Binomial Theorem · JEE Main previous year questions

2024 13 questions

1. Q82024 · 9 Apr 2024 · Shift 1

   The coefficient of $x^{70}$ in $x^2(1+x{)}^{98}+x^3(1+x{)}^{97}+x^4(1+x{)}^{96}+\cdots +x^{54}(1+x{)}^{46}$ is ${}^{99}{\mathrm{C}}_p-{}^{46}{\mathrm{C}}_q$. Then a possible value of $p+q$ is :

   1. A.61
   2. B.83
   3. C.55
   4. D.68
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   Answer: B

2. Q192024 · 9 Apr 2024 · Shift 2

   The sum of the coefficients of $x^{2/3}$ and $x^{-2/5}$ in the binomial expansion of ${\left(x^{2/3}+\frac{1}{2}{x}^{-2/5}\right)}^9$ is

   1. A.19/4
   2. B.69/16
   3. C.63/16
   4. D.21/4
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   Answer: D

3. Q202024 · 8 Apr 2024 · Shift 2

   If the term independent of $x$ in the expansion of ${\left(\sqrt{a}{x}^2+\frac{1}{2x^3}\right)}^{10}$ is 105 , then $a^2$ is equal to :

   1. A.6
   2. B.4
   3. C.2
   4. D.9
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   Answer: B

4. Q182024 · 6 Apr 2024 · Shift 2

   Let $0\le r\le n$. If ${}^{n+1}{\mathrm{C}}_{r+1}:{}^n{\mathrm{C}}_r:{}^{n-1}{\mathrm{C}}_{r-1}=55:35:21$, then $2n+5r$ is equal to :

   1. A.62
   2. B.60
   3. C.55
   4. D.50
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   Answer: D

5. Q292024 · 6 Apr 2024 · Shift 2· Numerical answer

   If $\mathrm{S}(x)=(1+x)+2(1+x{)}^2+3(1+x{)}^3+\cdots +60(1+x{)}^{60},x\ne 0$, and $(60{)}^2\mathrm{S}(60)=a(b{)}^b+b$, where $a,b\in N$, then $(a+b)$ equal to ____.

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   Answer: 3660

6. Q62024 · 5 Apr 2024 · Shift 2

   If the constant term in the expansion of ${\left(\frac{\sqrt[5]{3}}{x}+\frac{2x}{\sqrt[3]{5}}\right)}^{12},x\ne 0$, is $\alpha \times 2^8\times \sqrt[5]{3}$, then $25\alpha$ is equal to :

   1. A.724
   2. B.742
   3. C.693
   4. D.639
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   Answer: C

7. Q152024 · 4 Apr 2024 · Shift 1

   The sum of all rational terms in the expansion of ${\left(2^{\frac{1}{5}}+5^{\frac{1}{3}}\right)}^{15}$ is equal to :

   1. A.633
   2. B.6131
   3. C.3133
   4. D.931
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   Answer: C

8. Q232024 · 1 Feb 2024 · Shift 1· Numerical answer

   If the coefficient of $x^{30}$ in the expansion of ${\left(1+\frac{1}{x}\right)}^6{\left(1+x^2\right)}^7{\left(1-x^3\right)}^8;x\ne 0$ is $\alpha$, then $|\alpha |$ equals ____.

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   Answer: 678

9. Q192024 · 1 Feb 2024 · Shift 2

   Let $m$ and $n$ be the coefficients of seventh and thirteenth terms respectively in the expansion of ${\left(\frac{1}{3}{x}^{\frac{1}{3}}+\frac{1}{2x^{\frac{2}{3}}}\right)}^{18}$. Then ${\left(\frac{n}{m}\right)}^{\frac{1}{3}}$ is :

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

10. Q222024 · 31 Jan 2024 · Shift 1· Numerical answer

    In the expansion of $(1+x)\left(1-x^2\right){\left(1+\frac{3}{x}+\frac{3}{x^2}+\frac{1}{x^3}\right)}^5,x\ne 0$, the sum of the coefficients of $x^3$ and $x^{-13}$ is equal to ____.

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    Answer: 118

11. Q282024 · 31 Jan 2024 · Shift 2· Numerical answer

    Let the coefficient of $x^r$ in the expansion of $(x+3{)}^{n-1}+(x+3{)}^{n-2}(x+2)+(x+3{)}^{n-3}(x+2{)}^2+\dots \dots \dots .+(x+2{)}^{n-1}$ be ${\alpha }_r$. If ${\sum }_{r=0}^n{\alpha }_r={\beta }^n-{\gamma }^n,\beta ,\gamma \in \mathbb{N}$, then the value of ${\beta }^2+{\gamma }^2$ equals ____.

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    Answer: 25

12. Q192024 · 27 Jan 2024 · Shift 1

    If A denotes the sum of all the coefficients in the expansion of ${\left(1-3x+10x^2\right)}^n$ and B denotes the sum of all the coefficients in the expansion of ${\left(1+x^2\right)}^n$, then :

    1. A.$\mathrm{B}={\mathrm{A}}^3$
    2. B.$3\mathrm{A}=\mathrm{B}$
    3. C.$A=3B$
    4. D.$A={\mathrm{B}}^3$
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    Answer: D

13. Q222024 · 27 Jan 2024 · Shift 2· Numerical answer

    The coefficient of $x^{2012}$ in the expansion of $(1-x{)}^{2008}(1+x+x^2{)}^{2007}$ is equal to _______.

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    Answer: 0