Binomial Theorem · JEE Main previous year questions

2023 25 questions

1. Q272023 · 13 Apr 2023 · Shift 1· Numerical answer

   Let $\alpha$ be the constant term in the binomial expansion of ${\left(\sqrt{x}-\frac{6}{x^{\frac{3}{2}}}\right)}^n$, $n\le 15$. If the sum of the coefficients of the remaining terms in the expansion is 649 and the coefficient of $x^{-n}$ is $\lambda \alpha$, then $\lambda$ is equal to ___________.

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   Answer: 36

2. Q72023 · 13 Apr 2023 · Shift 2

   The coefficient of $x^5$ in the expansion of ${\left(2x^3-\frac{1}{3x^2}\right)}^5$ is :

   1. A.$\frac{26}{3}$
   2. B.$\frac{80}{9}$
   3. C.$9$
   4. D.$8$
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   Answer: B

3. Q182023 · 12 Apr 2023 · Shift 1

   The sum, of the coefficients of the first 50 terms in the binomial expansion of $(1-x{)}^{100}$, is equal to

   1. A.${}^{99}{\mathrm{C}}_{49}$
   2. B.${}^{101}{\mathrm{C}}_{50}$
   3. C.$-{}^{99}{\mathrm{C}}_{49}$
   4. D.$-{}^{101}{\mathrm{C}}_{50}$
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   Answer: C

4. Q262023 · 11 Apr 2023 · Shift 1· Numerical answer

   The mean of the coefficients of $x,x^2,\dots ,x^7$ in the binomial expansion of $(2+x{)}^9$ is _________.

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   Answer: 2736

5. Q282023 · 11 Apr 2023 · Shift 1· Numerical answer

   The number of integral terms in the expansion of ${\left(3^{\frac{1}{2}}+5^{\frac{1}{4}}\right)}^{680}$ is equal to _________.

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   Answer: 171

6. Q22023 · 11 Apr 2023 · Shift 2

   The sum of the coefficients of three consecutive terms in the binomial expansion of $(1+x{)}^{n+2}$, which are in the ratio $1:3:5$, is equal to :

   1. A.$63$
   2. B.$92$
   3. C.$25$
   4. D.$41$
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   Answer: A

7. Q72023 · 11 Apr 2023 · Shift 2

   If the $1011^{\text{th}}$ term from the end in the binomial expansion of ${\left(\frac{4x}{5}-\frac{5}{2x}\right)}^{2022}$ is $1024$ times $1011^{\text{th}}$ term from the beginning, then $|x|$ is equal to

   1. A.$\frac{5}{16}$
   2. B.$8$
   3. C.$12$
   4. D.$15$
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   Answer: A

8. Q62023 · 10 Apr 2023 · Shift 1

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

   1. A.22
   2. B.33
   3. C.44
   4. D.11
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   Answer: A

9. Q262023 · 10 Apr 2023 · Shift 1· Numerical answer

   The coefficient of $x^7$ in $(1-x+2x^3{)}^{10}$ is _________.

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   Answer: 960

10. Q52023 · 10 Apr 2023 · Shift 2

    If the coefficients of $x$ and $x^2$ in $(1+x{)}^p(1-x{)}^q$ are 4 and $-5$ respectively, then $2p+3q$ is equal to :

    1. A.66
    2. B.60
    3. C.69
    4. D.63
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    Answer: D

11. Q172023 · 8 Apr 2023 · Shift 1

    If the coefficients of three consecutive terms in the expansion of $(1+x{)}^n$ are in the ratio $1:5:20$, then the coefficient of the fourth term is

    1. A.$3654$
    2. B.$1827$
    3. C.$5481$
    4. D.$2436$
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    Answer: A

12. Q122023 · 6 Apr 2023 · Shift 1

    If the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of ${\left(\sqrt[4]{2}+\frac{1}{\sqrt[4]{3}}\right)}^n$ is $\sqrt{6}:1$, then the third term from the beginning is :

    1. A.$30\sqrt{2}$
    2. B.$60\sqrt{3}$
    3. C.$60\sqrt{2}$
    4. D.$30\sqrt{3}$
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    Answer: B

13. Q292023 · 6 Apr 2023 · Shift 1· Numerical answer

    The coefficient of $x^{18}$ in the expansion of ${\left(x^4-\frac{1}{x^3}\right)}^{15}$ is ________.

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    Answer: 5005

14. Q12023 · 6 Apr 2023 · Shift 2

    If the coefficient of $x^7$ in ${\left(a{x}^2+\frac{1}{2bx}\right)}^{11}$ and $x^{-7}$ in ${\left(ax-\frac{1}{3b{x}^2}\right)}^{11}$ are equal, then :

    1. A.$243ab=64$
    2. B.$32ab=729$
    3. C.$64ab=243$
    4. D.$729ab=32$
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    Answer: D

15. Q232023 · 31 Jan 2023 · Shift 1· Numerical answer

    Let $\alpha >0$, be the smallest number such that the expansion of ${\left(x^{\frac{2}{3}}+\frac{2}{x^3}\right)}^{30}$ has a term $\beta x^{-\alpha },\beta \in \mathbb{N}$. Then $\alpha$ is equal to ___________.

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    Answer: 2

16. Q222023 · 31 Jan 2023 · Shift 2· Numerical answer

    The coefficient of $x^{-6}$, in the expansion of ${\left(\frac{4x}{5}+\frac{5}{2x^2}\right)}^9$, is

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    Answer: 5040

17. Q272023 · 31 Jan 2023 · Shift 2· Numerical answer

    If the constant term in the binomial expansion of ${\left(\frac{x^{\frac{5}{2}}}{2}-\frac{4}{x^l}\right)}^9$ is $-84$ and the coefficient of $x^{-3l}$ is $2^{\alpha }\beta$, where $\beta <0$ is an odd number, then $|\alpha l-\beta |$ is equal to ___________.

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    Answer: 98

18. Q52023 · 30 Jan 2023 · Shift 1

    If the coefficient of $x^{15}$ in the expansion of ${\left(a{x}^3+\frac{1}{b{x}^{1/3}}\right)}^{15}$ is equal to the coefficient of $x^{-15}$ in the expansion of ${\left(a{x}^{1/3}-\frac{1}{b{x}^3}\right)}^{15}$, where $a$ and $b$ are positive real numbers, then for each such ordered pair $(a,b)$ :

    1. A.$a=3b$
    2. B.$ab=1$
    3. C.$ab=3$
    4. D.$a=b$
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    Answer: B

19. Q162023 · 30 Jan 2023 · Shift 1

    The coefficient of $x^{301}$ in $(1+x{)}^{500}+x(1+x{)}^{499}+x^2(1+x{)}^{498}+\dots +x^{500}$ is :

    1. A.${}^{500}{C}_{300}$
    2. B.${}^{501}{C}_{200}$
    3. C.${}^{500}{C}_{301}$
    4. D.${}^{501}{C}_{302}$
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    Answer: B

20. Q232023 · 29 Jan 2023 · Shift 1· Numerical answer

    Let the coefficients of three consecutive terms in the binomial expansion of $(1+2x{)}^n$ be in the ratio 2 : 5 : 8. Then the coefficient of the term, which is in the middle of those three terms, is ____.

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    Answer: 1120

21. Q122023 · 29 Jan 2023 · Shift 2

    Let K be the sum of the coefficients of the odd powers of $x$ in the expansion of $(1+x{)}^{99}$. Let $a$ be the middle term in the expansion of ${\left(2+\frac{1}{\sqrt{2}}\right)}^{200}$. If $\frac{{}^{200}{C}_{99}K}{a}=\frac{2^{l}m}{n}$, where m and n are odd numbers, then the ordered pair $(l,\mathrm{n})$ is equal to

    1. A.(50, 101)
    2. B.(50, 51)
    3. C.(51, 101)
    4. D.(51, 99)
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    Answer: A

22. Q92023 · 25 Jan 2023 · Shift 1

    Let $y(x)=(1+x)(1+x^2)(1+x^4)(1+x^8)(1+x^{16})$. Then $y^{\prime }-y^{\prime \prime }$ at $x=-1$ is equal to

    1. A.$496$
    2. B.$976$
    3. C.$464$
    4. D.$944$
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    Answer: A

23. Q172023 · 25 Jan 2023 · Shift 1

    If $a_r$ is the coefficient of $x^{10-r}$ in the Binomial expansion of $(1+x{)}^{10}$, then $\sum _{r=1}^{10}{r}^3{\left(\frac{a_r}{a_{r-1}}\right)}^2$ is equal to

    1. A.$3025$
    2. B.$4895$
    3. C.$5445$
    4. D.$1210$
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    Answer: D

24. Q212023 · 25 Jan 2023 · Shift 1· Numerical answer

    The constant term in the expansion of ${\left(2x+\frac{1}{x^7}+3x^2\right)}^5$ is ____.

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    Answer: 1080

25. Q242023 · 24 Jan 2023 · Shift 2· Numerical answer

    Let the sum of the coefficients of the first three terms in the expansion of ${\left(x-\frac{3}{x^2}\right)}^n,x\ne 0,n\in \mathbb{N}$, be 376. Then the coefficient of $x^4$ is _______.

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    Answer: 405