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Binomial Theorem

JEE Mains Maths. Try each one first, then open its answer.

Top 10 JEE Mains PYQs to practise first

The questions and kinds of question asked most often come first. Each one says how often and in which papers.

  1. Question 1

    This type asked 11 times: 2021 Paper 4, 2021 Paper 7, 14 Jun 2022, 26 Jun 2022, 25 Jan 2023, 31 Jan 2023, 13 Apr 2023, 29 Jan 2024, 9 Apr 2024, 29 Jan 2025, 7 Apr 2025. This one: 7 Apr 2025 See all 11 of this type

    54
    The remainder when ((64)(64))(64)\left( (64)^{(64)} \right)^{(64)} is divided by 7 is equal to.

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  2. Question 2

    This type asked 8 times: 2021 Paper 10, 2021 Paper 18, 2021 Paper 25, 24 Jan 2023, 8 Apr 2024, 3 Apr 2025, 4 Apr 2025, 7 Apr 2025. This one: 7 Apr 2025 See all 8 of this type

    150
    The sum of the series
    2×1× 20C4−3×2× 20C5+4×3× 20C6−5×4× 20C7+…+18×17× 20C20, is equal to .\begin{matrix} & 2 \times 1 \times \ ^{20}C_{4} - 3 \times 2 \times \ ^{20}C_{5} + 4 \times 3 \times \ ^{20}C_{6} - 5 \times 4 \times \\ & \ ^{20}C_{7} + \ldots + 18 \times 17 \times \ ^{20}C_{20},\text{~is equal to~}\text{.} \end{matrix}

  3. Question 3

    This type asked 7 times: 27 Jun 2022, 31 Jan 2023, 6 Apr 2023, 8 Apr 2023, 13 Apr 2023, 9 Apr 2024, 5 Apr 2026 S2. This one: 5 Apr 2026 S2 See all 7 of this type

    57
    The coefficient of x2x^{2} in the expansion of (2x2+1x)10,x≠0\left( 2x^{2}+\frac{1}{x} \right)^{10},x \neq 0, is :

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  4. Question 4

    This type asked 7 times: 2021 Paper 11, 28 Jun 2022, 27 Jan 2024, 31 Jan 2024, 1 Feb 2024, 5 Apr 2024, 21 Jan 2026 S1. This one: 21 Jan 2026 S1 See all 7 of this type

    66
    If the coefficient of xx in the expansion of (ax2+bx+c)(1−2x)26\left( ax^{2}+ bx + c \right)(1 - 2x)^{26} is -56 and the coefficients of x2x^{2} and x3x^{3} are both zero, then a+b+ca + b + c is equal to

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  5. Question 5

    This type asked 7 times: 2021 Paper 2, 2021 Paper 8, 2021 Paper 19, 2021 Paper 20, 26 Jul 2022, 24 Jan 2023, 30 Jan 2024. This one: 30 Jan 2024 See all 7 of this type

    174
    Let α=∑k=0n(( nCk)2k+1)\alpha=\sum_{k = 0}^{n} \left( \frac{\left( \ ^{n}C_{k} \right)^{2}}{k + 1} \right) and β=∑k=0n−1( nCk nCk+1k+2)\beta=\sum_{k = 0}^{n - 1} \left( \frac{\ ^{n}C_{k}\ ^{n}C_{k + 1}}{k + 2} \right). If 5α=6β5\alpha= 6\beta, then nn equals

  6. Question 6

    This type asked 6 times: 12 Apr 2023, 29 Jan 2024, 22 Jan 2025, 22 Jan 2026 S2, 23 Jan 2026 S1, 8 Apr 2026 S2. This one: 8 Apr 2026 S2 See all 6 of this type

    58
    If 26(233( 13C2)+255( 12C4)+277( 12C6)+...+21313( 12C12))=313−α26\left( \frac{2^{3}}{3}\left( \ ^{13}C_{2} \right)+\frac{2^{5}}{5}\left( \ ^{12}C_{4} \right)+\frac{2^{7}}{7}\left( \ ^{12}C_{6} \right)+ ... +\frac{2^{13}}{13}\left( \ ^{12}C_{12} \right) \right)=3^{13}-\alpha, then α\alpha is equal to :

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  7. Question 7

    This type asked 6 times: 2021 Paper 22, 1 Feb 2023, 8 Apr 2023, 5 Apr 2024, 8 Apr 2024, 4 Apr 2026 S2. This one: 4 Apr 2026 S2 See all 6 of this type

    59
    In the expansion of (9x−13x)18,x>0\left( 9x-\frac{1}{3\sqrt{x}} \right)^{18},x> 0, if the term independent of x is (221)k(221)k. then k is equal to:

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  8. Question 8

    This type asked 6 times: 25 Jul 2022, 27 Jul 2022, 28 Jul 2022, 30 Jan 2023, 1 Feb 2023, 10 Apr 2023. This one: 10 Apr 2023 See all 6 of this type

    169
    Let the number (22)2022+(2022)22(22)^{2022}+ (2022)^{22} leave the remainder α\alpha when divided by 3 and β\beta when divided by 7. Then (α2+β2)\left( \alpha^{2}+\beta^{2} \right) is equal to

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  9. Question 9

    Asked 5 times: 2021 Paper 16*, 29 Jan 2023*, 30 Jan 2023*, 6 Apr 2023*, 10 Apr 2023 (* = numbers or wording changed that year) See all 5 times it was asked

    75
    If the coefficient of x7x^{7} in (ax−1bx2)13\left( ax -\frac{1}{bx^{2}} \right)^{13} and the coefficient of x−5x^{- 5} in (ax+1bx2)13\left( ax +\frac{1}{bx^{2}} \right)^{13} are equal, then a4b4a^{4}b^{4} is equal to:

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  10. Question 10

    This type asked 5 times: 25 Jun 2022, 30 Jan 2023, 31 Jan 2024, 9 Apr 2024, 28 Jan 2026 S2. This one: 28 Jan 2026 S2 See all 5 of this type

    52
    The sum of the coefficients of x499x^{499} and x500x^{500} in (1+x)1000+x(1+x)999+x2(1+x)998+……..+x1000(1 +x)^{1000}+x(1 +x)^{999}+x^{2}(1 +x)^{998}+ \ldots\ldots.. +x^{1000} is

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