Question

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 ${\left(\alpha x^{\frac{1}{9}}+\beta x^{-\frac{1}{6}}\right)}^{10}$ is $10k$, then $k$ is equal to

JEE Main 2020 (02 Sep Shift 1)

Options

• A: $336$
• B: $352$
• C: $84$
• D: $176$

Question

If $\left\{\mathrm{p}\right\}$ denotes the fractional part of the number $\mathrm{p},$ then $\left\{\frac{3^{200}}{8}\right\}$ is equal to

JEE Main 2020 (06 Sep Shift 1)

Options

• A: $\frac{5}{8}$
• B: $\frac{7}{8}$
• C: $\frac{3}{8}$
• D: $\frac{1}{8}$

Question

Let ${\left(2x^2+3x+4\right)}^{10}=\sum _{r=0}^{20}{a}_r{x}^r.$ Then $\frac{a_7}{a_{13}}$ is equal to ______

JEE Main 2020 (04 Sep Shift 1)

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Question

If $b$ is very small as compared to the value of $a$, so that the cube and other higher powers of $\frac{b}{a}$ can be neglected in the identity

$\frac{1}{a-b}+\frac{1}{a-2b}+\frac{1}{a-3b}+\dots .+\frac{1}{a-nb}=\alpha n+\beta n^2+\gamma n^3$

then the value of $\gamma$ is :

JEE Main 2021 (25 Jul Shift 1)

Options

• A: $\frac{a^2+b}{3a^3}$
• B: $\frac{a+b}{3a^2}$
• C: $\frac{b^2}{3a^3}$
• D: $\frac{a+b^2}{3a^3}$

Question

The number of elements in the set $\{n\in \{1,2,3,\dots ,100\}\mid \left.(11{)}^n>(10{)}^n+(9{)}^n\right\}$ is ___________.

JEE Main 2021 (22 Jul Shift 1)

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Question

The remainder when ${\left(11\right)}^{1011}+{\left(1011\right)}^{11}$ is divided by $9$ is _____ .

JEE Main 2022 (25 Jul Shift 2)

Options

• A: $1$
• B: $8$
• C: $6$
• D: $4$

Question

If $\frac{1}{2·3^{10}}+\frac{1}{2^2·3^9}+\dots +\frac{1}{2^{10}·3}=\frac{K}{2^{10}·3^{10}}$, then the remainder when $K$ is divided by $6$ is

JEE Main 2022 (25 Jun Shift 1)

Options

• A: $2$
• B: $3$
• C: $4$
• D: $5$

Question

The term independent of $x$ in the expression of $\left(1-x^2+3x^3\right){\left(\frac{5}{2}{x}^3-\frac{1}{5x^2}\right)}^{11},x\ne 0$ is

JEE Main 2022 (28 Jun Shift 2)

Options

• A: $\frac{7}{40}$
• B: $\frac{33}{200}$
• C: $\frac{39}{200}$
• D: $\frac{11}{50}$

Question

If the constant term in the expansion of ${\left(3x^3-2x^2+\frac{5}{x^5}\right)}^{10}$ is $2^k.l$, where $l$ is an odd integer, then the value of $k$ is equal to

JEE Main 2022 (29 Jun Shift 1)

Options

• A: $6$
• B: $7$
• C: $8$
• D: $9$

Question

Among the statements :

$\left(S1\right) : {2023}^{2022}-{1999}^{2022}$ is divisible by $8$.

$\left(S2\right) : 13(13{)}^n-11n-13$ is divisible by $144$ for infinitely many $n\in \mathrm{\mathbb{N}}$

JEE Main 2023 (06 Apr Shift 2)

Options

• A: Only $\left(S2\right)$ is correct
• B: Only $\left(S1\right)$ is correct
• C: Both $\left(S1\right)$ and $\left(S2\right)$ are correct
• D: Both $\left(S1\right)$ and $\left(S2\right)$ are incorrect

Question

The remainder on dividing $5^{99}$ by $11$ is _____ .

JEE Main 2023 (31 Jan Shift 1)

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Question

The remainder, when $7^{103}$ is divided by $17,$ is

JEE Main 2023 (13 Apr Shift 2)

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Question

If the constant term in the expansion of $\left(1+2 x-3 x^3\right)\left(\frac{3}{2} x^2-\frac{1}{3 x}\right)^9$ is $\mathrm{p}$, then $108 \mathrm{p}$ is equal to

JEE Main 2024 (05 Apr Shift 1)

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Question

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

JEE Main 2024 (08 Apr Shift 2)

Options

• A: 2
• B: 4
• C: 6
• D: 9

Question

The remainder when $428^{2024}$ is divided by 21 is__________

JEE Main 2024 (09 Apr Shift 1)

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Question

Let $m \text{and} n$ 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:

JEE Main 2024 (01 Feb Shift 2)

Options

• A: $\frac{4}{9}$
• B: $\frac{1}{9}$
• C: $\frac{1}{4}$
• D: $\frac{9}{4}$

Question

Number of integral terms in the expansion of ${\left\{7^{\left(\frac{1}{2}\right)}+{11}^{\left(\frac{1}{6}\right)}\right\}}^{824}$ is equal to ______.

JEE Main 2024 (30 Jan Shift 1)

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Question

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

JEE Main 2024 (31 Jan Shift 1)

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Question

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 $\left|\alpha \right|$ equals _________.

JEE Main 2024 (01 Feb Shift 1)

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Question

Remainder when ${64}^{{32}^{32}}$ is divided by $9$ is equal to _____.

JEE Main 2024 (29 Jan Shift 2)

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Question

${}^{n-1}C_r=\left(k^2-8\right){}^{n}C_{r+1}$ if and only if :

JEE Main 2024 (27 Jan Shift 1)

Options

• A: $2\sqrt{2}<k\le 3$
• B: $2\sqrt{3}<\mathrm{k}\le 3\sqrt{2}$
• C: $2\sqrt{3}<\mathrm{k}<3\sqrt{3}$
• D: $2\sqrt{2}<\mathrm{k}<2\sqrt{3}$