α−β is equal to
[29-Jun-2022-Shift-2]
Let n≥5 be an integer. If 9n−8n−1=64α and 6n−5n−1=25β, then α−β is equal to
[29-Jun-2022-Shift-2]
Q.77 Correct
Q.77 In-correct
Q.77 Unattempt
Let the coefficients of x−1 and x−3 in the expansion of (2x‌
1
5
−‌
1
x‌
1
5
)15,x>0, be m and n respectively. If r is a positive integer such that mn2=‌15Cr⋅2r, then the value of r is equal to___
[29-Jun-2022-Shift-2]
Let the coefficients of x−1 and x−3 in the expansion of (2x‌
1
5
−‌
1
x‌
1
5
)15,x>0, be m and n respectively. If r is a positive integer such that mn2=‌15Cr⋅2r, then the value of r is equal to___
[29-Jun-2022-Shift-2]
Q.78 Correct
Q.78 In-correct
Q.78 Unattempt
If the maximum value of the term independent of t in the expansion of (t2x‌
1
5
+‌
(1−x)
1
10
t
)15,x≥slant0, is K, then 8K is equal to
[25-Jul-2022-Shift-1]
If the maximum value of the term independent of t in the expansion of (t2x‌
1
5
+‌
(1−x)
1
10
t
)15,x≥slant0, is K, then 8K is equal to
[25-Jul-2022-Shift-1]
Q.79 Correct
Q.79 In-correct
Q.79 Unattempt
The remainder when (11)1011+(1011)11 is divided by 9 is
[25-Jul-2022-Shift-2]
The remainder when (11)1011+(1011)11 is divided by 9 is
[25-Jul-2022-Shift-2]
Q.80 Correct
Q.80 In-correct
Q.80 Unattempt
If the coefficients of x and x2 in the expansion of (1+x)p(1−x)q,p,q≤15, are −3 and −5 respectively, then the coefficient of x3 is equal to _________.
[26-Jul-2022-Shift-1]
If the coefficients of x and x2 in the expansion of (1+x)p(1−x)q,p,q≤15, are −3 and −5 respectively, then the coefficient of x3 is equal to _________.
[26-Jul-2022-Shift-1]
Q.81 Correct
Q.81 In-correct
Q.81 Unattempt
n
∑
i,j=0
i≠j
‌nCi‌nCj is equal to
[26-Jul-2022-Shift-2]
n
∑
i,j=0
i≠j
‌nCi‌nCj is equal to
[26-Jul-2022-Shift-2]
Q.82 Correct
Q.82 In-correct
Q.82 Unattempt
The remainder when (2021)2022+(2022)2021 is divided by 7 is
[27-Jul-2022-Shift-1]
The remainder when (2021)2022+(2022)2021 is divided by 7 is
[27-Jul-2022-Shift-1]
Q.83 Correct
Q.83 In-correct
Q.83 Unattempt
Let for the 9‌th ‌ term in the binomial expansion of (3+6x)n, in the increasing powers of 6x, to be the greatest for x=‌
3
2
, the least value of n is n0. If k is the ratio of the coefficient of x6 to the coefficient of x3, then k+n0 is equal to:
[27-Jul-2022-Shift-2]
Let for the 9‌th ‌ term in the binomial expansion of (3+6x)n, in the increasing powers of 6x, to be the greatest for x=‌
3
2
, the least value of n is n0. If k is the ratio of the coefficient of x6 to the coefficient of x3, then k+n0 is equal to:
[27-Jul-2022-Shift-2]
Q.84 Correct
Q.84 In-correct
Q.84 Unattempt
The remainder when 72022+320222 is divided by 5 is :
[28-Jul-2022-Shift-1]
The remainder when 72022+320222 is divided by 5 is :
[28-Jul-2022-Shift-1]
Q.85 Correct
Q.85 In-correct
Q.85 Unattempt
Let the coefficients of the middle terms in the expansion of (‌
1
√6
+βx)4,(1−3βx)2 and (1−‌
β
2
x)6,β>0, respectively form the first three terms of an A.P. If d is the common difference of this A.P. , then 50−‌
2d
β2
 is equal to _________.
[28-Jul-2022-Shift-2]
Let the coefficients of the middle terms in the expansion of (‌
1
√6
+βx)4,(1−3βx)2 and (1−‌
β
2
x)6,β>0, respectively form the first three terms of an A.P. If d is the common difference of this A.P. , then 50−‌
2d
β2
 is equal to _________.
[28-Jul-2022-Shift-2]
Q.86 Correct
Q.86 In-correct
Q.86 Unattempt
If 1+(2+‌49C1+‌49C2+...+‌49C49)(‌50C2+‌50C4+...+‌50C50) is equal to 2n⋅m, where m is odd, then n+m is equal to _______.
[28-Jul-2022-Shift-2]
If 1+(2+‌49C1+‌49C2+...+‌49C49)(‌50C2+‌50C4+...+‌50C50) is equal to 2n⋅m, where m is odd, then n+m is equal to _______.
[28-Jul-2022-Shift-2]
Q.87 Correct
Q.87 In-correct
Q.87 Unattempt
Let the ratio of the fifth term from the beginning to the fifth term from the end in the binomial expansion of (‌4√2+‌
1
‌4√3
)n, in the increasing powers of ‌
1
‌4√3
 be ‌4√6:1. If the sixth term from the beginning is ‌
α
‌4√3
, then α is equal to ________.
[29-Jul-2022-Shift-1]
Let the ratio of the fifth term from the beginning to the fifth term from the end in the binomial expansion of (‌4√2+‌
1
‌4√3
)n, in the increasing powers of ‌
1
‌4√3
 be ‌4√6:1. If the sixth term from the beginning is ‌
α
‌4√3
, then α is equal to ________.
[29-Jul-2022-Shift-1]
Q.88 Correct
Q.88 In-correct
Q.88 Unattempt
If
10
∑
k=1
K2(10CK)2=22000L, then L is equal to _______.
[29-Jul-2022-Shift-2]
If
10
∑
k=1
K2(10CK)2=22000L, then L is equal to _______.
[29-Jul-2022-Shift-2]
Q.89 Correct
Q.89 In-correct
Q.89 Unattempt
If n≥2 is a positive integer, then the sum of the series
‌n+1C2+2(‌2C2+‌3C2+‌4C2+...+‌nC2) is
[2021, 24 Feb. Shift-II]
If n≥2 is a positive integer, then the sum of the series
‌n+1C2+2(‌2C2+‌3C2+‌4C2+...+‌nC2) is
[2021, 24 Feb. Shift-II]
Q.90 Correct
Q.90 In-correct
Q.90 Unattempt
For integers n and r, let (
n
r
)={
‌nCr‌‌ if ‌n≥r≥0
0‌‌ otherwise ‌.
 The maximum value of k for which the sum,
k
∑
i=0
(
10
i
)(
5
k−i
)+
k+1
∑
i=0
(
12
i
)(
13
k+1−i
)
exists, is equal to
[2021, 24 Feb Shift-II]
For integers n and r, let (
n
r
)={
‌nCr‌‌ if ‌n≥r≥0
0‌‌ otherwise ‌.
 The maximum value of k for which the sum,
k
∑
i=0
(
10
i
)(
5
k−i
)+
k+1
∑
i=0
(
12
i
)(
13
k+1−i
)
exists, is equal to
[2021, 24 Feb Shift-II]
Q.91 Correct
Q.91 In-correct
Q.91 Unattempt
The value of −‌15C1+2⋅‌15C2−3⋅‌15C3+... −15⋅‌15C15+‌14C1+‌14C3+‌14C5 +...+‌14C11‌ is ‌‌‌
[2021, 24 Feb. Shift-l]
The value of −‌15C1+2⋅‌15C2−3⋅‌15C3+... −15⋅‌15C15+‌14C1+‌14C3+‌14C5 +...+‌14C11‌ is ‌‌‌
[2021, 24 Feb. Shift-l]
Q.92 Correct
Q.92 In-correct
Q.92 Unattempt
If the remainder when x is divided by 4 is 3 , then the remainder when (2020+x)2022 is divided by 8 is
[2021, 25 Feb. Shift-II]
If the remainder when x is divided by 4 is 3 , then the remainder when (2020+x)2022 is divided by 8 is
[2021, 25 Feb. Shift-II]
Q.93 Correct
Q.93 In-correct
Q.93 Unattempt
Let m,n∈N and gcd(2,n)=1. If
30(
30
0
)+29(
30
28
)+...+2(
30
28
)+1(
30
29
) =n⋅2m, then n+m is ......... (Here, (
n
k
)=‌nCk)‌‌
[2021, 26 Feb. Shift-I]
Let m,n∈N and gcd(2,n)=1. If
30(
30
0
)+29(
30
28
)+...+2(
30
28
)+1(
30
29
) =n⋅2m, then n+m is ......... (Here, (
n
k
)=‌nCk)‌‌
[2021, 26 Feb. Shift-I]
Q.94 Correct
Q.94 In-correct
Q.94 Unattempt
The maximum value of the term independent of ' t ' in the expansion of (tx1∕5+‌
(1−x)1∕10
t
)10, where x∈(0,1) is ‌‌
[2021, 26 Feb. Shift-l]
The maximum value of the term independent of ' t ' in the expansion of (tx1∕5+‌
(1−x)1∕10
t
)10, where x∈(0,1) is ‌‌
[2021, 26 Feb. Shift-l]
Q.95 Correct
Q.95 In-correct
Q.95 Unattempt
The term independent of x in the expansion of [‌
x+1
x2∕3−x1∕3+1
−‌
x−1
x−x1∕2
]10,x≠1, is equal to
[2021, 18 March Shift-II]
The term independent of x in the expansion of [‌
x+1
x2∕3−x1∕3+1
−‌
x−1
x−x1∕2
]10,x≠1, is equal to
[2021, 18 March Shift-II]
Q.96 Correct
Q.96 In-correct
Q.96 Unattempt
Let ‌nCr denote the binomial coefficient of xr in the expansion of (1+x)n.
If
10
∑
k=0
(22+3k)nCk=α⋅310+β⋅210
α,β∈R, then α+β is equal to
[2021, 18 March Shift-II]
Let ‌nCr denote the binomial coefficient of xr in the expansion of (1+x)n.
If
10
∑
k=0
(22+3k)nCk=α⋅310+β⋅210
α,β∈R, then α+β is equal to
[2021, 18 March Shift-II]
Q.97 Correct
Q.97 In-correct
Q.97 Unattempt
If the fourth term in the expansion of (x+xlog2x)7 is 4480 , then the value of x, where x∈N is equal to
[2021, 17 March Shift-I]
If the fourth term in the expansion of (x+xlog2x)7 is 4480 , then the value of x, where x∈N is equal to
[2021, 17 March Shift-I]
Q.98 Correct
Q.98 In-correct
Q.98 Unattempt
If (2021)3762 is divided by 17 , then the remainder is
[2021, 17 March Shift-I]
If (2021)3762 is divided by 17 , then the remainder is
[2021, 17 March Shift-I]
Q.99 Correct
Q.99 In-correct
Q.99 Unattempt
Let the coefficients of third, fourth and fifth terms in the expansion of (x+‌
a
x2
)n,x≠0, be in the ratio 12:8:3. Then, the term independent of x in the expansion, is equal to
[2021, 17 March Shift-II]
Let the coefficients of third, fourth and fifth terms in the expansion of (x+‌
a
x2
)n,x≠0, be in the ratio 12:8:3. Then, the term independent of x in the expansion, is equal to
[2021, 17 March Shift-II]
Q.100 Correct
Q.100 In-correct
Q.100 Unattempt
If n is the number of irrational terms in the expansion of (31∕4+51∕8)60, then (n−1) is divisible by
[2021, 16 Mar Shift-I]
If n is the number of irrational terms in the expansion of (31∕4+51∕8)60, then (n−1) is divisible by
[2021, 16 Mar Shift-I]
Q.101 Correct
Q.101 In-correct
Q.101 Unattempt
Let [x] denote greatest integer less than or equal to x. If for n∈N, (1−x+x3)n=
3n
∑
j=0
ajxj, then
[‌
3n
2
]
∑
j=0
a2j+4‌
[‌
3n−1
2
]
∑
j=0
a2j+1 is equal to
[2021, 16 March Shift-I]
Let [x] denote greatest integer less than or equal to x. If for n∈N, (1−x+x3)n=
3n
∑
j=0
ajxj, then
[‌
3n
2
]
∑
j=0
a2j+4‌
[‌
3n−1
2
]
∑
j=0
a2j+1 is equal to
[2021, 16 March Shift-I]
Q.102 Correct
Q.102 In-correct
Q.102 Unattempt
The value of
6
∑
r=0
(‌6Cr⋅‌6C6−r) is equal to
[2021, 17 March Shift-II]
The value of
6
∑
r=0
(‌6Cr⋅‌6C6−r) is equal to
[2021, 17 March Shift-II]
Q.103 Correct
Q.103 In-correct
Q.103 Unattempt
Let n be a positive integer. Let
A=
n
∑
k
(−1)k‌nCk[
(‌
1
2
)k+(‌
3
4
)k+(‌
7
8
)k
+(‌
15
16
)k+(‌
31
32
)k
]
If 63A=1−‌
1
230
, then n is equal to
[2021, 16 March Shift-II]
Let n be a positive integer. Let
A=
n
∑
k
(−1)k‌nCk[
(‌
1
2
)k+(‌
3
4
)k+(‌
7
8
)k
+(‌
15
16
)k+(‌
31
32
)k
]
If 63A=1−‌
1
230
, then n is equal to
[2021, 16 March Shift-II]
Q.104 Correct
Q.104 In-correct
Q.104 Unattempt
The term independent of x in the expansion of (‌
x+1
x23−x1∕3+1
−‌
x−1
x−x1∕2
)10, where x≠0,1 is equal to .......
[2021, 2 July Shift I]
The term independent of x in the expansion of (‌
x+1
x23−x1∕3+1
−‌
x−1
x−x1∕2
)10, where x≠0,1 is equal to .......
[2021, 2 July Shift I]
Q.105 Correct
Q.105 In-correct
Q.105 Unattempt
If b is very small as compared to the value of a, so that the cube and other higher powers of ‌
b
a
 can be neglected in the identity
‌
1
a−b
+‌
1
a−2b
+‌
1
a−3b
+...+‌
1
a−nb
 =αn+βn2+γn3, then the value of γ is
[2021, 25 July Shift-I]
If b is very small as compared to the value of a, so that the cube and other higher powers of ‌
b
a
 can be neglected in the identity
‌
1
a−b
+‌
1
a−2b
+‌
1
a−3b
+...+‌
1
a−nb
 =αn+βn2+γn3, then the value of γ is
[2021, 25 July Shift-I]
Q.106 Correct
Q.106 In-correct
Q.106 Unattempt
The sum of all those terms which are rational numbers in the expansion of (21∕3+31∕4)12 is
[2021, 25 July Shift-II]
The sum of all those terms which are rational numbers in the expansion of (21∕3+31∕4)12 is
[2021, 25 July Shift-II]
Q.107 Correct
Q.107 In-correct
Q.107 Unattempt
If the greatest value of the term independent of x in the expansion of (x‌sin‌α+a‌
cos‌α
x
)10 is ‌
10!
(5!)2
, then the value of a is equal to
[2021, 25 July Shift-II]
If the greatest value of the term independent of x in the expansion of (x‌sin‌α+a‌
cos‌α
x
)10 is ‌
10!
(5!)2
, then the value of a is equal to
[2021, 25 July Shift-II]
Q.108 Correct
Q.108 In-correct
Q.108 Unattempt
For the natural numbers m,n, if (1−y)m(1+y)n
=1+a1y+a2y2+......+am+nym+n and
a1=a2=10, then the value of
(m+n) is equal to
[2021, 20 July Shift-II]
For the natural numbers m,n, if (1−y)m(1+y)n
=1+a1y+a2y2+......+am+nym+n and
a1=a2=10, then the value of
(m+n) is equal to
[2021, 20 July Shift-II]
Q.109 Correct
Q.109 In-correct
Q.109 Unattempt
The coefficient of x256 in the expansion of (1−x)101(x2+x+1)100 is
[2021, 20 July Shift-I]
The coefficient of x256 in the expansion of (1−x)101(x2+x+1)100 is
[2021, 20 July Shift-I]
Q.110 Correct
Q.110 In-correct
Q.110 Unattempt
The number of rational terms in the binomial expansion of (41∕4+51∕6)120 is .......
[2021, 20 July Shift-I]
The number of rational terms in the binomial expansion of (41∕4+51∕6)120 is .......
[2021, 20 July Shift-I]
Q.111 Correct
Q.111 In-correct
Q.111 Unattempt
If the constant term, in Binomial expansion of (2xr+‌
1
x2
)10 is 180 , then r is equal to
[2021, 22 July Shift-II]
If the constant term, in Binomial expansion of (2xr+‌
1
x2
)10 is 180 , then r is equal to
[2021, 22 July Shift-II]
Q.112 Correct
Q.112 In-correct
Q.112 Unattempt
The number of elements in the set {n∈{1,2,3,...,100}:(11)n>(10)n+(9)n} is
[2021, 22 July Shift-II]
The number of elements in the set {n∈{1,2,3,...,100}:(11)n>(10)n+(9)n} is
[2021, 22 July Shift-II]
Q.113 Correct
Q.113 In-correct
Q.113 Unattempt
The lowest integer which is greater than (1+‌
1
10100
)10100 is
[2021, 25 July Shift-11]
The lowest integer which is greater than (1+‌
1
10100
)10100 is
[2021, 25 July Shift-11]
Q.114 Correct
Q.114 In-correct
Q.114 Unattempt
If the co-efficient of x7 and x8 in the expansion of (2+‌
x
3
)n are equal, then the value of n is equal to ..........
[2021, 25 July Shift-II]
If the co-efficient of x7 and x8 in the expansion of (2+‌
x
3
)n are equal, then the value of n is equal to ..........
[2021, 25 July Shift-II]
Q.115 Correct
Q.115 In-correct
Q.115 Unattempt
20
∑
k=0
(‌20Ck)2 is equal to
[2021, 27 Aug. Shift-I]
20
∑
k=0
(‌20Ck)2 is equal to
[2021, 27 Aug. Shift-I]
Q.116 Correct
Q.116 In-correct
Q.116 Unattempt
If ‌20Cr is the coefficient of xr in the expansion of (1+x)20, then the value of
20
∑
r=0
r2‌20Cr is equal to
[2021, 26 Aug. Shift-I]
If ‌20Cr is the coefficient of xr in the expansion of (1+x)20, then the value of
20
∑
r=0
r2‌20Cr is equal to
[2021, 26 Aug. Shift-I]
Q.117 Correct
Q.117 In-correct
Q.117 Unattempt
Let (
n
k
) denotes ‌nCk and
[
n
k
]={
(
n
k
)		‌‌ if ‌0≤k≤n
0‌‌ otherwise ‌.

If Ak=
9
∑
i=0
(
9
i
)[
12
12−k+i
]
+
8
∑
i=0
(
8
i
)[
13
13−k+i
]
and A4−A3=190p, then p is equal to
[2021, 26 Aug. Shift-II]
Let (
n
k
) denotes ‌nCk and
[
n
k
]={
(
n
k
)		‌‌ if ‌0≤k≤n
0‌‌ otherwise ‌.

If Ak=
9
∑
i=0
(
9
i
)[
12
12−k+i
]
+
8
∑
i=0
(
8
i
)[
13
13−k+i
]
and A4−A3=190p, then p is equal to
[2021, 26 Aug. Shift-II]
Q.118 Correct
Q.118 In-correct
Q.118 Unattempt
If the coefficients of x7 in (x2+‌
1
bx
)11 and x−7 in (x−‌
1
bx2
)11,b≠0, are equal, then the value of b is equal to
[2021, 27 July Shift-1]
If the coefficients of x7 in (x2+‌
1
bx
)11 and x−7 in (x−‌
1
bx2
)11,b≠0, are equal, then the value of b is equal to
[2021, 27 July Shift-1]
Q.119 Correct
Q.119 In-correct
Q.119 Unattempt
A possible value of ' x, for which the ninth term in the expansion of
{3log3√25x−1+7+3(−‌
1
8
)log3(5x−1+1)}10
is equal to 180 , is
[2021, 27 July Shift-II]
A possible value of ' x, for which the ninth term in the expansion of
{3log3√25x−1+7+3(−‌
1
8
)log3(5x−1+1)}10
is equal to 180 , is
[2021, 27 July Shift-II]
Q.120 Correct
Q.120 In-correct
Q.120 Unattempt
The ratio of the coefficient of the middle term in the expansion of (1+x)20 and the sum of the coefficients of two middle terms in expansion of (1+x)19 is .......
[2021, 25 July Shift-I]
The ratio of the coefficient of the middle term in the expansion of (1+x)20 and the sum of the coefficients of two middle terms in expansion of (1+x)19 is .......
[2021, 25 July Shift-I]
Q.121 Correct
Q.121 In-correct
Q.121 Unattempt
If (‌
36
44
)k is the term, independent of x, in the binomial expansion of (‌
x
4
−‌
12
x2
)12, then k is equal to
[2021, 31 Aug. Shift-I]
If (‌
36
44
)k is the term, independent of x, in the binomial expansion of (‌
x
4
−‌
12
x2
)12, then k is equal to
[2021, 31 Aug. Shift-I]
Q.122 Correct
Q.122 In-correct
Q.122 Unattempt
3×722+2×1022−44 when divided by 18 leaves the remainder
[2021, 27 Aug. Shift-II]
3×722+2×1022−44 when divided by 18 leaves the remainder
[2021, 27 Aug. Shift-II]
Q.123 Correct
Q.123 In-correct
Q.123 Unattempt
If the coefficient of a7b8 in the expansion of (a+2b+4ab)10 is k⋅216, then k is equal to
[2021, 31 Aug. Shift-II]
If the coefficient of a7b8 in the expansion of (a+2b+4ab)10 is k⋅216, then k is equal to
[2021, 31 Aug. Shift-II]
Q.124 Correct
Q.124 In-correct
Q.124 Unattempt
If the sum of the coefficients of all even powers of x in the product (1+x+x2+...+x2n)(1−x+x2−x3+...+x2n) is 61, then n is equal to _________.
[NA Jan. 7, 2020 (I)]
If the sum of the coefficients of all even powers of x in the product (1+x+x2+...+x2n)(1−x+x2−x3+...+x2n) is 61, then n is equal to _________.
[NA Jan. 7, 2020 (I)]
Q.125 Correct
Q.125 In-correct
Q.125 Unattempt
The coefficient of x4 in the expansion of (1+x+x2)10 is ________.
[NA Jan. 9, 2020 (I)]
The coefficient of x4 in the expansion of (1+x+x2)10 is ________.
[NA Jan. 9, 2020 (I)]
Q.126 Correct
Q.126 In-correct
Q.126 Unattempt
If α and β be the coefficients of x4 and x2 respectively in the expansion of (x+√x2−1)6+(x−√x2−1)6, then:
[Jan. 8, 2020 (II)]
If α and β be the coefficients of x4 and x2 respectively in the expansion of (x+√x2−1)6+(x−√x2−1)6, then:
[Jan. 8, 2020 (II)]
Q.127 Correct
Q.127 In-correct
Q.127 Unattempt
In the expansion of (‌
x
cos‌θ
+‌
1
x‌sin‌θ
)16, if l1 is the least value of the term independent of x when ‌
π
8
≤θ≤‌
π
4
 and l2 is the least value of the term independent of x when ‌
π
16
≤θ≤‌
π
8
, then the ratio l2:l1 is equal to :
[Jan. 9, 2020 (II)]
In the expansion of (‌
x
cos‌θ
+‌
1
x‌sin‌θ
)16, if l1 is the least value of the term independent of x when ‌
π
8
≤θ≤‌
π
4
 and l2 is the least value of the term independent of x when ‌
π
16
≤θ≤‌
π
8
, then the ratio l2:l1 is equal to :
[Jan. 9, 2020 (II)]
Q.128 Correct
Q.128 In-correct
Q.128 Unattempt
If {p} denotes the fractional part of the number p, then {‌
3200
8
}, is equal to :
[Sep. 06, 2020 (I)]
If {p} denotes the fractional part of the number p, then {‌
3200
8
}, is equal to :
[Sep. 06, 2020 (I)]
Q.129 Correct
Q.129 In-correct
Q.129 Unattempt
The natural number m, for which the coefficient of x in the binomial expansion of (xm+‌
1
x2
)22 is 1540, is _______.
[NA Sep. 05, 2020 (I)]
The natural number m, for which the coefficient of x in the binomial expansion of (xm+‌
1
x2
)22 is 1540, is _______.
[NA Sep. 05, 2020 (I)]
Q.130 Correct
Q.130 In-correct
Q.130 Unattempt
The coefficient of x4 in the expansion of (1+x+x2+x3)6 in powers of x, is __________.
[NA Sep. 04, 2020 (I)]
The coefficient of x4 in the expansion of (1+x+x2+x3)6 in powers of x, is __________.
[NA Sep. 04, 2020 (I)]
Q.131 Correct
Q.131 In-correct
Q.131 Unattempt
Let (2x2+3x+4)10=
20
∑
r=0
arxr. Then ‌
a7
a12
 is equal to __________.
[NA Sep. 04, 2020 (I)]
Let (2x2+3x+4)10=
20
∑
r=0
arxr. Then ‌
a7
a12
 is equal to __________.
[NA Sep. 04, 2020 (I)]
Q.132 Correct
Q.132 In-correct
Q.132 Unattempt
If the constant term in the binomial expansion of (√x‌
k
x2
)10 is 405, then |k| equals:
[Sep. 06, 2020 (II)]
If the constant term in the binomial expansion of (√x‌
k
x2
)10 is 405, then |k| equals:
[Sep. 06, 2020 (II)]
Q.133 Correct
Q.133 In-correct
Q.133 Unattempt
If for some positive integer n, the coefficients of three consecutive terms in the binomial expansion of (1+x)n+5 are in the ratio 5:10:14, then the largest coefficient in this expansion is :
[Sep. 04, 2020 (II)]
If for some positive integer n, the coefficients of three consecutive terms in the binomial expansion of (1+x)n+5 are in the ratio 5:10:14, then the largest coefficient in this expansion is :
[Sep. 04, 2020 (II)]
Q.134 Correct
Q.134 In-correct
Q.134 Unattempt
If the number of integral terms in the expansion of (31∕2+51∕8)n is exactly 33, then the least value of n is :
[Sep. 03, 2020 (I)]
If the number of integral terms in the expansion of (31∕2+51∕8)n is exactly 33, then the least value of n is :
[Sep. 03, 2020 (I)]
Q.135 Correct
Q.135 In-correct
Q.135 Unattempt
If the term independent of x in the expansion of (‌
3
2
x2−‌
1
3x
)9 is k, then 18k is equal to :
[Sep. 03, 2020 (II)]
If the term independent of x in the expansion of (‌
3
2
x2−‌
1
3x
)9 is k, then 18k is equal to :
[Sep. 03, 2020 (II)]
Q.136 Correct
Q.136 In-correct
Q.136 Unattempt
Let α>0,β>0 be such that α3+β2=4. If the maximum value of the term independent of x in the binomial expansion of (αx‌
1
9
+βx−‌
1
6
)10 is 10k, then k is equal to :
[Sep. 02, 2020 (I)]
Let α>0,β>0 be such that α3+β2=4. If the maximum value of the term independent of x in the binomial expansion of (αx‌
1
9
+βx−‌
1
6
)10 is 10k, then k is equal to :
[Sep. 02, 2020 (I)]
Q.137 Correct
Q.137 In-correct
Q.137 Unattempt
For a positive integer n,(1+‌
1
x
)n is expanded in increasing powers of x. If three consecutive coefficients in this expansion are in the ratio, 2:5:12, then n is equal to ________.
[NA Sep. 02, 2020 (II)]
For a positive integer n,(1+‌
1
x
)n is expanded in increasing powers of x. If three consecutive coefficients in this expansion are in the ratio, 2:5:12, then n is equal to ________.
[NA Sep. 02, 2020 (II)]
Q.138 Correct
Q.138 In-correct
Q.138 Unattempt
The value of
20
∑
r=0
‌50−rC6 is equal to:
[Sep. 04, 2020 (I)]
The value of
20
∑
r=0
‌50−rC6 is equal to:
[Sep. 04, 2020 (I)]
Q.139 Correct
Q.139 In-correct
Q.139 Unattempt
Let (x+10)50+(x−10)50=a0+a1x+a2x2+....+a50x50 for all x∈R; then ‌
a2
a0
 is equal to:
[Jan. 11, 2019 (II)]
Let (x+10)50+(x−10)50=a0+a1x+a2x2+....+a50x50 for all x∈R; then ‌
a2
a0
 is equal to:
[Jan. 11, 2019 (II)]
Q.140 Correct
Q.140 In-correct
Q.140 Unattempt
If the third term in the binomial expansion of (1+xlog2x)5 equals 2560 , then a possible value of x is:
[Jan. 10, 2019 (I)]
If the third term in the binomial expansion of (1+xlog2x)5 equals 2560 , then a possible value of x is:
[Jan. 10, 2019 (I)]
Q.141 Correct
Q.141 In-correct
Q.141 Unattempt
The positive value of λ for which the co-efficient of x2 in the expression x2(√x+‌
λ
x2
)10 is 720, is:
[Jan. 10, 2019 (II)]
The positive value of λ for which the co-efficient of x2 in the expression x2(√x+‌
λ
x2
)10 is 720, is:
[Jan. 10, 2019 (II)]
Q.142 Correct
Q.142 In-correct
Q.142 Unattempt
If the fractional part of the number ‌
2403
15
 is ‌
k
15
, then k is equal to:
[Jan. 9, 2019 (I)]
If the fractional part of the number ‌
2403
15
 is ‌
k
15
, then k is equal to:
[Jan. 9, 2019 (I)]
Q.143 Correct
Q.143 In-correct
Q.143 Unattempt
The total number is irrational terms in the binomial expansion of (7‌
1
5
−3‌
1
10
)60 is :
Jan. 12, 2019 (II)]
The total number is irrational terms in the binomial expansion of (7‌
1
5
−3‌
1
10
)60 is :
Jan. 12, 2019 (II)]
Q.144 Correct
Q.144 In-correct
Q.144 Unattempt
A ratio of the 5‌th ‌ term from the begining to the 5 th term from the end in the binomial expansion of (2‌
1
3
+‌
1
2(3)‌
1
3
)10 is:
[Jan. 12, 2019 (I)]
A ratio of the 5‌th ‌ term from the begining to the 5 th term from the end in the binomial expansion of (2‌
1
3
+‌
1
2(3)‌
1
3
)10 is:
[Jan. 12, 2019 (I)]
Q.145 Correct
Q.145 In-correct
Q.145 Unattempt
The sum of the real values of x for which the middle term in the binomial expansion of (‌
x3
3
+‌
3
x
)8 equals 5670 is :
[Jan. 11, 2019 (I)]
The sum of the real values of x for which the middle term in the binomial expansion of (‌
x3
3
+‌
3
x
)8 equals 5670 is :
[Jan. 11, 2019 (I)]
Q.146 Correct
Q.146 In-correct
Q.146 Unattempt
The value of r for which ‌20Cr‌20C0+‌20Cr−1‌20C1+‌20Cr−2‌20C2+...+‌20C0‌20Cr is maximum, is :
[Jan. 11, 2019 (I)]
The value of r for which ‌20Cr‌20C0+‌20Cr−1‌20C1+‌20Cr−2‌20C2+...+‌20C0‌20Cr is maximum, is :
[Jan. 11, 2019 (I)]
Q.147 Correct
Q.147 In-correct
Q.147 Unattempt
If
∑
r=0
25{‌50Cr⋅‌50−rC25−r}=K(‌50C25), then K is equal to:
[Jan. 10, 2019 (II)]
If
∑
r=0
25{‌50Cr⋅‌50−rC25−r}=K(‌50C25), then K is equal to:
[Jan. 10, 2019 (II)]
Q.148 Correct
Q.148 In-correct
Q.148 Unattempt
The coefficient of t4 in the expansion of (‌
1−t6
1−t
)3
[Jan. 09, 2019 (II)]
The coefficient of t4 in the expansion of (‌
1−t6
1−t
)3
[Jan. 09, 2019 (II)]
Q.149 Correct
Q.149 In-correct
Q.149 Unattempt
The smallest natural number n, such that the coefficient of x in the expansion of (x2+‌
1
x3
)n is ‌nC23, is :
[April 10, 2019 (II)]
The smallest natural number n, such that the coefficient of x in the expansion of (x2+‌
1
x3
)n is ‌nC23, is :
[April 10, 2019 (II)]
Q.150 Correct
Q.150 In-correct
Q.150 Unattempt
If the fourth term in the Binomial expansion of (‌
2
x
+xlog8x)6(x>0) is 20×87, then a value of x is:
[April 9, 2019 (I)]
If the fourth term in the Binomial expansion of (‌
2
x
+xlog8x)6(x>0) is 20×87, then a value of x is:
[April 9, 2019 (I)]
Q.151 Correct
Q.151 In-correct
Q.151 Unattempt
If some three consecutive coefficients in the binomial expansion of (x+1)n in powers of x are in the ratio 2: 15: 70 , then the average of these three coefficients is:
[April 09, 2019 (II)]
If some three consecutive coefficients in the binomial expansion of (x+1)n in powers of x are in the ratio 2: 15: 70 , then the average of these three coefficients is:
[April 09, 2019 (II)]
Q.152 Correct
Q.152 In-correct
Q.152 Unattempt
The sum of the co-efficients of all even degree terms in x in the expansion of (x+√x3−1)6+(x−√x3−1)6,(x> 1) is equal to :
[April 8, 2019 (I)]
The sum of the co-efficients of all even degree terms in x in the expansion of (x+√x3−1)6+(x−√x3−1)6,(x> 1) is equal to :
[April 8, 2019 (I)]
Q.153 Correct
Q.153 In-correct
Q.153 Unattempt
If the fourth term in the binomial expansion of (√‌
1
x1+log10x
+x‌
1
12
)6 is equal to 200, and x>1, then the value of x is:
[April 08, 2019 (II)]
If the fourth term in the binomial expansion of (√‌
1
x1+log10x
+x‌
1
12
)6 is equal to 200, and x>1, then the value of x is:
[April 08, 2019 (II)]
Q.154 Correct
Q.154 In-correct
Q.154 Unattempt
The term independent of x in the expansion of (‌
1
60
−‌
x8
81
)⋅(2x2−‌
3
x2
)6 is equal to :
[NA April 12, 2019 (II)]
The term independent of x in the expansion of (‌
1
60
−‌
x8
81
)⋅(2x2−‌
3
x2
)6 is equal to :
[NA April 12, 2019 (II)]
Q.155 Correct
Q.155 In-correct
Q.155 Unattempt
If20C1+(22)20C2+(32)20C3+.........+(202)20C20=A(2β), then the ordered pair (A,β) is equal to :
[April 12, 2019 (II)]
If20C1+(22)20C2+(32)20C3+.........+(202)20C20=A(2β), then the ordered pair (A,β) is equal to :
[April 12, 2019 (II)]
Q.156 Correct
Q.156 In-correct
Q.156 Unattempt
The coefficient of x18 in the product (1+x)(1−x)10 (1+x+x2)9 is :
[April 12, 2019 (I)]
The coefficient of x18 in the product (1+x)(1−x)10 (1+x+x2)9 is :
[April 12, 2019 (I)]
Q.157 Correct
Q.157 In-correct
Q.157 Unattempt
If the coefficients of x2 and x3 are both zero, in the expansion of the expression (1+ax+bx2)(1−3x)15 in powers of x, then the ordered pair (a,b) is equal to:
[April 10, 2019 (I)]
If the coefficients of x2 and x3 are both zero, in the expansion of the expression (1+ax+bx2)(1−3x)15 in powers of x, then the ordered pair (a,b) is equal to:
[April 10, 2019 (I)]
Q.158 Correct
Q.158 In-correct
Q.158 Unattempt
The sum of the series 2.20C0+5⋅‌20C1+8⋅‌.20C2+11‌20C3+...+62‌.20C20 is equal to :
[April 8, 2019 (I)]
The sum of the series 2.20C0+5⋅‌20C1+8⋅‌.20C2+11‌20C3+...+62‌.20C20 is equal to :
[April 8, 2019 (I)]
Q.159 Correct
Q.159 In-correct
Q.159 Unattempt
The coefficient of x10 in the expansion of (1+x)2(1+x2)3 (1+x3)4 is equal to
[Online April 15, 2018]
The coefficient of x10 in the expansion of (1+x)2(1+x2)3 (1+x3)4 is equal to
[Online April 15, 2018]
Q.160 Correct
Q.160 In-correct
Q.160 Unattempt
If n is the degree of the polynomial, [‌
1
√5x3+1−√5x3−1
]8+[‌
1
√5x3+1+√5x3−1
]8 and m is the coefficient of xn in it, then the ordered pair (n,m) is equal to
[Online April 15, 2018]
If n is the degree of the polynomial, [‌
1
√5x3+1−√5x3−1
]8+[‌
1
√5x3+1+√5x3−1
]8 and m is the coefficient of xn in it, then the ordered pair (n,m) is equal to
[Online April 15, 2018]
Q.161 Correct
Q.161 In-correct
Q.161 Unattempt
The coefficient of x2 in the expansion of the product (2−x2)⋅((1+2x+3x2)6+(1−4x2)6) is
[Online April 16, 2018]
The coefficient of x2 in the expansion of the product (2−x2)⋅((1+2x+3x2)6+(1−4x2)6) is
[Online April 16, 2018]
Q.162 Correct
Q.162 In-correct
Q.162 Unattempt
The sum of the co-efficients of all odd degree terms in the expansion of (x+√x3−1)5+(x−√x3−1)5,(x>1) is :
[2018]
The sum of the co-efficients of all odd degree terms in the expansion of (x+√x3−1)5+(x−√x3−1)5,(x>1) is :
[2018]
Q.163 Correct
Q.163 In-correct
Q.163 Unattempt
If (27)999 is divided by 7, then the remainder is:
[Online April 8, 2017]
If (27)999 is divided by 7, then the remainder is:
[Online April 8, 2017]
Q.164 Correct
Q.164 In-correct
Q.164 Unattempt
The coefficient of x−5 in the binomial expansion of (‌
x+1
x‌
2
3
−x‌
1
3
+1
−‌
x−1
x−x‌
1
2
)10 where x≠0,1, is
[Online April 9, 2017]
The coefficient of x−5 in the binomial expansion of (‌
x+1
x‌
2
3
−x‌
1
3
+1
−‌
x−1
x−x‌
1
2
)10 where x≠0,1, is
[Online April 9, 2017]
Q.165 Correct
Q.165 In-correct
Q.165 Unattempt
The value of (‌21C1−‌10C1)+(‌21C2−‌10C2)+(‌21C3−‌10C3)+(‌21C4−‌10C4) +....+(‌21C10−‌10C10) is:
[2017]
The value of (‌21C1−‌10C1)+(‌21C2−‌10C2)+(‌21C3−‌10C3)+(‌21C4−‌10C4) +....+(‌21C10−‌10C10) is:
[2017]
Q.166 Correct
Q.166 In-correct
Q.166 Unattempt
If the coefficients of x−2 and x−4 in the expansion of (x‌
1
3
+‌
1
2x‌
1
3
)18,(x>0), are m and n respectively, then ‌
m
n
 is equal to :
[Online April 10, 2016]
If the coefficients of x−2 and x−4 in the expansion of (x‌
1
3
+‌
1
2x‌
1
3
)18,(x>0), are m and n respectively, then ‌
m
n
 is equal to :
[Online April 10, 2016]
Q.167 Correct
Q.167 In-correct
Q.167 Unattempt
If the number of terms in the expansion of (1−‌
2
x
+‌
4
x2
)n, x≠0, is 28, then the sum of the coefficients of all the terms in this expansion, is:
[2016]
If the number of terms in the expansion of (1−‌
2
x
+‌
4
x2
)n, x≠0, is 28, then the sum of the coefficients of all the terms in this expansion, is:
[2016]
Q.168 Correct
Q.168 In-correct
Q.168 Unattempt
If the coefficients of the three successive terms in the binomial expansion of (1+x)n are in the ratio 1:7:42, then the first of these terms in the expansion is:
[Online April 10, 2015]
If the coefficients of the three successive terms in the binomial expansion of (1+x)n are in the ratio 1:7:42, then the first of these terms in the expansion is:
[Online April 10, 2015]
Q.169 Correct
Q.169 In-correct
Q.169 Unattempt
The term independent of x in the binomial expansion of (1−‌
1
x
+3x5)(2x2−‌
1
x
)8 is :
[Online April 11, 2015]
The term independent of x in the binomial expansion of (1−‌
1
x
+3x5)(2x2−‌
1
x
)8 is :
[Online April 11, 2015]
Q.170 Correct
Q.170 In-correct
Q.170 Unattempt
The sum of coefficients of integral power of x in the binomial expansion (1−2√x)50 is :
[2015]
The sum of coefficients of integral power of x in the binomial expansion (1−2√x)50 is :
[2015]
Q.171 Correct
Q.171 In-correct
Q.171 Unattempt
If the coefficents of x3 and x4 in the expansion of (1+ax+bx2)(1−2x)18 in powers of x are both zero, then (a,b) is equal to:
[2014]
If the coefficents of x3 and x4 in the expansion of (1+ax+bx2)(1−2x)18 in powers of x are both zero, then (a,b) is equal to:
[2014]
Q.172 Correct
Q.172 In-correct
Q.172 Unattempt
If X={4n−3n−1:n∈N} and Y={9(n−1):n∈N}, where N is the set of natural numbers, then X∪Y is equal to:
[2014]
If X={4n−3n−1:n∈N} and Y={9(n−1):n∈N}, where N is the set of natural numbers, then X∪Y is equal to:
[2014]
Q.173 Correct
Q.173 In-correct
Q.173 Unattempt
The number of terms in the expansion of (1+x)101(1+x2−x)100 in powers of x is:
[Online April 9, 2014]
The number of terms in the expansion of (1+x)101(1+x2−x)100 in powers of x is:
[Online April 9, 2014]
Q.174 Correct
Q.174 In-correct
Q.174 Unattempt
If 1+x4+x5=
5
∑
i=0
ai(1+x)i, for all x in R, then a2 is:
[Online April 12, 2014]
If 1+x4+x5=
5
∑
i=0
ai(1+x)i, for all x in R, then a2 is:
[Online April 12, 2014]
Q.175 Correct
Q.175 In-correct
Q.175 Unattempt
If (2+‌
x
3
)55 is expanded in the ascending powers of x and the coefficients of powers of x in two consecutive terms of the expansion are equal, then these terms are:
[Online April 12, 2014]
If (2+‌
x
3
)55 is expanded in the ascending powers of x and the coefficients of powers of x in two consecutive terms of the expansion are equal, then these terms are:
[Online April 12, 2014]
Q.176 Correct
Q.176 In-correct
Q.176 Unattempt
The coefficient of x1012 in the expansion of (1+xn+x253)10, (where n≤22 is any positive integer ), is
[Online April 19, 2014]
The coefficient of x1012 in the expansion of (1+xn+x253)10, (where n≤22 is any positive integer ), is
[Online April 19, 2014]
Q.177 Correct
Q.177 In-correct
Q.177 Unattempt
If the 7 th term in the binomial expansion of (‌
3
√[3]84
+√3‌ln‌x)9,x>0, is equal to 729, then x can be
[Online April 22, 2013
If the 7 th term in the binomial expansion of (‌
3
√[3]84
+√3‌ln‌x)9,x>0, is equal to 729, then x can be
[Online April 22, 2013
Q.178 Correct
Q.178 In-correct
Q.178 Unattempt
If for positive integers r>1,n>2, the coefficients of the (3r)‌th ‌ and (r+2)‌th ‌ powers of x in the expansion of (1+x)2n are equal, then n is equal to :
[Online April 25, 2013]
If for positive integers r>1,n>2, the coefficients of the (3r)‌th ‌ and (r+2)‌th ‌ powers of x in the expansion of (1+x)2n are equal, then n is equal to :
[Online April 25, 2013]
Q.179 Correct
Q.179 In-correct
Q.179 Unattempt
The sum of the rational terms in the binomial expansion of (2‌
1
2
+3‌
1
5
)10 is:
[Online April 23, 2013]
The sum of the rational terms in the binomial expansion of (2‌
1
2
+3‌
1
5
)10 is:
[Online April 23, 2013]
Q.180 Correct
Q.180 In-correct
Q.180 Unattempt
The term independent of x in expansion of (‌
x+1
x2∕3−x1∕3+1
−‌
x−1
x−x1∕2
)10 is
[2013]
The term independent of x in expansion of (‌
x+1
x2∕3−x1∕3+1
−‌
x−1
x−x1∕2
)10 is
[2013]
Q.181 Correct
Q.181 In-correct
Q.181 Unattempt
The ratio of the coefficient of x15 to the term independent of x in the expansion of (x2+‌
2
x
)15 is :
[Online April 9, 2013]
The ratio of the coefficient of x15 to the term independent of x in the expansion of (x2+‌
2
x
)15 is :
[Online April 9, 2013]
Q.182 Correct
Q.182 In-correct
Q.182 Unattempt
If n is a positive integer, then (√3+1)2n−(√3−1)2n is
[2012]
If n is a positive integer, then (√3+1)2n−(√3−1)2n is
[2012]
Q.183 Correct
Q.183 In-correct
Q.183 Unattempt
Iff(y)=1−(y−1)+(y−1)2−(y−1)3
‌‌+...−(y−1)17 then the coefficient of y2 in it is [O
[Online May 7, 2012]
Iff(y)=1−(y−1)+(y−1)2−(y−1)3
‌‌+...−(y−1)17 then the coefficient of y2 in it is [O
[Online May 7, 2012]
Q.184 Correct
Q.184 In-correct
Q.184 Unattempt
The number of terms in the expansion of (y1∕5+x1∕10)55, in which powers of x and y are free from radical signs are
[Online May 12, 2012]
The number of terms in the expansion of (y1∕5+x1∕10)55, in which powers of x and y are free from radical signs are
[Online May 12, 2012]
Q.185 Correct
Q.185 In-correct
Q.185 Unattempt
The middle term in the expansion of (1−‌
1
x
)n(1−x)n in powers of x is
[Online May 26, 2012]
The middle term in the expansion of (1−‌
1
x
)n(1−x)n in powers of x is
[Online May 26, 2012]
Q.186 Correct
Q.186 In-correct
Q.186 Unattempt
Statement -1: For each natural number n,(n+1)7−1 is divisible by 7 .
Statement -2: For each natural number n,n7−n is divisible by 7 .
[2011 RS]
Statement -1: For each natural number n,(n+1)7−1 is divisible by 7 .
Statement -2: For each natural number n,n7−n is divisible by 7 .
[2011 RS]
Q.187 Correct
Q.187 In-correct
Q.187 Unattempt
The coefficient of x7 in the expansion of (1−x−x2+x3)6 is
[2011]
The coefficient of x7 in the expansion of (1−x−x2+x3)6 is
[2011]
Q.188 Correct
Q.188 In-correct
Q.188 Unattempt
Let S1=
10
∑
j=1
j(j−1)10CJ,S2=
10
∑
j=1
j10Cj and S3=
10
∑
j=1
j210Cj.
Statement -1: S3=55×29
Statement -2: S1=90×28 and S2=10×28
[2010]
Let S1=
10
∑
j=1
j(j−1)10CJ,S2=
10
∑
j=1
j10Cj and S3=
10
∑
j=1
j210Cj.
Statement -1: S3=55×29
Statement -2: S1=90×28 and S2=10×28
[2010]
Q.189 Correct
Q.189 In-correct
Q.189 Unattempt
The remainder left out when 82n−(62)2n+1 is divided by 9 is:
[2009]
The remainder left out when 82n−(62)2n+1 is divided by 9 is:
[2009]
Q.190 Correct
Q.190 In-correct
Q.190 Unattempt
Statement -1:
n
∑
r=0
(r+1)‌nCr=(n+2)2n−1
Statement-2:
n
∑
r=0
(r+1)‌nCrxr=(1+x)n+nx(1+x)n−1.
[2008]
Statement -1:
n
∑
r=0
(r+1)‌nCr=(n+2)2n−1
Statement-2:
n
∑
r=0
(r+1)‌nCrxr=(1+x)n+nx(1+x)n−1.
[2008]
Q.191 Correct
Q.191 In-correct
Q.191 Unattempt
In a shop there are five types of ice-creams available. A child buys six ice-creams.
Statement-1 : The number of different ways the child can buy the six ice-creams is ‌10C5
Statement -2: The number of different ways the child can buy the six ice-creams is equal to the number of different ways of arranging 6A 's and 4B 's in a row.
[2008]
In a shop there are five types of ice-creams available. A child buys six ice-creams.
Statement-1 : The number of different ways the child can buy the six ice-creams is ‌10C5
Statement -2: The number of different ways the child can buy the six ice-creams is equal to the number of different ways of arranging 6A 's and 4B 's in a row.
[2008]
Q.192 Correct
Q.192 In-correct
Q.192 Unattempt
In the binomial expansion of (a−b)n,n≥5, the sum of 5‌th ‌ and 6‌th ‌ terms is zero, then a/b equals
[2007]
In the binomial expansion of (a−b)n,n≥5, the sum of 5‌th ‌ and 6‌th ‌ terms is zero, then a/b equals
[2007]
Q.193 Correct
Q.193 In-correct
Q.193 Unattempt
The sum of the series ‌20C0−‌20C1+‌20C2−‌20C3+.....−.....+‌20C10 is
[2007]
The sum of the series ‌20C0−‌20C1+‌20C2−‌20C3+.....−.....+‌20C10 is
[2007]
Q.194 Correct
Q.194 In-correct
Q.194 Unattempt
For natural numbers m,n if (1−y)m(1+y)n =1+a1y+a2y2+....... and a1=a2=10, then (m,n) is
[2006]
For natural numbers m,n if (1−y)m(1+y)n =1+a1y+a2y2+....... and a1=a2=10, then (m,n) is
[2006]
Q.195 Correct
Q.195 In-correct
Q.195 Unattempt
If the coefficient of x7 in [ax2+(‌
1
bx
)]11 equals the coefficient of x−7 in [ax−(‌
1
bx2
)]11, then a and b satisfy the relation
[2005]
If the coefficient of x7 in [ax2+(‌
1
bx
)]11 equals the coefficient of x−7 in [ax−(‌
1
bx2
)]11, then a and b satisfy the relation
[2005]
Q.196 Correct
Q.196 In-correct
Q.196 Unattempt
If x is so small that x3 and higher powers of x may be neglected, then ‌
(1+x)
3
2
−(1+
1
2
x)3
(1−x)‌
1
2
 may be approximated as
[2005]
If x is so small that x3 and higher powers of x may be neglected, then ‌
(1+x)
3
2
−(1+
1
2
x)3
(1−x)‌
1
2
 may be approximated as
[2005]
Q.197 Correct
Q.197 In-correct
Q.197 Unattempt
The coefficient of xn in expansion of (1+x)(1−x)n is
[2004]
The coefficient of xn in expansion of (1+x)(1−x)n is
[2004]
Q.198 Correct
Q.198 In-correct
Q.198 Unattempt
The coefficient of the middle term in the binomial expansion in powers of x of (1+αx)4 and of (1−αx)6 is the same if α equals
[2004]
The coefficient of the middle term in the binomial expansion in powers of x of (1+αx)4 and of (1−αx)6 is the same if α equals
[2004]
Q.199 Correct
Q.199 In-correct
Q.199 Unattempt
The number of integral terms in the expansion of (√3+8√5)256 is
[2003]
The number of integral terms in the expansion of (√3+8√5)256 is
[2003]
Q.200 Correct
Q.200 In-correct
Q.200 Unattempt
If x is positive, the first negative term in the expansion of (1+x)27∕5 is
[2003]
If x is positive, the first negative term in the expansion of (1+x)27∕5 is
[2003]
Q.201 Correct
Q.201 In-correct
Q.201 Unattempt
r and n are positive integers r>1,n>2 and coefficient of (r+2)‌th ‌ term and 3r‌th ‌ term in the expansion of (1+x)2n are equal, then n equals
[2002]
r and n are positive integers r>1,n>2 and coefficient of (r+2)‌th ‌ term and 3r‌th ‌ term in the expansion of (1+x)2n are equal, then n equals
[2002]
Q.202 Correct
Q.202 In-correct
Q.202 Unattempt
The coefficients of xp and xq in the expansion of (1+x)p+q are
[2002]
The coefficients of xp and xq in the expansion of (1+x)p+q are
[2002]
Q.203 Correct
Q.203 In-correct
Q.203 Unattempt
The positive integer just greater than (1+0.0001)10000 is
[2002]
The positive integer just greater than (1+0.0001)10000 is
[2002]
Q.204 Correct
Q.204 In-correct
Q.204 Unattempt
If the sum of the coefficients in the expansion of (a+b)n is 4096, then the greatest coefficient in the expansion is
[2002]
If the sum of the coefficients in the expansion of (a+b)n is 4096, then the greatest coefficient in the expansion is
[2002]
Get latest Exam Updates
& Study Material Alerts!
No, Thanks
Click on Allow to receive notifications
×
Open Now