Binomial Theorem PYQs - Last 5 Years
JEE Main / Mathematics / Algebra / 139 recent questions
MathematicsAlgebra2022-2026
Practice 139 JEE Main Mathematics questions from Binomial Theorem. Use the year-wise and type-wise breakdown to prioritize recent PYQs, then continue into the question list below.
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2022-2026
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139PYQs
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INTEGER42.4%
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#1 Medium97
#2 Hard29
#3 Easy13
139 in last 5 years139 in last 10 years
Last 5 Years Binomial Theorem Questions
Showing 39 of 139 filtered questions.
1Binomial Theorem
The sum, of the coefficients of the first 50 terms in the binomial expansion of \((1-x)^{100}\), is equal to
MCQ+4 / -12023
2Binomial Theorem
If \(\frac{1}{n+1}{ }^{n} \mathrm{C}_{n}+\frac{1}{n}{ }^{n} \mathrm{C}_{n-1}+\ldots+\frac{1}{2}{ }^{n} \mathrm{C}_{1}+{ }^{n} \mathrm{C}_{0}=\frac{1023}{10}\) then \(n\) is equal to :
MCQ+4 / -12023
3Binomial Theorem
The number of integral terms in the expansion of \(\left(3^{\frac{1}{2}}+5^{\frac{1}{4}}\right)^{680}\) is equal to ___________.
INTEGER+4 / -12023
4Binomial Theorem
The mean of the coefficients of \(x, x^{2}, \ldots, x^{7}\) in the binomial expansion of \((2+x)^{9}\) is ___________.
INTEGER+4 / -12023
5Binomial Theorem
If the \(1011^{\text {th }}\) term from the end in the binominal expansion of \(\left(\frac{4 x}{5}-\frac{5}{2 x}\right)^{2022}\) is 1024 times \(1011^{\text {th }}\)R term from the beginning, then \(|x|\) is equal to
MCQ+4 / -12023
6Binomial Theorem
The sum of the coefficients of three consecutive terms in the binomial expansion of \((1+\mathrm{x})^{\mathrm{n}+2}\), which are in the ratio \(1: 3: 5\), is equal to :
MCQ+4 / -12023
7Binomial Theorem
The coefficient of \(x^7\) in \({(1 - x + 2{x^3})^{10}}\) is ___________.
INTEGER+4 / -12023
8Binomial Theorem
If the coefficient of \({x^7}\) in \({\left( {ax - {1 \over {b{x^2}}}} \right)^{13}}\) and the coefficient of \({x^{ - 5}}\) in \({\left( {ax + {1 \over {b{x^2}}}} \right)^{13}}\) are equal, then \({a^4}{b^4}\) is equal to :
MCQ+4 / -12023
9Binomial Theorem
If the coefficients of \(x\) and \(x^{2}\) in \((1+x)^{\mathrm{p}}(1-x)^{\mathrm{q}}\) are 4 and \(-\)5 respectively, then \(2 p+3 q\) is equal to :
MCQ+4 / -12023
10Binomial Theorem
Let the number \((22)^{2022}+(2022)^{22}\) leave the remainder \(\alpha\) when divided by 3 and \(\beta\) when divided by 7. Then \(\left(\alpha^{2}+\beta^{2}\right)\) is equal to :
MCQ+4 / -12023
11Binomial Theorem
For two positive real numbers a and b such that \({1 \over {{a^2}}} + {1 \over {{b^3}}} = 4\), then minimum value of the constant term in the expansion of \({\left( {a{x^{{1 \over 8}}} + b{x^{ - {1 \over {12}}}}} \right)^{10}}\) is :
MCQ+4 / -12022
12Binomial Theorem
If the constant term in the expansion of
\({\left( {3{x^3} - 2{x^2} + {5 \over {{x^5}}}} \right)^{10}}\) is 2k.l, where l is an odd integer, then the value of k is equal to:
\({\left( {3{x^3} - 2{x^2} + {5 \over {{x^5}}}} \right)^{10}}\) is 2k.l, where l is an odd integer, then the value of k is equal to:
MCQ+4 / -12022
13Binomial Theorem
Let the coefficients of x\(-\)1 and x\(-\)3 in the expansion of \({\left( {2{x^{{1 \over 5}}} - {1 \over {{x^{{1 \over 5}}}}}} \right)^{15}},x > 0\), be m and n respectively. If r is a positive integer such that $$m{n^2} = {}^{15}{C_r}\,.\,...
INTEGER+4 / -12022
14Binomial Theorem
Let n \(\ge\) 5 be an integer. If 9n \(-\) 8n \(-\) 1 = 64\(\alpha\) and 6n \(-\) 5n \(-\) 1 = 25\(\beta\), then \(\alpha\) \(-\) \(\beta\) is equal to
MCQ+4 / -12022
15Binomial Theorem
Let the ratio of the fifth term from the beginning to the fifth term from the end in the binomial expansion of \(\left(\sqrt[4]{2}+\frac{1}{\sqrt[4]{3}}\right)^{\mathrm{n}}\), in the increasing powers of \(\frac{1}{\sqrt[4]{3}}\) be $$\sqrt...
INTEGER+4 / -12022
16Binomial Theorem
\(\text { If } \sum\limits_{k=1}^{10} K^{2}\left(10_{C_{K}}\right)^{2}=22000 L \text {, then } L \text { is equal to }\) ________.
INTEGER+4 / -12022
17Binomial Theorem
\(\sum\limits_{r=1}^{20}\left(r^{2}+1\right)(r !)\) is equal to
MCQ+4 / -12022
18Binomial Theorem
The number of positive integers k such that the constant term in the binomial expansion of \({\left( {2{x^3} + {3 \over {{x^k}}}} \right)^{12}}\), x \(\ne\) 0 is 28 . l, where l is an odd integer, is ______________.
INTEGER+4 / -12022
19Binomial Theorem
If \(\sum\limits_{k = 1}^{31} {\left( {{}^{31}{C_k}} \right)\left( {{}^{31}{C_{k - 1}}} \right) - \sum\limits_{k = 1}^{30} {\left( {{}^{30}{C_k}} \right)\left( {{}^{30}{C_{k - 1}}} \right) = {{\alpha (60!)} \over {(30!)(31!)}}} }\), where ...
MCQ+4 / -12022
20Binomial Theorem
The term independent of x in the expansion of \((1 - {x^2} + 3{x^3}){\left( {{5 \over 2}{x^3} - {1 \over {5{x^2}}}} \right)^{11}},\,x \ne 0\) is :
MCQ+4 / -12022
21Binomial Theorem
The remainder when \(7^{2022}+3^{2022}\) is divided by 5 is :
MCQ+4 / -12022
22Binomial Theorem
If \(1 + (2 + {}^{49}{C_1} + {}^{49}{C_2} + \,\,...\,\, + \,\,{}^{49}{C_{49}})({}^{50}{C_2} + {}^{50}{C_4} + \,\,...\,\, + \,\,{}^{50}{C_{50}})\) is equal to \(2^{\mathrm{n}} \cdot \mathrm{m}\), where \(\mathrm{m}\) is odd, then $$\mathrm{n...
INTEGER+4 / -12022
23Binomial Theorem
Let the coefficients of the middle terms in the expansion of \(\left(\frac{1}{\sqrt{6}}+\beta x\right)^{4},(1-3 \beta x)^{2}\) and \(\left(1-\frac{\beta}{2} x\right)^{6}, \beta>0\), respectively form the first three terms of an A.P. If d is...
INTEGER+4 / -12022
24Binomial Theorem
If the coefficient of x10 in the binomial expansion of \({\left( {{{\sqrt x } \over {{5^{{1 \over 4}}}}} + {{\sqrt 5 } \over {{x^{{1 \over 3}}}}}} \right)^{60}}\) is \({5^k}\,.\,l\), where l, k \(\in\) N and l is co-prime to 5, then k is eq...
INTEGER+4 / -12022
25Binomial Theorem
If the sum of the coefficients of all the positive powers of x, in the Binomial expansion of \({\left( {{x^n} + {2 \over {{x^5}}}} \right)^7}\) is 939, then the sum of all the possible integral values of n is _________.
INTEGER+4 / -12022
26Binomial Theorem
The remainder when \((2021)^{2022}+(2022)^{2021}\) is divided by 7 is
MCQ+4 / -12022
27Binomial Theorem
Let for the \(9^{\text {th }}\) term in the binomial expansion of \((3+6 x)^{\mathrm{n}}\), in the increasing powers of \(6 x\), to be the greatest for \(x=\frac{3}{2}\), the least value of \(\mathrm{n}\) is \(\mathrm{n}_{0}\). If $$\mathrm...
INTEGER+4 / -12022
28Binomial Theorem
The remainder when (2021)2023 is divided by 7 is :
MCQ+4 / -12022
29Binomial Theorem
If \(\left( {{}^{40}{C_0}} \right) + \left( {{}^{41}{C_1}} \right) + \left( {{}^{42}{C_2}} \right) + \,\,.....\,\, + \,\,\left( {{}^{60}{C_{20}}} \right) = {m \over n}{}^{60}{C_{20}}\) m and n are coprime, then m + n is equal to ___________...
INTEGER+4 / -12022
30Binomial Theorem
If the coefficients of \(x\) and \(x^{2}\) in the expansion of \((1+x)^{\mathrm{p}}(1-x)^{\mathrm{q}}, \mathrm{p}, \mathrm{q} \leq 15\), are \(-3\) and \(-5\) respectively, then the coefficient of \(x^{3}\) is equal to _____________.
INTEGER+4 / -12022
31Binomial Theorem
\(\sum\limits_{\matrix{
{i,j = 0} \cr
{i \ne j} \cr
} }^n {{}^n{C_i}\,{}^n{C_j}}\) is equal to
MCQ+4 / -12022
32Binomial Theorem
Let Cr denote the binomial coefficient of xr in the expansion of \({(1 + x)^{10}}\). If for \(\alpha\), \(\beta\) \(\in\) R, \({C_1} + 3.2{C_2} + 5.3{C_3} +\) ....... upto 10 terms $$ = {{\alpha \times {2^{11}}} \over {{2^\beta } - 1}}\le...
INTEGER+4 / -12022
33Binomial Theorem
If \({1 \over {2\,.\,{3^{10}}}} + {1 \over {{2^2}\,.\,{3^9}}} + \,\,.....\,\, + \,\,{1 \over {{2^{10}}\,.\,3}} = {K \over {{2^{10}}\,.\,{3^{10}}}}\), then the remainder when K is divided by 6 is :
MCQ+4 / -12022
34Binomial Theorem
If the sum of the co-efficient of all the positive even powers of x in the binomial expansion of \({\left( {2{x^3} + {3 \over x}} \right)^{10}}\) is \({5^{10}} - \beta \,.\,{3^9}\), then \(\beta\) is equal to ____________.
INTEGER+4 / -12022
35Binomial Theorem
The coefficient of x101 in the expression \({(5 + x)^{500}} + x{(5 + x)^{499}} + {x^2}{(5 + x)^{498}} + \,\,.....\,\, + \,\,{x^{500}}\), x > 0, is
MCQ+4 / -12022
36Binomial Theorem
If the maximum value of the term independent of \(t\) in the expansion of \(\left(\mathrm{t}^{2} x^{\frac{1}{5}}+\frac{(1-x)^{\frac{1}{10}}}{\mathrm{t}}\right)^{15}, x \geqslant 0\), is \(\mathrm{K}\), then \(8 \mathrm{~K}\) is equal to ___...
INTEGER+4 / -12022
37Binomial Theorem
The remainder when \((11)^{1011}+(1011)^{11}\) is divided by 9 is
MCQ+4 / -12022
38Binomial Theorem
The remainder when 32022 is divided by 5 is :
MCQ+4 / -12022
39Binomial Theorem
The remainder on dividing 1 + 3 + 32 + 33 + ..... + 32021 by 50 is _________.
INTEGER+4 / -12022