Binomial Theorem PYQs - Last 10 Years

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MathematicsAlgebra2017-2026

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Practice 224 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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Mathematics / Algebra

2017-2026

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224

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2017-2026

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224PYQs

MCQ61.2%

INTEGER38.8%

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#1 Medium167

#2 Hard37

#3 Easy20

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Last 10 Years Binomial Theorem Questions

Showing 50 of 224 filtered questions.

1Binomial Theorem

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 _____________.

INTEGER+4 / -12021

2Binomial Theorem

If b is very small as compared to the value of a, so that the cube and other higher powers of \({b \over a}\) can be neglected in the identity $${1 \over {a - b}} + {1 \over {a - 2b}} + {1 \over {a - 3b}} + ..... + {1 \over {a - nb}} = \alp...

3Binomial Theorem

If the co-efficient of x7 and x8 in the expansion of \({\left( {2 + {x \over 3}} \right)^n}\) are equal, then the value of n is equal to _____________.

INTEGER+4 / -12021

4Binomial Theorem

Let n\(\in\)N and [x] denote the greatest integer less than or equal to x. If the sum of (n + 1) terms \({}^n{C_0},3.{}^n{C_1},5.{}^n{C_2},7.{}^n{C_3},.....\) is equal to 2100 . 101, then \(2\left[ {{{n - 1} \over 2}} \right]\) is equal to ...

INTEGER+4 / -12021

5Binomial Theorem

The lowest integer which is greater than \({\left( {1 + {1 \over {{{10}^{100}}}}} \right)^{{{10}^{100}}}}\) is ______________.

6Binomial Theorem

If the greatest value of the term independent of 'x' in the expansion of \({\left( {x\sin \alpha + a{{\cos \alpha } \over x}} \right)^{10}}\) is \({{10!} \over {{{(5!)}^2}}}\), then the value of 'a' is equal to :

7Binomial Theorem

The sum of all those terms which are rational numbers in the expansion of (21/3 + 31/4)12 is :

8Binomial Theorem

The total number of two digit numbers 'n', such that 3n + 7n is a multiple of 10, is __________.

INTEGER+4 / -12021

9Binomial Theorem

If the remainder when x is divided by 4 is 3, then the remainder when (2020 + x)2022 is divided by 8 is __________.

INTEGER+4 / -12021

10Binomial Theorem

The value of
-15C1 + 2.15C2 – 3.15C3 + ... - 15.15C15 + 14C1 + 14C3 + 14C5 + ...+ 14C11 is :

11Binomial Theorem

For integers n and r, let \(\left( {\matrix{ n \cr r \cr } } \right) = \left\{ {\matrix{ {{}^n{C_r},} & {if\,n \ge r \ge 0} \cr {0,} & {otherwise} \cr } } \right.\) The maximum value of k for which the sum $$\sum\lim...

INTEGER+4 / -12021

12Binomial Theorem

If \(n \ge 2\) is a positive integer, then the sum of the series \({}^{n + 1}{C_2} + 2\left( {{}^2{C_2} + {}^3{C_2} + {}^4{C_2} + ... + {}^n{C_2}} \right)\) is :

13Binomial Theorem

The number of elements in the set {n \(\in\) {1, 2, 3, ......., 100} | (11)n > (10)n + (9)n} is ______________.

INTEGER+4 / -12021

14Binomial Theorem

If the constant term, in binomial expansion of \({\left( {2{x^r} + {1 \over {{x^2}}}} \right)^{10}}\) is 180, then r is equal to __________________.

INTEGER+4 / -12021

15Binomial Theorem

The number of rational terms in the binomial expansion of \({\left( {{4^{{1 \over 4}}} + {5^{{1 \over 6}}}} \right)^{120}}\) is _______________.

INTEGER+4 / -12021

16Binomial Theorem

The coefficient of x256 in the expansion of (1 \(-\) x)101 (x2 + x + 1)100 is :

17Binomial Theorem

For the natural numbers m, n, if \({(1 - y)^m}{(1 + y)^n} = 1 + {a_1}y + {a_2}{y^2} + .... + {a_{m + n}}{y^{m + n}}\) and \({a_1} = {a_2} = 10\), then the value of (m + n) is equal to :

18Binomial Theorem

If the sum of the coefficients in the expansion of (x + y)n is 4096, then the greatest coefficient in the expansion is _____________.

INTEGER+4 / -12021

19Binomial Theorem

Let (1 + x + 2x2)20 = a0 + a1x + a2x2 + .... + a40x40. Then a1 + a3 + a5 + ..... + a37 is equal to

20Binomial Theorem

Let \({}^n{C_r}\) denote the binomial coefficient of xr in the expansion of (1 + x)n.
If \(\sum\limits_{k = 0}^{10} {({2^2} + 3k)} {}^{10}{C_k} = \alpha {.3^{10}} + \beta {.2^{10}},\alpha ,\beta \in R\), then \(\alpha\) + \(\beta\) is eq...

INTEGER+4 / -12021

21Binomial Theorem

The term independent of x in the expansion of \({\left[ {{{x + 1} \over {{x^{2/3}} - {x^{1/3}} + 1}} - {{x - 1} \over {x - {x^{1/2}}}}} \right]^{10}}\), x \(\ne\) 1, is equal to ____________.

INTEGER+4 / -12021

22Binomial Theorem

If the fourth term in the expansion of \({(x + {x^{{{\log }_2}x}})^7}\) is 4480, then the value of x where x\(\in\)N is equal to :

23Binomial Theorem

If (2021)3762 is divided by 17, then the remainder is __________.

INTEGER+4 / -12021

24Binomial Theorem

The value of \(\sum\limits_{r = 0}^6 {\left( {{}^6{C_r}\,.\,{}^6{C_{6 - r}}} \right)}\) is equal to :

25Binomial Theorem

Let the coefficients of third, fourth and fifth terms in the expansion of \({\left( {x + {a \over {{x^2}}}} \right)^n},x \ne 0\), be in the ratio 12 : 8 : 3. Then the term independent of x in the expansion, is equal to ___________.

INTEGER+4 / -12021

26Binomial Theorem

Let [ x ] denote greatest integer less than or equal to x. If for n\(\in\)N, \({(1 - x + {x^3})^n} = \sum\limits_{j = 0}^{3n} {{a_j}{x^j}}\), then $$\sum\limits_{j = 0}^{\left[ {{{3n} \over 2}} \right]} {{a_{2j}} + 4} \sum\limits_{j = 0}^{...

27Binomial Theorem

If n is the number of irrational terms in the expansion of \({\left( {{3^{1/4}} + {5^{1/8}}} \right)^{60}}\), then (n \(-\) 1) is divisible by :

28Binomial Theorem

Let n be a positive integer. Let $$A = \sum\limits_{k = 0}^n {{{( - 1)}^k}{}^n{C_k}\left[ {{{\left( {{1 \over 2}} \right)}^k} + {{\left( {{3 \over 4}} \right)}^k} + {{\left( {{7 \over 8}} \right)}^k} + {{\left( {{{15} \over {16}}} \right)}^...

INTEGER+4 / -12021

29Binomial Theorem

The coefficient of x4 is the expansion of
(1 + x + x2)10 is _____.

INTEGER+4 / -02020

30Binomial Theorem

If Cr \(\equiv\) 25Cr and
C0 + 5.C1 + 9.C2 + .... + (101).C25 = 225.k, then k is equal to _____.

INTEGER+4 / -02020

31Binomial Theorem

In the expansion of \({\left( {{x \over {\cos \theta }} + {1 \over {x\sin \theta }}} \right)^{16}}\), if \({\ell _1}\) is
the least value of the term independent of x
when \({\pi \over 8} \le \theta \le {\pi \over 4}\) and \({\ell _2}\) ...

32Binomial Theorem

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

33Binomial Theorem

The greatest positive integer k, for which 49k + 1 is a factor of the sum 49125 + 49124 + ..... + 492 + 49 + 1, is:

34Binomial Theorem

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 _______.

INTEGER+4 / -02020

35Binomial Theorem

The coefficient of x7
in the expression
(1 + x)10 + x(1 + x)9
+ x2(1 + x)8
+ ......+ x10 is:

36Binomial Theorem

If {p} denotes the fractional part of the number p, then
\(\left\{ {{{{3^{200}}} \over 8}} \right\}\), is equal to :

37Binomial Theorem

If the constant term in the binomial expansion
of \({\left( {\sqrt x - {k \over {{x^2}}}} \right)^{10}}\) is 405, then |k| equals :

38Binomial Theorem

The natural number m, for which the coefficient of x in the binomial expansion of
\({\left( {{x^m} + {1 \over {{x^2}}}} \right)^{22}}\) is 1540, is .............

INTEGER+4 / -02020

39Binomial Theorem

The coefficient of x4
in the expansion of
(1 + x + x2
+ x3)6
in powers of x, is ______.

INTEGER+4 / -02020

40Binomial Theorem

The value of \(\sum\limits_{r = 0}^{20} {{}^{50 - r}{C_6}}\) is equal to:

41Binomial Theorem

Let \({\left( {2{x^2} + 3x + 4} \right)^{10}} = \sum\limits_{r = 0}^{20} {{a_r}{x^r}}\)
Then \({{{a_7}} \over {{a_{13}}}}\) is equal to ______.

INTEGER+4 / -02020

42Binomial Theorem

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 :

43Binomial Theorem

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 :

44Binomial Theorem

If the term independent of x in the expansion of
\({\left( {{3 \over 2}{x^2} - {1 \over {3x}}} \right)^9}\) is k, then 18 k is equal to :

45Binomial Theorem

Let
\(\alpha\) > 0,
\(\beta\) > 0 be such that
\(\alpha\)3 + \(\beta\)2 = 4. If the
maximum value of the term independent of x in
the binomial expansion of
$${\left( {\alpha {x^{{1 \over 9}}} + \beta {x^{ - {1 \over 6}}}} \right)^{10}}$...

46Binomial Theorem

For a positive integer n,
\({\left( {1 + {1 \over x}} \right)^n}\) 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________.

INTEGER+4 / -02020

47Binomial Theorem

If the fractional part of the number \(\left\{ {{{{2^{403}}} \over {15}}} \right\} is \, {k \over {15}}\), then k is equal to :

48Binomial Theorem

The coefficient of t4 in the expansion of \({\left( {{{1 - {t^6}} \over {1 - t}}} \right)^3}\) is :

49Binomial Theorem

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 :

50Binomial Theorem

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 :-