Binomial Theorem
360 Questions
Start JEE Mains Test
2020
Q301
TS-EAMCET
MCQ
20 May 2026
Let $x \in \mathbf{R}$ be so small that the powers of $x$ beyond two are insignificant and negligibly small. For such $x$, if $(1-x)^3(2+x)^6$ is approximated by $a+b x+c x^2$, then $a+b+c=$
A.
-80
B.
144
C.
80
D.
127
2020
Q302
TS-EAMCET
MCQ
20 May 2026
For $0 < x < 1$, the expansion of $\left(1+\frac{1}{x}\right)^{\frac{1}{2}}$ is
A.
$1+\frac{1}{2 x}-\frac{1}{2!}\left(\frac{1}{2 x}\right)^2+\frac{1 \cdot 3}{3!}\left(\frac{1}{2 x}\right)^3-\frac{1 \cdot 3 \cdot 5}{4!}\left(\frac{1}{2 x}\right)^4+\ldots \infty$
B.
$\frac{1}{\sqrt{x}}+\frac{1}{2} \sqrt{x}-\frac{1}{2!} \frac{x \sqrt{x}}{2^2}+\frac{1 \cdot 3}{3!} \frac{x^2 \sqrt{x}}{2^3}-\ldots . \infty$
C.
$1+\frac{1}{\sqrt{x}}+\frac{1}{2} x \sqrt{x}+\frac{1}{2!} \frac{x^2 \sqrt{x}}{2^3}+\frac{1 \cdot 3}{3!} \frac{x^3 \sqrt{x}}{2^4}+\ldots . \infty$
D.
$\frac{1}{\sqrt{x}}+\frac{1}{2 x \sqrt{x}}-\frac{1}{2!}\left(\frac{1}{2 x}\right)^2 \frac{1}{\sqrt{x}}+\frac{1 \cdot 3}{3!}\left(\frac{1}{2 x}\right)^3 \frac{1}{\sqrt{x}}-\ldots \ldots \infty$
2020
Q303
BITSAT
MCQ
11 Jun 2026
The value of ${}^{47}{C_4} + \sum\limits_{r = 1}^5 {{}^{52 - r}{C_3}} $ is equal to
A.
${}^{47}{C_6}$
B.
${}^{52}{C_5}$
C.
${}^{52}{C_4}$
D.
None of these
2020
Q304
BITSAT
MCQ
11 Jun 2026
The coefficient of x8 in the polynomial (x $-$ 1) (x $-$ 2) ..... (x $-$ 10)
A.
2640
B.
1320
C.
1370
D.
2740
2019
Q305
JEE Mains
MCQ
14 Mar 2026
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
Q306
JEE Mains
MCQ
14 Mar 2026
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 :
$\left( {{1 \over {60}} - {{{x^8}} \over {81}}} \right).{\left( {2{x^2} - {3 \over {{x^2}}}} \right)^6}$ is equal to :
A.
36
B.
- 108
C.
- 36
D.
- 72
2019
Q307
JEE Mains
MCQ
14 Mar 2026
The coefficient of x18 in the product
(1 + x) (1 – x)10 (1 + x + x2)9 is :
(1 + x) (1 – x)10 (1 + x + x2)9 is :
A.
126
B.
- 84
C.
- 126
D.
84
2019
Q308
JEE Mains
MCQ
14 Mar 2026
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
Q309
JEE Mains
MCQ
14 Mar 2026
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
Q310
JEE Mains
MCQ
14 Mar 2026
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
Q311
JEE Mains
MCQ
14 Mar 2026
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
Q312
JEE Mains
MCQ
14 Mar 2026
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 :
${\left( {\sqrt {{x^{\left( {{1 \over {1 + {{\log }_{10}}x}}} \right)}}} + {x^{{1 \over {12}}}}} \right)^6}$ is equal to 200, and x > 1, then the value of x is :
A.
100
B.
103
C.
10
D.
104
2019
Q313
JEE Mains
MCQ
14 Mar 2026
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:
${\left( {x + \sqrt {{x^3} - 1} } \right)^6}$ + ${\left( {x - \sqrt {{x^3} - 1} } \right)^6}$, (x > 1) is equal to:
A.
32
B.
26
C.
29
D.
24
2019
Q314
JEE Mains
MCQ
14 Mar 2026
The sum of the series
2.20C0 + 5.20C1 + 8.20C2 + 11.20C3 + ... +62.20C20 is equal to :
2.20C0 + 5.20C1 + 8.20C2 + 11.20C3 + ... +62.20C20 is equal to :
A.
225
B.
224
C.
226
D.
223
2019
Q315
JEE Mains
MCQ
14 Mar 2026
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
Q316
JEE Mains
MCQ
14 Mar 2026
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
Q317
JEE Mains
MCQ
14 Mar 2026
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
Q318
JEE Mains
MCQ
14 Mar 2026
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
Q319
JEE Mains
MCQ
14 Mar 2026
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
Q320
JEE Mains
MCQ
14 Mar 2026
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
Q321
JEE Mains
MCQ
14 Mar 2026
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
Q322
JEE Mains
MCQ
14 Mar 2026
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
Q323
JEE Mains
MCQ
14 Mar 2026
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 -
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
Q324
JEE Mains
MCQ
14 Mar 2026
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
Q325
JEE Mains
MCQ
14 Mar 2026
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
Q326
JEE Mains
MCQ
14 Mar 2026
The coefficient of x2 in the expansion of the product
(2$-$x2) .((1 + 2x + 3x2)6 + (1 $-$ 4x2)6) is :
(2$-$x2) .((1 + 2x + 3x2)6 + (1 $-$ 4x2)6) is :
A.
107
B.
106
C.
108
D.
155
2018
Q327
JEE Mains
MCQ
14 Mar 2026
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
${\left( {x + \sqrt {{x^3} - 1} } \right)^5} + {\left( {x - \sqrt {{x^3} - 1} } \right)^5}$, $\left( {x > 1} \right)$ is
A.
2
B.
-1
C.
0
D.
1
2018
Q328
JEE Mains
MCQ
14 Mar 2026
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
Q329
JEE Mains
MCQ
14 Mar 2026
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 :
${\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
Q330
JEE Mains
MCQ
14 Mar 2026
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 :
${\left( {{{x + 1} \over {{x^{{2 \over 3}}} - {x^{{1 \over 3}}} + 1}} - {{x - 1} \over {x - {x^{{1 \over 2}}}}}} \right)^{10}},$ where x $ \ne $ 0, 1, is :
A.
1
B.
4
C.
$-$ 4
D.
$-$ 1
2017
Q331
JEE Mains
MCQ
14 Mar 2026
If (27)999 is divided by 7, then the remainder is :
A.
1
B.
2
C.
3
D.
6
2017
Q332
JEE Mains
MCQ
14 Mar 2026
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)$$ + .... + \left( {{}^{21}{C_{10}} - {}^{10}{C_{10}}} \right)$ is
$\left( {{}^{21}{C_4} - {}^{10}{C_4}} \right)$$ + .... + \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
Q333
JEE Mains
MCQ
14 Mar 2026
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
Q334
JEE Mains
MCQ
14 Mar 2026
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 :
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
Q335
JEE Mains
MCQ
14 Mar 2026
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
Q336
JEE Mains
MCQ
14 Mar 2026
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
Q337
JEE Mains
MCQ
14 Mar 2026
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
Q338
JEE Mains
MCQ
14 Mar 2026
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
${\left( {{{x + 1} \over {{x^{2/3}} - {x^{1/3}} + 1}} - {{x - 1} \over {x - {x^{1/2}}}}} \right)^{10}}$ is
A.
4
B.
120
C.
210
D.
310
2012
Q339
JEE Mains
MCQ
14 Mar 2026
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
Q340
JEE Mains
MCQ
14 Mar 2026
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
Q341
JEE Mains
MCQ
14 Mar 2026
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
${{s_3} = \sum\limits_{j = 1}^{10} {{j^2}.{}^{10}{C_j}.} }$
${{s_2} = \sum\limits_{j = 1}^{10} {} } j.{}^{10}{C_j}$ 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
Q342
JEE Mains
MCQ
14 Mar 2026
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
Q343
JEE Mains
MCQ
14 Mar 2026
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}}.} $
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
Q344
JEE Mains
MCQ
14 Mar 2026
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
Q345
JEE Mains
MCQ
14 Mar 2026
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
Q346
JEE Mains
MCQ
14 Mar 2026
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
Q347
JEE Mains
MCQ
14 Mar 2026
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
Q348
JEE Mains
MCQ
14 Mar 2026
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
Q349
JEE Mains
MCQ
14 Mar 2026
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
Q350
JEE Mains
MCQ
14 Mar 2026
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.
${x \over 2} - {3 \over 8}{x^2}$