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Binomial Theorem

244 Questions
All Exams JEE Mains JEE Advanced
2020 JEE Mains MCQ
JEE Main 2020 (Online) 8th January Evening Slot
If $\alpha $ and $\beta $ be the coefficients of x4 and x2 respectively in the expansion of
${\left( {x + \sqrt {{x^2} - 1} } \right)^6} + {\left( {x - \sqrt {{x^2} - 1} } \right)^6}$, then
A.
$\alpha + \beta = 60$
B.
$\alpha - \beta = 60$
C.
$\alpha + \beta = -30$
D.
$\alpha - \beta = -132$
2020 JEE Mains MCQ
JEE Main 2020 (Online) 7th January Evening Slot
The coefficient of x7 in the expression
(1 + x)10 + x(1 + x)9 + x2(1 + x)8 + ......+ x10 is:
A.
120
B.
330
C.
420
D.
210
2020 JEE Mains MCQ
JEE Main 2020 (Online) 7th January Morning Slot
The greatest positive integer k, for which 49k + 1 is a factor of the sum
49125 + 49124 + ..... + 492 + 49 + 1, is:
A.
32
B.
60
C.
63
D.
65
2019 JEE Mains MCQ
JEE Main 2019 (Online) 12th April Evening Slot
If 20C1 + (22) 20C2 + (32) 20C3 + ..... + (202 ) 20C20 = A(2$\beta $), then the ordered pair (A, $\beta $) is equal to :
A.
(420, 19)
B.
(420, 18)
C.
(380, 18)
D.
(380, 19)
2019 JEE Mains MCQ
JEE Main 2019 (Online) 12th April Evening Slot
The term independent of x in the expansion of
$\left( {{1 \over {60}} - {{{x^8}} \over {81}}} \right).{\left( {2{x^2} - {3 \over {{x^2}}}} \right)^6}$ is equal to :
A.
36
B.
- 108
C.
- 36
D.
- 72
2019 JEE Mains MCQ
JEE Main 2019 (Online) 12th April Morning Slot
The coefficient of x18 in the product
(1 + x) (1 – x)10 (1 + x + x2)9 is :
A.
126
B.
- 84
C.
- 126
D.
84
2019 JEE Mains MCQ
JEE Main 2019 (Online) 10th April Evening Slot
The smallest natural number n, such that the coefficient of x in the expansion of ${\left( {{x^2} + {1 \over {{x^3}}}} \right)^n}$ is nC23, is :
A.
23
B.
58
C.
38
D.
35
2019 JEE Mains MCQ
JEE Main 2019 (Online) 10th April Morning Slot
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 :
A.
(28, 861)
B.
(28, 315)
C.
(–21, 714)
D.
(–54, 315)
2019 JEE Mains MCQ
JEE Main 2019 (Online) 9th April Evening Slot
If some three consecutive in the binomial expansion of (x + 1)n is powers of x are in the ratio 2 : 15 : 70, then the average of these three coefficient is :-
A.
625
B.
227
C.
964
D.
232
2019 JEE Mains MCQ
JEE Main 2019 (Online) 9th April Morning Slot
If the fourth term in the binomial expansion of ${\left( {{2 \over x} + {x^{{{\log }_8}x}}} \right)^6}$ (x > 0) is 20 × 87, then a value of x is :
A.
8–2
B.
82
C.
83
D.
8
2019 JEE Mains MCQ
JEE Main 2019 (Online) 8th April Evening Slot
If the fourth term in the binomial expansion of
${\left( {\sqrt {{x^{\left( {{1 \over {1 + {{\log }_{10}}x}}} \right)}}} + {x^{{1 \over {12}}}}} \right)^6}$ is equal to 200, and x > 1, then the value of x is :
A.
100
B.
103
C.
10
D.
104
2019 JEE Mains MCQ
JEE Main 2019 (Online) 8th April Morning Slot
The sum of the co-efficients of all even degree terms in x in the expansion of
${\left( {x + \sqrt {{x^3} - 1} } \right)^6}$ + ${\left( {x - \sqrt {{x^3} - 1} } \right)^6}$, (x > 1) is equal to:
A.
32
B.
26
C.
29
D.
24
2019 JEE Mains MCQ
JEE Main 2019 (Online) 8th April Morning Slot
The sum of the series

2.20C0 + 5.20C1 + 8.20C2 + 11.20C3 + ... +62.20C20 is equal to :
A.
225
B.
224
C.
226
D.
223
2019 JEE Mains MCQ
JEE Main 2019 (Online) 12th January Evening Slot
The total number of irrational terms in the binomial expansion of (71/5 – 31/10)60 is :
A.
54
B.
55
C.
49
D.
48
2019 JEE Mains MCQ
JEE Main 2019 (Online) 12th January Morning Slot
A ratio of the 5th term from the beginning to the 5th term from the end in the binomial expansion of ${\left( {{2^{1/3}} + {1 \over {2{{\left( 3 \right)}^{1/3}}}}} \right)^{10}}$ is :
A.
1 : 2(6)1/3
B.
1 : 4(6)1/3
C.
2(36)1/3 : 1
D.
4(36)1/3 : 1
2019 JEE Mains MCQ
JEE Main 2019 (Online) 11th January Evening Slot
Let (x + 10)50 + (x $-$ 10)50 = a0 + a1x + a2x2 + . . . . + a50x50, for all x $ \in $ R; then ${{{a_2}} \over {{a_0}}}$ is equal to
A.
12.25
B.
12.75
C.
12.00
D.
12.50
2019 JEE Mains MCQ
JEE Main 2019 (Online) 11th January Evening Slot
Let Sn = 1 + q + q2 + . . . . . + qn and Tn = 1 + $\left( {{{q + 1} \over 2}} \right) + {\left( {{{q + 1} \over 2}} \right)^2}$ + . . . . . .+ ${\left( {{{q + 1} \over 2}} \right)^n}$ where q is a real number and q $ \ne $ 1. If   101C1 + 101C2 . S1 + .... + 101C101 . S100 = $\alpha $T100 then $\alpha $ is equal to
A.
202
B.
200
C.
2100
D.
299
2019 JEE Mains MCQ
JEE Main 2019 (Online) 11th January Morning Slot
The sum of the real values of x for which the middle term in the binomial expansion of ${\left( {{{{x^3}} \over 3} + {3 \over x}} \right)^8}$ equals 5670 is :
A.
0
B.
8
C.
6
D.
4
2019 JEE Mains MCQ
JEE Main 2019 (Online) 11th January Morning Slot
The value of r for which 20Cr 20C0 + 20Cr$-$1 20C1 + 20Cr$-$2 20C2 + . . . . .+ 20C0 20Cr  is maximum, is
A.
20
B.
15
C.
10
D.
11
2019 JEE Mains MCQ
JEE Main 2019 (Online) 10th January Evening Slot
The positive value of $\lambda $ for which the co-efficient of x2 in the expression x2 ${\left( {\sqrt x + {\lambda \over {{x^2}}}} \right)^{10}}$ is 720, is -
A.
4
B.
$2\sqrt 2 $
C.
3
D.
$\sqrt 5 $
2019 JEE Mains MCQ
JEE Main 2019 (Online) 10th January Morning Slot
If  ${\sum\limits_{i = 1}^{20} {\left( {{{{}^{20}{C_{i - 1}}} \over {{}^{20}{C_i} + {}^{20}{C_{i - 1}}}}} \right)} ^3} = {k \over {21}}$  then k is equal to
A.
100
B.
200
C.
50
D.
400
2019 JEE Mains MCQ
JEE Main 2019 (Online) 10th January Morning Slot
If the third term in the binomial expansion
of ${\left( {1 + {x^{{{\log }_2}x}}} \right)^5}$ equals 2560, then a possible value of x is -
A.
$2\sqrt 2 $
B.
$4\sqrt 2 $
C.
${1 \over 8}$
D.
${1 \over 4}$
2019 JEE Mains MCQ
JEE Main 2019 (Online) 9th January Evening Slot
The coefficient of t4 in the expansion of ${\left( {{{1 - {t^6}} \over {1 - t}}} \right)^3}$ is :
A.
14
B.
15
C.
10
D.
12
2019 JEE Mains MCQ
JEE Main 2019 (Online) 9th January Morning Slot
If the fractional part of the number $\left\{ {{{{2^{403}}} \over {15}}} \right\} is \, {k \over {15}}$, then k is equal to :
A.
8
B.
14
C.
6
D.
1
2018 JEE Mains MCQ
JEE Main 2018 (Online) 16th April Morning Slot
The coefficient of x2 in the expansion of the product
(2$-$x2) .((1 + 2x + 3x2)6 + (1 $-$ 4x2)6) is :
A.
107
B.
106
C.
108
D.
155
2018 JEE Mains MCQ
JEE Main 2018 (Offline)
The sum of the co-efficients of all odd degree terms in the expansion of

${\left( {x + \sqrt {{x^3} - 1} } \right)^5} + {\left( {x - \sqrt {{x^3} - 1} } \right)^5}$, $\left( {x > 1} \right)$ is
A.
2
B.
-1
C.
0
D.
1
2018 JEE Mains MCQ
JEE Main 2018 (Online) 15th April Evening Slot
The coefficien of x10 in the expansion of (1 + x)2(1 + x2)3(1 + x3)4 is equal to :
A.
52
B.
56
C.
50
D.
44
2018 JEE Mains MCQ
JEE Main 2018 (Online) 15th April Morning Slot
If n is the degree of the polynomial,

${\left[ {{2 \over {\sqrt {5{x^3} + 1} - \sqrt {5{x^3} - 1} }}} \right]^8} + $ ${\left[ {{2 \over {\sqrt {5{x^3} + 1} + \sqrt {5{x^3} - 1} }}} \right]^8}$

and m is the coefficient of xn in it, then the ordered pair (n, m) is equal to :
A.
(24, (10)8)
B.
(8, 5(10)4)
C.
(12, (20)4)
D.
(12, 8(10)4)
2017 JEE Mains MCQ
JEE Main 2017 (Online) 9th April Morning Slot
The coefficient of x−5 in the binomial expansion of

${\left( {{{x + 1} \over {{x^{{2 \over 3}}} - {x^{{1 \over 3}}} + 1}} - {{x - 1} \over {x - {x^{{1 \over 2}}}}}} \right)^{10}},$ where x $ \ne $ 0, 1, is :
A.
1
B.
4
C.
$-$ 4
D.
$-$ 1
2017 JEE Mains MCQ
JEE Main 2017 (Online) 8th April Morning Slot
If (27)999 is divided by 7, then the remainder is :
A.
1
B.
2
C.
3
D.
6
2017 JEE Mains MCQ
JEE Main 2017 (Offline)
The value of $\left( {{}^{21}{C_1} - {}^{10}{C_1}} \right) + \left( {{}^{21}{C_2} - {}^{10}{C_2}} \right) + \left( {{}^{21}{C_3} - {}^{10}{C_3}} \right)$
$\left( {{}^{21}{C_4} - {}^{10}{C_4}} \right)$$ + .... + \left( {{}^{21}{C_{10}} - {}^{10}{C_{10}}} \right)$ is
A.
${2^{21}} - {2^{10}}$
B.
${2^{20}} - {2^{9}}$
C.
${2^{20}} - {2^{10}}$
D.
${2^{21}} - {2^{11}}$
2016 JEE Mains MCQ
JEE Main 2016 (Online) 10th April Morning Slot
If the coefficients of x−2 and x−4 in the expansion of ${\left( {{x^{{1 \over 3}}} + {1 \over {2{x^{{1 \over 3}}}}}} \right)^{18}},\left( {x > 0} \right),$ are m and n respectively, then ${m \over n}$ is equal to :
A.
182
B.
${4 \over 5}$
C.
${5 \over 4}$
D.
27
2016 JEE Mains MCQ
JEE Main 2016 (Online) 9th April Morning Slot
For x $ \in $ R, x $ \ne $ -1,

if (1 + x)2016 + x(1 + x)2015 + x2(1 + x)2014 + . . . . + x2016 =

$\sum\limits_{i = 0}^{2016} {{a_i}} \,{x^i},\,\,$ then a17 is equal to :
A.
${{2017!} \over {17!\,\,\,2000!}}$
B.
${{2016!} \over {17!\,\,\,1999!}}$
C.
${{2017!} \over {2000!}}$
D.
${{2016!} \over {16!}}$
2016 JEE Mains MCQ
JEE Main 2016 (Offline)
If the number of terms in the expansion of ${\left( {1 - {2 \over x} + {4 \over {{x^2}}}} \right)^n},\,x \ne 0,$ is 28, then the sum of the coefficients of all the terms in this expansion, is :
A.
243
B.
729
C.
64
D.
2187
2015 JEE Mains MCQ
JEE Main 2015 (Offline)
The sum of coefficients of integral power of $x$ in the binomial expansion ${\left( {1 - 2\sqrt x } \right)^{50}}$ is :
A.
${1 \over 2}\left( {{3^{50}} - 1} \right)$
B.
${1 \over 2}\left( {{2^{50}} + 1} \right)$
C.
${1 \over 2}\left( {{3^{50}} + 1} \right)$
D.
${1 \over 2}\left( {{3^{50}}} \right)$
2014 JEE Mains MCQ
JEE Main 2014 (Offline)
If the coefficints of ${x^3}$ and ${x^4}$ in the expansion of $\left( {1 + ax + b{x^2}} \right){\left( {1 - 2x} \right)^{18}}$ in powers of $x$ are both zero, then $\left( {a,\,b} \right)$ is equal to:
A.
$\left( {14,{{272} \over 3}} \right)$
B.
$\left( {16,{{272} \over 3}} \right)$
C.
$\left( {16,{{251} \over 3}} \right)$
D.
$\left( {14,{{251} \over 3}} \right)$
2013 JEE Mains MCQ
JEE Main 2013 (Offline)
The term independent of $x$ in expansion of
${\left( {{{x + 1} \over {{x^{2/3}} - {x^{1/3}} + 1}} - {{x - 1} \over {x - {x^{1/2}}}}} \right)^{10}}$ is
A.
4
B.
120
C.
210
D.
310
2012 JEE Mains MCQ
AIEEE 2012
If $n$ is a positive integer, then ${\left( {\sqrt 3 + 1} \right)^{2n}} - {\left( {\sqrt 3 - 1} \right)^{2n}}$ is :
A.
an irrational number
B.
an odd positive integer
C.
an even positive integer
D.
a rational number other than positive integers
2011 JEE Mains MCQ
AIEEE 2011
The coefficient of ${x^7}$ in the expansion of ${\left( {1 - x - {x^2} + {x^3}} \right)^6}$ is
A.
$-132$
B.
$-144$
C.
$132$
D.
$144$
2010 JEE Mains MCQ
AIEEE 2010
Let ${s_1} = \sum\limits_{j = 1}^{10} {j\left( {j - 1} \right){}^{10}} {C_j}$,

${{s_2} = \sum\limits_{j = 1}^{10} {} } j.{}^{10}{C_j}$ and

${{s_3} = \sum\limits_{j = 1}^{10} {{j^2}.{}^{10}{C_j}.} }$

Statement-1 : ${{S_3} = 55 \times {2^9}}$.
Statement-2 : ${{S_1} = 90 \times {2^8}}$ and ${{S_2} = 10 \times {2^8}}$.

A.
Statement - 1 is true, Statement- 2 is true; Statement - 2 is not a correct explanation for Statement - 1.
B.
Statement - 1 is true, Statement-2 is false.
C.
Statement - 1 is false, Statement-2 is true.
D.
Statement - 1 is true, Statement-2 is true: -Statement - 2 is a correct explanation for Statement - 1.
2009 JEE Mains MCQ
AIEEE 2009
The remainder left out when ${8^{2n}} - {\left( {62} \right)^{2n + 1}}$ is divided by 9 is :
A.
2
B.
7
C.
8
D.
0
2008 JEE Mains MCQ
AIEEE 2008
Statement - 1 : $\sum\limits_{r = 0}^n {\left( {r + 1} \right)\,{}^n{C_r} = \left( {n + 2} \right){2^{n - 1}}.} $
Statement - 2 : $\sum\limits_{r = 0}^n {\left( {r + 1} \right)\,{}^n{C_r}{x^r} = {{\left( {1 + x} \right)}^n} + nx{{\left( {1 + x} \right)}^{n - 1}}.} $
A.
Statement - 1 is false, Statement - 2 is true
B.
Statement - 1 is true, Statement - 2 is true; Statement - 2 is a correct explanation for Statement - 1
C.
Statement - 1 is true, Statement - 2 is true; Statement - 2 is not a correct explanation for Statement - 1
D.
Statement - 1 is true, Statement - 2 is false
2007 JEE Mains MCQ
AIEEE 2007
The sum of the series ${}^{20}{C_0} - {}^{20}{C_1} + {}^{20}{C_2} - {}^{20}{C_3} + .....\, - \,.....\, + {}^{20}{C_{10}}$ is
A.
$0$
B.
${}^{20}{C_{10}}$
C.
$ - {}^{20}{C_{10}}$
D.
${1 \over 2}{}^{20}{C_{10}}$
2007 JEE Mains MCQ
AIEEE 2007
In the binomial expansion of ${\left( {a - b} \right)^n},\,\,\,n \ge 5,$ the sum of ${5^{th}}$ and ${6^{th}}$ terms is zero, then $a/b$ equals
A.
${{n - 5} \over 6}$
B.
${{n - 4} \over 5}$
C.
${5 \over {n - 4}}$
D.
${6 \over {n - 5}}$
2006 JEE Mains MCQ
AIEEE 2006
For natural numbers $m$ , $n$, if ${\left( {1 - y} \right)^m}{\left( {1 + y} \right)^n}\,\, = 1 + {a_1}y + {a_2}{y^2} + ..........$ and ${a_1} = {a_2} = 10,$ then $\left( {m,\,n} \right)$ is
A.
$\left( {20,\,45} \right)$
B.
$\left( {35,\,20} \right)$
C.
$\left( {45,\,35} \right)$
D.
$\left( {35,\,45} \right)$
2006 JEE Mains MCQ
AIEEE 2006
If the expansion in powers of $x$ of the function ${1 \over {\left( {1 - ax} \right)\left( {1 - bx} \right)}}$ is ${a_0} + {a_1}x + {a_2}{x^2} + {a_3}{x^3}.....$ then ${a_n}$ is
A.
${{{b^n} - {a^n}} \over {b - a}}$
B.
${{{a^n} - {b^n}} \over {b - a}}$
C.
${{{a^{n + 1}} - {b^{n + 1}}} \over {b - a}}$
D.
${{{b^{n + 1}} - {a^{n + 1}}} \over {b - a}}$
2005 JEE Mains MCQ
AIEEE 2005
The value of $\,{}^{50}{C_4} + \sum\limits_{r = 1}^6 {^{56 - r}} {C_3}$ is
A.
${}^{55}{C_4}$
B.
${}^{55}{C_3}$
C.
${}^{56}{C_3}$
D.
${}^{56}{C_4}$
2005 JEE Mains MCQ
AIEEE 2005
If the coefficients of rth, (r+1)th, and (r + 2)th terms in the binomial expansion of ${{\rm{(1 + y )}}^m}$ are in A.P., then m and r satisfy the equation
A.
${m^2} - m(4r - 1) + 4\,{r^2} - 2 = 0$
B.
${m^2} - m(4r + 1) + 4\,{r^2} + 2 = 0$
C.
${m^2} - m(4r + 1) + 4\,{r^2} - 2 = 0$
D.
${m^2} - m(4r - 1) + 4\,{r^2} + 2 = 0$
2005 JEE Mains MCQ
AIEEE 2005
If $x$ is so small that ${x^3}$ and higher powers of $x$ may be neglected, then ${{{{\left( {1 + x} \right)}^{{3 \over 2}}} - {{\left( {1 + {1 \over 2}x} \right)}^3}} \over {{{\left( {1 - x} \right)}^{{1 \over 2}}}}}$ may be approximated as
A.
$1 - {3 \over 8}{x^2}$
B.
$3x + {3 \over 8}{x^2}$
C.
$ - {3 \over 8}{x^2}$
D.
${x \over 2} - {3 \over 8}{x^2}$
2005 JEE Mains MCQ
AIEEE 2005
If the coefficient of ${x^7}$ in ${\left[ {a{x^2} + \left( {{1 \over {bx}}} \right)} \right]^{11}}$ equals the coefficient of ${x^{ - 7}}$ in ${\left[ {ax - \left( {{1 \over {b{x^2}}}} \right)} \right]^{11}}$, then $a$ and $b$ satisfy the relation
A.
$a - b = 1$
B.
$a + b = 1$
C.
${a \over b} = 1$
D.
$ab = 1$