JEE MAIN Binomial Theorem Previous Year Question Paper and Solution with PDF

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JEE MAIN Previous Year Questions (PYQs)

JEE MAIN Binomial Theorem PYQ

Qus : 1 JEE MAIN PYQ

In the expansion of $\left(\sqrt[3]{2}+\dfrac{1}{\sqrt[3]{3}}\right)^{n},\ n\in\mathbb{N}$, if the ratio of $15^{\text{th}}$ term from the beginning to the $15^{\text{th}}$ term from the end is $\dfrac{1}{6}$, then the value of ${}^nC_3$ is

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (4 April Morning Shift) PYQ

Solution

$T_r=\binom{n}{r-1}a^{,n-r+1}b^{,r-1}$ for $(a+b)^n$.

$15^{\text{th}}$ from the beginning: $T_{15}^{(beg)}=\binom{n}{14}a^{,n-14}b^{14}$.

$15^{\text{th}}$ from the end (swap $a,b$): $T_{15}^{(end)}=\binom{n}{14}b^{,n-14}a^{14}$.

Given $\dfrac{T_{15}^{(beg)}}{T_{15}^{(end)}}=\dfrac{1}{6}$,

coefficients cancel: $\left(\dfrac{a}{b}\right)^{n-28}=\dfrac{1}{6}$.

Here $a=2^{1/3},\ b=3^{-1/3}$

$\ \Rightarrow\ \dfrac{a}{b}=2^{1/3}\cdot 3^{1/3}=6^{1/3}$.

So $(6^{1/3})^{,n-28}=6^{-1}$

$\ \Rightarrow\ n-28=-3\ \Rightarrow\ n=25$.

Therefore, $\binom{n}{3}=\binom{25}{3}=\dfrac{25\cdot24\cdot23}{6}=2300$.

Qus : 2 JEE MAIN PYQ

The coefficient of $x^{18}$ in the product $(1+x)(1-x)^{10}(1+x+x^{2})^{9}$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (12 April Morning Shift) PYQ

Solution

Qus : 3 JEE MAIN PYQ

If the coefficients of $x^4$, $x^5$, and $x^6$ in the expansion of $(1+x)^n$ are in arithmetic progression, then the maximum value of $n$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (4 April Evening Shift) PYQ

Solution

Qus : 4 JEE MAIN PYQ

The positive value of $\lambda$ for which the coefficient of $x^2$ in the expression $x^2 \left( \sqrt{x} + \dfrac{\lambda}{x^2} \right)^{10}$ is $720$, is –

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (10 January Evening Shift) PYQ

Solution

Qus : 5 JEE MAIN PYQ

The term independent of $x$ in the expansion of $\left(\dfrac{1}{60} - \dfrac{x^{8}}{81}\right)\left(2x^{2} - \dfrac{3}{x^{2}}\right)^{6}$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (12 April Evening Shift) PYQ

Solution

Qus : 6 JEE MAIN PYQ

If the coefficients of x⁷ in ${\left( {{x^2} + {1 \over {bx}}} \right)^{11}}$ and x^$-$7 in ${\left( {{x} - {1 \over {bx^2}}} \right)^{11}}$, b $\ne$ 0, are equal, then the value of b is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (27 July Morning Shift) PYQ

Solution

Qus : 7 JEE MAIN PYQ

If $\displaystyle \sum_{r=0}^{25} \left\{ {^{50}C_{r}} \cdot {^{\,50-r}C_{\,25-r}} \right\} = K \binom{50}{25}$, then $K$ is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (10 January Evening Shift) PYQ

Solution

Qus : 8 JEE MAIN PYQ

If $1^{2}\cdot{^{15}C_{1}}+2^{2}\cdot{^{15}C_{2}}+3^{2}\cdot{^{15}C_{3}}+\cdots+15^{2}\cdot{^{15}C_{15}}=2^{m}\cdot3^{n}\cdot5^{k}$, where $m,n,k\in\mathbb{N}$, then $m+n+k$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (4 April Evening Shift) PYQ

Solution

Qus : 9 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (23 January Evening Shift) PYQ

Solution

Qus : 10 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 6 September 2020 (Evening) PYQ

Solution

Qus : 11 JEE MAIN PYQ

The term independent of $x$ in expansion of $\left(\dfrac{x+1}{x^{2/3}-x^{1/3}+1}-\dfrac{x-1}{x-x^{1/2}}\right)^{10}$ is

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2013 (Offline) PYQ

Solution

Qus : 12 JEE MAIN PYQ

The sum of the real values of $x$ for which the middle term in the binomial expansion of $\left(\dfrac{x^{3}}{3}+\dfrac{3}{x}\right)^{8}$ equals $5670$ is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (11 January Morning Shift) PYQ

Solution

Qus : 13 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (6 April Morning Shift) PYQ

Solution

Qus : 14 JEE MAIN PYQ

For some $ n \ne 10 $, let the coefficients of the 5th, 6th and 7th terms in the binomial expansion of $ (1 + x)^{n+4} $ be in A.P. Then the largest coefficient in the expansion of $ (1 + x)^{n+4} $ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (24 January Morning Shift) PYQ

Solution

Qus : 15 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (6 April Evening Shift) PYQ

Solution

Qus : 16 JEE MAIN PYQ

The remainder when 3²⁰²² is divided by 5 is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (24 June Morning Shift) PYQ

Solution

Qus : 17 JEE MAIN PYQ

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 ${1 \over {a - b}} + {1 \over {a - 2b}} + {1 \over {a - 3b}} + ..... + {1 \over {a - nb}} = \alpha n + \beta {n^2} + \gamma {n^3}$, then the value of $\gamma$ is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (25 February Morning Shift) PYQ

Solution

Qus : 18 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (5 April Evening Shift) PYQ

Solution

Qus : 19 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (27 January Morning Shift) PYQ

Solution

Qus : 20 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (27 January Morning Shift) PYQ

Solution

Qus : 21 JEE MAIN PYQ

Suppose $A$ and $B$ are the coefficients of $30^{\text{th}}$ and $12^{\text{th}}$ terms respectively in the binomial expansion of $(1+x)^{2n-1}$. If $2A=5B$, then $n$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (24 January Evening Shift) PYQ

Solution

Qus : 22 JEE MAIN PYQ

$25^{190}-19^{190}-8^{190}+2^{190}$ is divisible by:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (8 April Evening Shift) PYQ

Solution

Qus : 23 JEE MAIN PYQ

A ratio of the $5^{\text{th}}$ term from the beginning to the $5^{\text{th}}$ term from the end in the binomial expansion of $\left(2^{1/3}+\dfrac{1}{2\cdot 3^{1/3}}\right)^{10}$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (12 January Morning Shift) PYQ

Solution

Qus : 24 JEE MAIN PYQ

The maximum value of the term independent of 't' in the expansion of ${\left( {t{x^{{1 \over 5}}} + {{{{(1 - x)}^{{1 \over {10}}}}} \over t}} \right)^{10}}$ where x$\in$(0, 1) is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (26 February Morning Shift) PYQ

Solution

Qus : 25 JEE MAIN PYQ

The absolute difference of the coefficients of $x^{10}$ and $x^{7}$ in the expansion of $\left(2x^{2}+\dfrac{1}{2x}\right)^{11}$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (8 April Evening Shift) PYQ

Solution

Qus : 26 JEE MAIN PYQ

The coefficient of $x^{7}$ in $\left(ax-\dfrac{1}{bx^{2}}\right)^{13}$ and the coefficient of $x^{-5}$ in $\left(ax+\dfrac{1}{bx^{2}}\right)^{13}$ are equal. Then $a^{4}b^{4}$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (10 April Morning Shift) PYQ

Solution

Qus : 27 JEE MAIN PYQ

The coefficient of x¹⁰¹ in the expression ${(5 + x)^{500}} + x{(5 + x)^{499}} + {x^2}{(5 + x)^{498}} + \,\,.....\,\, + \,\,{x^{500}}$, x > 0, is

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (25 June Evening Shift) PYQ

Solution

Qus : 28 JEE MAIN PYQ

If ⁿC₄, ⁿC₅ and ⁿC₆ are in A.P., then n can be :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (12 January Evening Shift) PYQ

Solution

Qus : 29 JEE MAIN PYQ

The total number of irrational terms in the binomial expansion of $(7^{1/5}-3^{1/10})^{60}$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (12 January Evening Shift) PYQ

Solution

Qus : 30 JEE MAIN PYQ

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:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (10 April Evening Shift) PYQ

Solution

Qus : 31 JEE MAIN PYQ

Let the coefficients of three consecutive terms $T_r, T_{r+1}$ and $T_{r+2}$ in the binomial expansion of $(a+b)^{12}$ be in a G.P. Let $p$ be the number of all possible values of $r$. Let $q$ be the sum of all rational terms in the binomial expansion of $(\sqrt{3}+\sqrt[3]{4})^{12}$. Then $p+q$ is equal to

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (28 January Evening Shift) PYQ

Solution

Qus : 32 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2 September 2020 (Morning) PYQ

Solution

Qus : 33 JEE MAIN PYQ

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 :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (16 March Morning Shift) PYQ

Solution

Qus : 34 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (6 April Evening Shift) PYQ

Solution

Qus : 35 JEE MAIN PYQ

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:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (11 April Evening Shift) PYQ

Solution

Qus : 36 JEE MAIN PYQ

The number of integral terms in the expansion of $\left(5^{\tfrac{1}{2}}+7^{\tfrac{1}{8}}\right)^{1016}$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (8 April Evening Shift) PYQ

Solution

Qus : 37 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (11 April Evening Shift) PYQ

Solution

Qus : 38 JEE MAIN PYQ

If the number of integral terms in the expansion of (3^1/2 + 5^1/8)ⁿ is exactly 33, then the least valueof n is

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 3 September 2020 (Morning) PYQ

Solution

Qus : 39 JEE MAIN PYQ

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 :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (17 March Morning Shift) PYQ

Solution

Qus : 40 JEE MAIN PYQ

If the fourth term in the binomial expansion of $\left(\sqrt{,x^{\frac{1}{1+\log_{10}x}}+x^{\frac{1}{12}},}\right)^{6}$ is equal to $200$, and $x>1$, then the value of $x$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (8 April Evening Shift) PYQ

Solution

Qus : 41 JEE MAIN PYQ

Suppose $2-p,\ p,\ 2-\alpha,\ \alpha$ are the coefficients of four consecutive terms in the expansion of $(1+x)^n$. Then the value of $\,p^2-\alpha^2+6\alpha+2p\,$ equals:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (30 January Evening Shift) PYQ

Solution

Qus : 42 JEE MAIN PYQ

The coefficient of $x^{-5}$ in the binomial expansion of $\left( \dfrac{x+1}{x^{\frac{2}{3}} - x^{\frac{1}{3}} + 1} ;-; \dfrac{x-1}{x - x^{\frac{1}{2}}} \right)^{10}$, where $x \neq 0,1$, is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2017 (9 April Morning Shift) PYQ

Solution

Qus : 43 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (8 April Evening Shift) PYQ

Solution

Qus : 44 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (9 April Morning Shift) PYQ

Solution

Qus : 45 JEE MAIN PYQ

Evaluate the sum: $\displaystyle \sum_{k=0}^{6} \binom{51-k}{3}$

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (25 January Evening Shift) PYQ

Solution

Qus : 46 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (28 June Evening Shift) PYQ

Solution

Qus : 47 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (13 April Evening Shift) PYQ

Solution

Qus : 48 JEE MAIN PYQ

If the fourth term in the binomial expansion of $\left(\dfrac{2}{x}+x^{\log_8 x}\right)^6$ $(x>0)$ is $20\times 8^7$, then a value of $x$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (9 April Morning Shift) PYQ

Solution

Qus : 49 JEE MAIN PYQ

For $x \in \mathbb{R}$, $f(x) = |\log 2 - \sin x|$ and $g(x) = f(f(x))$, then:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2016 (Offline) PYQ

Solution

Qus : 50 JEE MAIN PYQ

The term independent of $x$ in the expansion of $\left(\frac{(x+1)}{\left(x^{2 / 3}+1-x^{1 / 3}\right)}-\frac{(x-1)}{\left(x-x^{1 / 2}\right)}\right)^{10}, x>1$, is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (2 April Morning Shift) PYQ

Solution

Qus : 51 JEE MAIN PYQ

Let $\left(a+bx+cx^2\right)^{10}=\displaystyle\sum_{i=0}^{20} p_i x^i,\ a,b,c\in\mathbb{N}.$ If $p_1=20$ and $p_2=210$, then $2(a+b+c)$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (15 April Morning Shift) PYQ

Solution

Qus : 52 JEE MAIN PYQ

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 :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 3 September 2020 (Evening) PYQ

Solution

Qus : 53 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (29 June Morning Shift) PYQ

Solution

Qus : 54 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (9 April Evening Shift) PYQ

Solution

Qus : 55 JEE MAIN PYQ

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 $\displaystyle \frac{\binom{200}{99} \, K}{a} = \frac{2^{\,\ell} \, m}{n}$, where $m$ and $n$ are odd numbers, then the ordered pair $(\ell,n)$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (29 January Evening Shift) PYQ

Solution

Qus : 56 JEE MAIN PYQ

If $\displaystyle \sum_{r=0}^{10} \left(\dfrac{10^{r+1}-1}{10^r}\right) , {}^{11}C_{r+1} = \dfrac{\alpha^{11} - 11^{11}}{10^{10}}$, then $\alpha$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (2 April Evening Shift) PYQ

Solution

Qus : 57 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (30 January Morning Shift) PYQ

Solution

Qus : 58 JEE MAIN PYQ

For two positive real numbers a and b such that ${1 \over {{a^2}}} + {1 \over {{b^3}}} = 4$, then minimum value of the constant term in the expansion of ${\left( {a{x^{{1 \over 8}}} + b{x^{ - {1 \over {12}}}}} \right)^{10}}$ is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (30 June Morning Shift) PYQ

Solution

Qus : 59 JEE MAIN PYQ

The sum of all rational terms in the expansion of $(2 + \sqrt{3})^8$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (3 April Morning Shift) PYQ

Solution

Qus : 60 JEE MAIN PYQ

If $\displaystyle \sum_{r=1}^{9} \left(\dfrac{r + 3}{2^r}\right) \cdot {^9C_r} = \alpha \left(\dfrac{3}{2}\right)^9 - \beta,; \alpha, \beta \in \mathbb{N}$, then $(\alpha + \beta)^2$ is equal to

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (3 April Morning Shift) PYQ

Solution

Qus : 61 JEE MAIN PYQ

If the coefficients of $x^{-2}$ and $x^{-4}$ in the expansion of $\left(x^{\tfrac13} + \dfrac{1}{2x^{\tfrac13}}\right)^{18},\ (x>0)$ are $m$ and $n$ respectively, then $\dfrac{m}{n}$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2016 (10 April Morning Shift) PYQ

Solution

Qus : 62 JEE MAIN PYQ

Let $\alpha, \beta, \gamma$ and $\delta$ be the coefficients of $x^7, x^5, x^3$ and $x$ respectively in the expansion of $\begin{aligned} & \left(x+\sqrt{x^3-1}\right)^5+\left(x-\sqrt{x^3-1}\right)^5, x>1 \text {. If } u \text { and } v \text { satisfy the equations } \\\\ & \alpha u+\beta v=18, \\\\ & \gamma u+\delta v=20, \end{aligned}$ then $\mathrm{u+v}$ equals :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (22 January Evening Shift) PYQ

Solution

Qus : 63 JEE MAIN PYQ

The smallest natural number $n$ such that the coefficient of $x$ in the expansion of $\left(x^{2}+\dfrac{1}{x^{3}}\right)^{n}$ is ${}^nC_{23}$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (10 April Evening Shift) PYQ

Solution

Qus : 64 JEE MAIN PYQ

Let $m$ and $n$ be the coefficients of the seventh and thirteenth terms respectively in the expansion of $\left(\dfrac{1}{3}x^{\tfrac13}+\dfrac{1}{2x^{\tfrac23}}\right)^{18}$. Then $\left(\dfrac{n}{m}\right)^{\tfrac13}$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (1 February Evening Shift) PYQ

Solution

Qus : 65 JEE MAIN PYQ

If the third term in the binomial expansion of $(1+x^{\log_{8}x})^{5}$ equals $2560$, then a possible value of $x$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (10 January Morning Shift) PYQ

Solution

Qus : 66 JEE MAIN PYQ

The sum of all those terms which are rational numbers in the expansion of (2^1/3 + 3^1/4)¹² is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (25 July Morning Shift) PYQ

Solution

Qus : 67 JEE MAIN PYQ

The sum of all those terms which are rational numbers in the expansion of (2^1/3 + 3^1/4)¹² is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (25 July Evening Shift) PYQ

Solution

Qus : 68 JEE MAIN PYQ

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:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (25 July Evening Shift) PYQ

Solution

Qus : 69 JEE MAIN PYQ

For an integer $n\ge 2$, if the arithmetic mean of all coefficients in the binomial expansion of $(x+y)^{2n-3}$ is $16$, then the distance of the point $P,(2n-1,\ n^{2}-4n)$ from the line $x+y=8$ is

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (4 April Morning Shift) PYQ

Solution

Qus : 70 JEE MAIN PYQ

The sum of all rational terms in the expansion of $\left(2^{\frac15}+5^{\frac13}\right)^{15}$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (4 April Morning Shift) PYQ

Solution

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