Binomial Theorem — JEE Main PYQs

If the coefficient of $x$ in the expansion of $\left(a{x}^2+bx+c\right)(1-2x{)}^{26}$ is -56 and the coefficients of ${\mathrm{x}}^2$ and ${\mathrm{x}}^3$ are both zero, then $\mathrm{a}+\mathrm{b}+\mathrm{c}$ is equal to

If …

The coefficient of $x^{48}$ in $(1+x)+2(1+x{)}^2+3(1+x{)}^3+\dots +100(1+x{)}^{100}$ is equal to:

Let ${\mathrm{C}}_{\mathrm{r}}$ denote the coefficient of ${\mathrm{x}}^{\mathrm{r}}$ in the binomial expansion of $(1+\mathrm{x}{)}^{\mathrm{n}},\mathrm{n}\in \mathbb{N},0\le \mathrm{r}\le \mathrm{n}$.

If …

The value of $\frac{{}^{100}{\mathrm{C}}_{50}}{51}+\frac{{}^{100}{\mathrm{C}}_{51}}{52}+\dots .+\frac{{}^{100}{\mathrm{C}}_{100}}{101}$ is :

The sum of all possible values of $n\in N$, so that the coefficients of $x,x^2$ and $x^3$ in the expansion of $(1+x^2{)}^2(1+x{)}^n$, are in arithmetic progression is:

Let $\mathrm{S}=\frac{1}{25!}+\frac{1}{3!23!}+\frac{1}{5!21!}+\dots$ up to 13 terms.

If $13\mathrm{S}=\frac{2^{\mathrm{k}}}{\mathrm{n}!},\mathrm{k}\in \mathrm{N}$, then $\mathrm{n}+\mathrm{k}$ is equal to

Given below two statements :

Statement I : $25^{13}+20^{13}+8^{13}+3^{13}$ is divisible by 7.
Statement II : The integral part of $(7+4\sqrt{3}{)}^{25}$ is an odd number.

In the light of the above statements, choose the correct answer from the options given below :

The sum of the coefficients of $x^{499}$ and $x^{500}$ in $(1+x{)}^{1000}+x(1+x{)}^{999}+x^2(1+x{)}^{998}+\dots \dots +x^{1000}$ is

The number of elements in the set $S=\{(r,k):k\in \mathbf{Z}$ and ${}^{36}{C}_{r+1}=\frac{6({}^{35}{C}_r)}{(k^2-3)}\}$, is

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

Let the smallest value of $\mathrm{k}\in \mathbb{N}$, for which the coefficient of ${\mathrm{x}}^3$ in $(1+\mathrm{x}{)}^3+(1+\mathrm{x}{)}^4+(1+\mathrm{x}{)}^5+\dots +(1+x{)}^{99}+(1+kx{)}^{100},x\ne 0$, is $\left(43n+\frac{101}{4}\right)$ ( …

If the sum of the coefficients of $x^7$ and $x^{14}$ in the expansion of ${\left(\frac{1}{x^3}-x^4\right)}^n,x\ne 0$, is zero, then the value of $n$ is ______.

The coefficient of $x^2$ in the expansion of ${\left(2{\mathrm{x}}^2+\frac{1}{\mathrm{x}}\right)}^{10},\mathrm{x}\ne 0$, is :

A variable $X$ takes values $0,0,2,6,12,20,\dots$. $\mathrm{n}(\mathrm{n}-1)$ with frequencies ${}^{\mathrm{n}}{\mathrm{C}}_0,{}^{\mathrm{n}}{\mathrm{C}}_1,{}^{\mathrm{n}}{\mathrm{C}}_2,{}^{\mathrm{n}}{\mathrm{C}}_3,{}^{\mathrm{n}}{\mathrm{C}}_4$, …

If ${\left(1-x^3\right)}^{10}={\sum }_{r=0}^{10}{a}_r{x}^r(1-x{)}^{30-2r}$, then $\frac{9a_9}{a_{10}}$ is equal to

If the coefficients of the middle terms in the binomial expansions of $(1+\alpha x{)}^{26}$ and $(1-\alpha x{)}^{28}$, $\alpha \ne 0$, are equal, then the value of $\alpha$ is:

If $26\left(\frac{2^3}{3}\left({}^{12}{\mathrm{C}}_2\right)+\frac{2^5}{5}\left({}^{12}{\mathrm{C}}_4\right)+\frac{2^7}{7}\left({}^{12}{\mathrm{C}}_6\right)+\dots .+\frac{2^{13}}{13}\left({}^{12}{\mathrm{C}}_{12}\right)\right)=3^{13}-\alpha$, then $\alpha$ is equ …

If for $3\le \mathrm{r}\le 30$, …

If ${\sum }_{\mathrm{r}=0}^5\frac{{}^{11}{\mathrm{C}}_{2\mathrm{r}+1}}{2\mathrm{r}+2}=\frac{\mathrm{m}}{\mathrm{n}}$, where $\gcd (\mathrm{m},\mathrm{n})=1$, then $\mathrm{m}-\mathrm{n}$ is equal to ____.

Let $\alpha ,\beta ,\gamma$ and $\delta$ be the coefficients of ${\mathrm{x}}^7,{\mathrm{x}}^5,{\mathrm{x}}^3$ and $x$ respectively in the expansion of ${\left(\mathrm{x}+\sqrt{{\mathrm{x}}^3-1}\right)}^5+{\left(\mathrm{x}-\sqrt{{\mathrm{x}}^3-1}\right)}^5,\mathrm{x}>1$. If u …

If ${\sum }_{\mathrm{r}=1}^{30}\frac{{\mathrm{r}}^2{\left({}^{30}{\mathrm{C}}_{\mathrm{r}}\right)}^2}{{}^{30}{\mathrm{C}}_{\mathrm{r}-1}}=\alpha \times 2^{29}$, then $\alpha$ is equal to
____ .

The sum of all rational terms in the expansion of ${\left(1+2^{1/3}+3^{1/2}\right)}^6$ is equal to

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

The remainder, when $7^{103}$ is divided by 23 , is equal to:

The least value of n for which the number of integral terms in the Binomial expansion of $(\sqrt[3]{7}+\sqrt[12]{11}{)}^{\mathrm{n}}$ is 183 , is :

Let the coefficients of three consecutive terms ${\mathrm{T}}_{\mathrm{r}}$, ${\mathrm{T}}_{\mathrm{r}+1}$ and ${\mathrm{T}}_{\mathrm{r}+2}$ in the binomial expansion of $(\mathrm{a}+\mathrm{b}{)}^{12}$ be in a G.P. and let p be the number of all possible values of r . Let q be the sum o …

If $\alpha =1+{\sum }_{r=1}^6(-3{)}^{r-1}{}^{12}{C}_{2r-1}$, then the distance of the point $(12,\sqrt{3})$ from the line $\alpha x-\sqrt{3}y+1=0$ is

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:

For some $n\ne 10$, let the coefficients of the $5^{\text{th }},6^{\text{th }}$ and $7^{\text{th }}$ 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 :

If ${\sum }_{r=0}^{10}\left(\frac{10^{r+1}-1}{10^r}\right)\cdot {}^{11}{C}_{r+1}=\frac{{\alpha }^{11}-11^{11}}{10^{10}}$, then $\alpha$ is equal to :

Let ${\left(1+x+x^2\right)}^{10}=a_0+a_{1}x+a_2{x}^2+\dots .+a_{20}{x}^{20}$. If $\left(a_1+a_3+a_5+\dots .+a_{19}\right)-11{\mathrm{a}}_2=121\mathrm{k}$, then k is equal to ______ .

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

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

If ${\sum }_{\mathrm{r}=1}^9\left(\frac{\mathrm{r}+3}{2^{\mathrm{r}}}\right)\cdot {}^9{\mathrm{C}}_{\mathrm{r}}=\alpha {\left(\frac{3}{2}\right)}^9-\beta$, where $\alpha ,\beta \in \mathrm{N}$, then $(\alpha +\beta {)}^2$ is equal to

If

where …

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

In the expansion of ${\left(\sqrt[3]{2}+\frac{1}{\sqrt[3]{3}}\right)}^n,\mathrm{n}\in \mathrm{N}$, if the ratio of $15^{\text{th }}$ term from the beginning to the $15^{\text{th }}$ term from the end is $\frac{1}{6}$, then the value of ${}^n{C}_3$ is:

The sum of the series
$2\times 1\times {}^{20}{\mathrm{C}}_4-3\times 2\times {}^{20}{\mathrm{C}}_5+4\times 3\times {}^{20}{\mathrm{C}}_6-5\times 4\times {}^{20}{\mathrm{C}}_7+\dots +18\times 17\times {}^{20}{\mathrm{C}}_{20}$, is equal to

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

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

The product of the last two digits of $(1919{)}^{1919}$ is

Let ${}^n{C}_{r-1}=28$, ${}^n{C}_r=56$ and ${}^n{C}_{r+1}=70$. Let $A(4\cos t,4\sin t)$, $B(2\sin t,-2\cos t)$ and $C(3r-n,r^2-n-1)$ be the vertices of a triangle ABC, where $t$ is a parameter. If $(3x-1{)}^2+(3y{)}^2=\lambda$ is the locus of the centroid of t …

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

Let $a=1+\frac{{}^2{C}_2}{3!}+\frac{{}^3{C}_2}{4!}+\frac{{}^4{C}_2}{5!}+\dots$, $b=1+\frac{{}^1{C}_0+{}^1{C}_1}{1!}+\frac{{}^2{C}_0+{}^2{C}_1+{}^2{C}_2}{2!}+\frac{{}^3{C}_0+{}^3{C}_1+{}^3{C}_2+{}^3{C}_3}{3!}+\dots$. Then $\frac{2b}{a^2}$ is equal …

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

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

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

If the second, third and fourth terms in the expansion of $(x+y{)}^n$ are 135, 30 and $\frac{10}{3}$, respectively, then $6(n^3+x^2+y)$ is equal to ____.

Let $\alpha ={\sum }_{\mathrm{r}=0}^{\mathrm{n}}{\left(4{\mathrm{r}}^2+2\mathrm{r}+1\right)}^{\mathrm{n}}{\mathrm{C}}_{\mathrm{r}}$
and $\beta =\left({\sum }_{\mathrm{r}=0}^{\mathrm{n}}\frac{{}^{\mathrm{n}}{\mathrm{C}}_{\mathrm{r}}}{\mathrm{r}+1}\right)+\frac{1}{\mathrm{n}+1}$. If …