Binomial Theorem
153 JEE Maths previous year questions on Binomial Theorem — options free on every question; 15 include the answer & explanation free, the rest unlock with PYQ Pass.
The remainder when \({\left((64{)}^{(64)}\right)}^{(64)}\) is divided by \(7\) is equal to
[JEE Main 2025, 7 Apr (Shift 1)]
\(1\)
\({64}^{64}=(63+1{)}^{64}=63{\lambda }_{1}+1\)
\({({(64)}^{(64)})}^{(64)}=(63\lambda +1{)}^{64}=63{\lambda }_{2}+1\)
If the \(5^{\text {th }}, 6^{\text {th }}\) and \(7^{\text {th }}\) term of the binomial expansion of \(\left(1+x^2\right)^{n+4}\) are in A.P. Then the greatest binomial coefficient in the expansion of \(\left(1+x^2\right)^{n+4}\) is(\(\mathrm{n}\neq 10\)) (24 Jan, Shift I, Memory Based)
35
\({T}_{5},{T}_{6}\&{T}_{7}areinA.P.\\ 2{T}_{6}={T}_{5}+{T}_{7}\)
\(2 \times{ }^{n+4} C_5={ }^{n+4} C_4+{}^{n+4}C_{6}\)
\(\Rightarrow \quad 2 \times \frac{(n+4)!}{5! \times (n-1)!}=\frac{(n+4)!}{4! \times n!}+\frac{(n+4)!}{6! \times (n-2)!}\)
\(\begin{aligned}& \Rightarrow \frac{2 }{5(n-1)}=\frac{1}{n(n-1)}+\frac{1}{6 \times 5} \\& \Rightarrow \frac{2}{5(n-1)}=\frac{1}{n(n-1)}+\frac{1}{30} \\& \Rightarrow 2 n=5+\frac{n(n-1)}{6}\end{aligned}\)
\(\begin{aligned}&\begin{array}{r}\quad 12 n=30+n^2-n \\\Rightarrow \quad n^2-13 n+30=0 \\\Rightarrow \quad(n-10)(n-3)=0 \\\quad n=10,(3) \\\end{array}\\&=\text { greatest binomial coeff }\\&={}^7 C_3=\frac{7!}{3!.4!}=7 \times 5=35\end{aligned}\)
If \(26\left(\frac{{2}^{3}}{3}\left({}^{12}C_{2}\right)+\frac{{2}^{5}}{5}\left({}^{12}C_{4}\right)+\frac{{2}^{7}}{7}\left({}^{12}C_{6}\right)+⋯+\frac{{2}^{13}}{13}\left({}^{12}C_{12}\right)\right)\)\(={3}^{13}-\alpha\), then \(\alpha\) is equal to:
[JEE Main 2026, 8 Apr (Shift 2)]
\(51\)
Let \(S=26\left(\frac{2^3}{3}({}^{12}C_2)+\frac{2^5}{5}({}^{12}C_4)+\frac{2^7}{7}({}^{12}C_6)+\cdots+\frac{2^{13}}{13}({}^{12}C_{12})\right)\)
\(S=26\sum_{r=1}^{6}\frac{2^{2r+1}}{2r+1}({}^{12}C_{2r})\)
Using \(\frac{{}^{12}C_{2r}}{2r+1}=\frac{{}^{13}C_{2r+1}}{13}\), we get:
\(S=26\sum_{r=1}^{6}\frac{2^{2r+1}}{13}({}^{13}C_{2r+1})\)
\(S=2\sum_{r=1}^{6}2^{2r+1}({}^{13}C_{2r+1})\)
\(S=2\left[\sum_{r=0}^{6}2^{2r+1}({}^{13}C_{2r+1})-2({}^{13}C_1)\right]\)
\(S=2\sum_{r=0}^{6}2^{2r+1}({}^{13}C_{2r+1})-52\)
\(\sum_{r=0}^{6}{}^{13}C_{2r+1}2^{2r+1}=\frac{(1+2)^{13}-(1-2)^{13}}{2}\)
\(S=2\cdot\frac{3^{13}-(-1)^{13}}{2}-52\)
\(S=3^{13}+1-52=3^{13}-51\)
Given \(S=3^{13}-\alpha\), so \(\alpha=51\)
The sum of the coefficient 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
[JEE Main 2024, 09 Apr (Shift 2)]
\(\frac{21}{4}\)
General term, \({T}_{r+1}={}^{9}C_{r}{({x}^{2/3})}^{9-r}{\left(\frac{{x}^{-2/5}}{2}\right)}^{r}\)
\(={}^{9}\mathrm{C}_{\mathrm{r}}{\left(\frac{1}{2}\right)}^{\mathrm{r}}{\left(\mathrm{r}\right)}^{\left(6-\frac{2\mathrm{r}}{3}-\frac{2\mathrm{r}}{5}\right)}\)
for coefficient of \({x}^{2/3}\),
put \(6-\frac{2r}{3}-\frac{2r}{5}=\frac{2}{3}\)
\(\Rightarrow r = 5\)
Coefficient of \({x}^{2/3}\) \(={}^{9}C_{5}{\left(\frac{1}{5}\right)}^{5}\)
For coefficient of \({x}^{-2/5}\)
put \(6-\frac{2r}{3}-\frac{2r}{5}=-\frac{2}{3}\)
\(\Rightarrow r = 6\)
Coefficient of \({x}^{-2/5}\) is \({}^{9}C_{6}{\left(\frac{1}{2}\right)}^{6}\)
sum \(={}^{9}C_{5}{\left(\frac{1}{2}\right)}^{5}+{}^{9}C_{6}{\left(\frac{1}{2}\right)}^{6}=\frac{21}{4}\)
The sum of the coefficient 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
[JEE Main 2024, 09 Apr (Shift 2)]
\(\frac{21}{4}\)
General term, \({T}_{r+1}={}^{9}C_{r}{({x}^{2/3})}^{9-r}{\left(\frac{{x}^{-2/5}}{2}\right)}^{r}\)
\(={}^{9}\mathrm{C}_{\mathrm{r}}{\left(\frac{1}{2}\right)}^{\mathrm{r}}{\left(\mathrm{r}\right)}^{\left(6-\frac{2\mathrm{r}}{3}-\frac{2\mathrm{r}}{5}\right)}\)
for coefficient of \({x}^{2/3}\),
put \(6-\frac{2r}{3}-\frac{2r}{5}=\frac{2}{3}\)
\(\Rightarrow r = 5\)
Coefficient of \({x}^{2/3}\) \(={}^{9}C_{5}{\left(\frac{1}{5}\right)}^{5}\)
For coefficient of \({x}^{-2/5}\)
put \(6-\frac{2r}{3}-\frac{2r}{5}=-\frac{2}{3}\)
\(\Rightarrow r = 6\)
Coefficient of \({x}^{-2/5}\) is \({}^{9}C_{6}{\left(\frac{1}{2}\right)}^{6}\)
sum \(={}^{9}C_{5}{\left(\frac{1}{2}\right)}^{5}+{}^{9}C_{6}{\left(\frac{1}{2}\right)}^{6}=\frac{21}{4}\)
Let \(C_r\) denote the coefficient of \(x^r\) in the binomial expansion of \((1+{x}^{n}),\text{ }n\in N,\text{ }0\leq r\leq n.\)
If \({P}_{n}={C}_{0}−{C}_{1}+\frac{{2}^{2}}{3}{C}_{2}−\frac{{2}^{3}}{4}{C}_{3}+...+\frac{{(−2)}^{n}}{n+1}{C}_{n},\) then the value of \(\sum _{n=1}^{25}\frac{1}{{P}_{2n}}\) equals
[JEE Main 2026, 22 Jan (Shift 2)]
\(675\)
\({P}_{n}={\sum }_{r=0}^{n}\frac{{}^{n}{C}_{r}{(−2)}^{r}}{r+1}\)
\(={\sum }_{r=0}^{n}\frac{1}{\left(n+1\right)}{}^{n+1}{C}_{r+1}{(−2)}^{r}\)
\(=\frac{−1}{2\left(n+1\right)}{\sum }_{r=0}^{n}{}^{n+1}{C}_{r+1}{(−2)}^{r+1}\)
\(=\frac{−1}{2\left(\text{n}+1\right)}\left[{(1−2)}^{\text{n}+1}−1\right]\)
\({\text{⇒P}}_{\text{n}}=\frac{1}{2\left(\text{n}+1\right)}\left[1−{(−1)}^{\text{n}+1}\right]\)
\({P}_{2n}=\frac{1}{2\left(2n+1\right)}\left[1−{(−1)}^{2n+1}\right]\)
\({P}_{2n}=\frac{1}{2n+1}\)
\(\sum_{n=1}^{25} \frac{1}{P_{2 n}}=\sum_{n=1}^{25}(2 n+1)=3+5+\ldots . .+51=\frac{25}{2}[51+3]=25 \times 27=675\)
If in the expansion of \(\ (1+x)^p(1-x)^q\) coefficient of \(x \\& x^2\) is \(1 \\&-2\) then find \(p^2+q^2\)
13
\(\begin{aligned}&& \text { coeff. of } \mathrm{x} \Rightarrow \mathrm{p}-\mathrm{q}=1 \Rightarrow \mathrm{p}=1+\mathrm{q} \\& \text { coeff. of } \mathrm{x}^2 \Rightarrow{ }^q C_2-p q+{ }^p C_2=-2 \\& \frac{p(p-1)}{2}-p q+\frac{(q-1) q}{2}=-2 \\& \Rightarrow \mathrm{p}^2-\mathrm{p}-2 \mathrm{pq}+ \mathrm{q}^2-\mathrm{q}=-4 \\& \Rightarrow \mathrm{q}=2 \\& \mathrm{p}=3&\text { So, } p^2+q^2=3^2+2^2=13\end{aligned}\)
Coefficient of \(x^{2012}\) in \((1-x)^{2008}\left(1+x+x^2\right)^{2007}\). (22 Jan, Shift I, Memory Based)
0
\(\begin{aligned}& (1-x)\left[(1-x)\left(1+x+x^2\right)\right]^{2007} \\& =(1-x)\left(1-x^3\right)^{2007} \\& =\left(1-x^3\right)^{2007}-x\left(1-x^3\right)^{2007}\end{aligned}\)
\([(1- \left.x^3\right)^{2007}\) contains \(3 \lambda\) types of exponents while \(x\left(1-x^3\right)^{2007}\) will have \((3 \lambda+1)\) type while 2012 is
\((3 \lambda+2)\) type] that is not possible \(\Rightarrow 0\)
Coefficient of \(x^{2012}\) in \(\left(1-x^3\right)^{2007}=0\)
Coefficient of \(x^{2011}\) in \(\left(1-x^3\right)^{2007}=0\)
\(\Rightarrow\) Coefficient of \(x^{2012}\) in \((1-x)^{2008}\left(1+x+x^2\right)^{2007}=0\)
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 \(\left(1+x^2\right)^2(1+x)^n\), are in arithmetic progression is:
[JEE Main 2026, 23 Jan (Shift 2)]
\(9\)
Given expansion:
\(\left({x}^{4}+2{x}^{2}+1\right)\left({}^{n}C_{0}{x}^{0}+{}^{n}C_{1}{x}^{1}+{}^{n}C_{2}{x}^{2}+{}^{n}C_{3}{x}^{3}+\ldots \right)\)
Coefficient \(x\Rightarrow {}^{n}C_{1}\),
coeff. of \({x}^{2}\Rightarrow 2+{}^{n}C_{2}\)\(=2+\frac{n(n-1)}{2}\)
Coeff. of \({x}^{3}=2.{}^{n}C_{1}+{}^{n}C_{3}\)
\(=2n+\frac{n(n-1)(n-2)}{6}\)
Now according to question
\(n+2n+\frac{n(n-1)(n-2)}{6}=2\left[2+\frac{n(n-1)}{2}\right]\\ \Rightarrow 3n+\frac{n(n-1)(n-2)}{6}=4+n(n-1)\\ \Rightarrow {n}^{3}-9{n}^{2}+26n-24=0\\ \Rightarrow n=2,3,4\)
Now checking for \(n = 2\)
Coeff. of \(x=2\), Coeff. of \(x^2=3\), Coeff. of \(x^3=4\)
are in A.P.
\(\Rightarrow n=2\) is also the correct choice
Required sum of values of \(n=2+3+4=9\)
The sum of the coefficients of \({x}^{499}\) and \({x}^{500}\) in \({\left(1+x\right)}^{1000}+x{\left(1+x\right)}^{999}+{x}^{2}{\left(1+x\right)}^{998}+...+{x}^{1000}\) is:
[JEE Main 2026, 28 Jan (Shift 2)]
\({}^{1002}{C}_{500}\)
Let \(S={(1+x)}^{1000}+x{(1+x)}^{999}+{x}^{2}{(1+x)}^{998}+\ldots .+{x}^{1000}\)
\(={(1+x)}^{1000}\frac{\left(1−{\left(\frac{x}{1+x}\right)}^{1001}\right)}{1−\frac{x}{1+x}}\)
\(={(1+x)}^{1001}−{x}^{1001}\)
The sum of the coefficients of \(x^{499}\) and \(x^{500}\)
\({=}^{1001}{C}_{499}{+}^{1001}{C}_{500}{=}^{1002}{C}_{500}\)
The value of \(\frac{{}^{100}C_{50}}{51}+\frac{{}^{100}C_{51}}{52}+\ldots +\frac{{}^{100}C_{100}}{101}\) is:
\(\frac{{2}^{100}}{101}\)
\(S=\sum _{r=50}^{100}\frac{{}^{100}C_{r}}{r+1}=\sum _{r=50}^{100}\frac{1}{r+1}\cdot \frac{r+1}{101}\cdot {}^{101}C_{r+1}\\ S=\frac{1}{101}\sum _{r=50}^{100}{}^{101}C_{r+1}\\ ∵\sum _{r=0}^{n}{}^{n}C_{r}={2}^{n},{}^{n}C_{r}={}^{n}C_{n-r}\\ \Rightarrow \sum _{r=0}^{50}{}^{101}C_{r}=\sum _{r=51}^{101}{}^{101}C_{r}=\frac{{2}^{101}}{2}={2}^{100}\\ ∴S=\frac{1}{101}\times \frac{{2}^{101}}{2}=\frac{{2}^{100}}{101}\)
Let the coefficient of three consecutive terms \({T}_{r},{T}_{r+1},\&{T}_{r+2}\)in the binomial expansion of \({(a+b)}^{12}\) be in a A.P. and let p be the no. of all possible values of r, let q be the sum of all rational term in the binomial expansion \({(\sqrt[4]{3}+\sqrt[3]{4})}^{12}\). then p+q is equal to:
283
\({(\sqrt[4]{3}+\sqrt[3]{4})}^{12}\)
| Exponent of \(\sqrt[4]{3}\) | Exponent of \(\sqrt[3]{4}\) | Term |
| 12 | 0 | 27 |
| 0 | 12 | 256 |
q = 27+256 = 283
Now, \({}^{12}\mathrm{C}_{\mathrm{r}-1}+{}^{12}\mathrm{C}_{\mathrm{r}+1}=2\cdot {}^{12}\mathrm{C}_{\mathrm{r}}\)
\(\frac{12!}{(r-1)!(13-r)!}+\frac{12!}{(r+1)!(11-r)!}=2\cdot \frac{12!}{r!(12-r)!}\)
\(\frac{1}{(13-r)(12-r)}+\frac{1}{(r+1)(r)}=\frac{2}{r(12-r)}\)
\(2{r}^{2}-24r+156=26r+26-2{r}^{2}-2r\)
\(2{\mathrm{r}}^{2}-24\mathrm{r}+65=0\)
No integral value of r.
\(p=0\\ p+q=283\)
If \(\sum _{\mathrm{r}=0}^{10}\left(\frac{{10}^{\mathrm{r}+1}-1}{{10}^{\mathrm{r}}}\right)\cdot {}^{11}\mathrm{C}_{\mathrm{r}+1}=\frac{{\alpha }^{11}-{11}^{11}}{{10}^{10}}\), then \(\alpha\) is equal to :
\(20\)
Let \(k=r+1\). When \(r=0\), \(k=1\); when \(r=10\), \(k=11\). The sum becomes:
\(\sum _{k=1}^{11}(\frac{11}{k})\)
Using the binomial theorem, \({\sum }_{k=0}^{11}(\frac{11}{k})={2}^{11}\). Thus:
\(\sum _{k=1}^{11}(\frac{11}{k})={2}^{11}−(\frac{11}{0})={2}^{11}−1\)
Multiplying by 10:
\(10\sum _{r=0}^{10}(\frac{11}{r+1})=10({2}^{11}−1)\)
Step 2: Evaluate the second sum \({\sum }_{r=0}^{10}\frac{(\frac{11}{r+1})}{{10}^{r}}\)Again, let \(k=r+1\), so \(r=k−1\). The sum becomes:
\(\sum _{k=1}^{11}\frac{(\frac{11}{k})}{{10}^{k−1}}=10\sum _{k=1}^{11}\frac{(\frac{11}{k})}{{10}^{k}}\)
Using the binomial theorem for \({(1+\frac{1}{10})}^{11}\):
\({(1+\frac{1}{10})}^{11}=\sum _{k=0}^{11}(\frac{11}{k}){(\frac{1}{10})}^{k}\text{ }⟹\text{ }\sum _{k=0}^{11}\frac{(\frac{11}{k})}{{10}^{k}}={(\frac{11}{10})}^{11}\)
Subtracting the \(k=0\) term:
\(\sum _{k=1}^{11}\frac{(\frac{11}{k})}{{10}^{k}}={(\frac{11}{10})}^{11}−1\)
Multiplying by 10:
\(10\sum _{k=1}^{11}\frac{(\frac{11}{k})}{{10}^{k}}=10(\frac{{11}^{11}}{{10}^{11}}−1)=\frac{{11}^{11}}{{10}^{10}}−10\)
Step 3: Combine the two sumsSubstitute the results back into the original expression:
\(10({2}^{11}−1)−(\frac{{11}^{11}}{{10}^{10}}−10)=10⋅{2}^{11}−10−\frac{{11}^{11}}{{10}^{10}}+10=10⋅{2}^{11}−\frac{{11}^{11}}{{10}^{10}}\)
Notice that \(10⋅{2}^{11}=\frac{(20{)}^{11}}{{10}^{10}}\) (since \({20}^{11}=(2⋅10{)}^{11}={2}^{11}⋅{10}^{11}\), so dividing by \({10}^{10}\) gives \({2}^{11}⋅10\)). Thus:
\(10⋅{2}^{11}−\frac{{11}^{11}}{{10}^{10}}=\frac{{20}^{11}−{11}^{11}}{{10}^{10}}\)
Comparing with the given form \(\frac{{\alpha }^{11}−{11}^{11}}{{10}^{10}}\), we see \(\alpha =20\).
The term independent of x in the expansion of \({\left(\frac{(\mathrm{x}+1)}{\left({\mathrm{x}}^{2/3}+1-{\mathrm{x}}^{1/3}\right)}-\frac{(\mathrm{x}-1)}{\left(\mathrm{x}-{\mathrm{x}}^{1/2}\right)}\right)}^{10},\mathrm{x}>1\) is:
[JEE Main 2025, 2 Apr (Shift 1)]
\(210\)
The first term is \(\frac{x+1}{{x}^{2\mathrm{/}3}−{x}^{1\mathrm{/}3}+1}\).
Substituting this into the first term: \(\frac{({x}^{1\mathrm{/}3}+1)({x}^{2\mathrm{/}3}−{x}^{1\mathrm{/}3}+1)}{{x}^{2\mathrm{/}3}−{x}^{1\mathrm{/}3}+1}={x}^{1\mathrm{/}3}+1\)
The second term is \(\frac{x−1}{x−{x}^{1\mathrm{/}2}}\).
Substituting these: \(\frac{(\sqrt{x}−1)(\sqrt{x}+1)}{\sqrt{x}(\sqrt{x}−1)}=\frac{\sqrt{x}+1}{\sqrt{x}}\)\(=\frac{{x}^{1\mathrm{/}2}+1}{{x}^{1\mathrm{/}2}}=1+{x}^{−1\mathrm{/}2}\)
\({(({x}^{1\mathrm{/}3}+1)−(1+{x}^{−1\mathrm{/}2}))}^{10}\)\(={({x}^{1\mathrm{/}3}−{x}^{−1\mathrm{/}2})}^{10}\)
\({T}_{r+1}=(\frac{10}{r})({x}^{1\mathrm{/}3}{)}^{10−r}(−{x}^{−1\mathrm{/}2}{)}^{r}\\ =(\frac{10}{r})(−1{)}^{r}{x}^{\frac{10−r}{3}−\frac{r}{2}}\)
For the term to be independent of \(x\), the exponent of \(x\) must be zero:
\(\frac{10−r}{3}−\frac{r}{2}=0\) :
\(2(10−r)−3r=0\)
\(20−2r−3r=0\)
\(20=5r\text{ }⟹\text{ }r=4\)
Substitute \(r=4\) into the expression for
\({T}_{r+1}\): \({T}_{5}=(\frac{10}{4})(−1{)}^{4}=\frac{10\times 9\times 8\times 7}{4\times 3\times 2\times 1}\times 1=10\times 3\times 7=210\)
The coefficient of \(x^2\) in the expansion of \(\left(2 x^2+\frac{1}{x}\right)^{10}, x \neq 0\), is:
[JEE Main 2026, 5 Apr (Shift 2)]
\(3360\)
\(\left(2 x^2+\frac{1}{x}\right)^{10}, x \neq 0 \)
\( T_{r+1}={ }^{10} C_r\left(2 x^2\right)^{10-r}\left(\frac{1}{x}\right)^r \)
\( T_{r+1}={ }^{10} C_r(2)^{10-r} \cdot(x)^{20-3 r}\)
We need to find coefficient of \(x^2\)
\(\therefore 20-3 r=2 \)
\( 3 r=18 \Rightarrow r=6 \)
\( \therefore\) Coefficient of \( x^2={ }^{10} C_6(2)^{10-6} \)
\( ={ }^{10} C_6(2)^4=210 \times 16=3360\)
If the coefficients of the middle terms in the binomial expansions of \((1+\alpha x{)}^{26}\) and \((1-\alpha x{)}^{28},\alpha \neq 0\), are equal, then value of \(\alpha\) is:
[JEE Main 2026, 6 Apr (Shift 1)]
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Let \(C_r\) denote the coefficient of \(x^r\) in the binomial expansion of \((1+{x}^{n}),\text{ }n\in N,\text{ }0\leq r\leq n.\)
If \({P}_{n}={C}_{0}−{C}_{1}+\frac{{2}^{2}}{3}{C}_{2}−\frac{{2}^{3}}{4}{C}_{3}+...+\frac{{(−2)}^{n}}{n+1}{C}_{n},\) then the value of \(\sum _{n=1}^{25}\frac{1}{{P}_{2n}}\) equals
[JEE Main 2026, 22 Jan (Shift 2)]
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If in the expansion of \((1+x)^p(1-x)^q\), the coefficients of x and \(x^2\) are 1 and -2 , respectively, then \(\mathrm{p}^2+\mathrm{q}^2\) is equal to :
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\(\text{ If }{3}^{107}\text{ is divided by }23\text{, then remainder is }\)
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Given below are 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:
[JEE Main 2026, 28 Jan (Shift 2)]
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Let \(\alpha, \beta, \gamma\) and \(\delta\) be the coefficients of \(x^7, x^5, x^3\) and \(x\) respectively in the expansion of \(\left(x+\sqrt{x^3-1}\right)^5+\left(x-\sqrt{x^3-1}\right)^5, x>1\). If \(u\) and \(v\) satisfy the equations
\(\begin{aligned}& \alpha u+\beta v=18 \\& \gamma u+\delta v=20\end{aligned}\)
then \(u+v\) equals :
[JEE Main 2025, 22 Jan (Shift 2)]
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If A denotes the sum of all the coefficients in the expansion of \(\left(1-3 x+10 x^2\right)^n\) and \(B\) denotes the sum of all the coefficients in the expansion of \(\left(1+x^2\right)^n\), then :
[JEE Main 2024, 27 Jan (Shift 1)]
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If A denotes the sum of all the coefficients in the expansion of \(\left(1-3 x+10 x^2\right)^n\) and \(B\) denotes the sum of all the coefficients in the expansion of \(\left(1+x^2\right)^n\), then :
[JEE Main 2024, 27 Jan (Shift 1)]
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If \({1}^{2}\cdot \left({}^{15}\mathrm{C}_{1}\right)+{2}^{2}\cdot \left({}^{15}\mathrm{C}_{2}\right)+{3}^{2}\cdot \left({}^{15}\mathrm{C}_{3}\right)+\ldots .+{15}^{2}\cdot \left({}^{15}\mathrm{C}_{15}\right)=\)\({2}^{\mathrm{m}}\cdot {3}^{\mathrm{n}}\cdot {5}^{\mathrm{k}}\), where \(\mathrm{m},\mathrm{n},\mathrm{k}\in \mathrm{N}\), then \(\mathrm{m}+\mathrm{n}+\mathrm{k}\) is equal to :-
[JEE Main 2025, 4 Apr (Shift 1)]
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The remainder when \({\left((64{)}^{(64)}\right)}^{(64)}\) is divided by \(7\) is equal to
[JEE Main 2025, 7 Apr (Shift 1)]
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The coefficient of \(x^2\) term in the binomial expansion of \(\left(\frac{1}{3} x^{\frac{1}{2}}+x^{-\frac{1}{4}}\right)^{10}\) is:
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If \(\sum _{r\text{ }=\text{ }1}^{9}\left(\frac{r+3}{{2}^{r}}\right)\text{ }.{\text{ }}^{9}{C}_{r}=\text{ }\alpha {\left(\frac{3}{2}\right)}^{9}−\text{ }\beta ,\alpha ,\beta \in \mathrm{N},\) then \((\alpha +\beta {)}^{2}\) is equal to
[JEE Main 2025, 3 Apr (Shift 1)]
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Coefficient of \(x^{2012}\) in \((1-x)^{2008}\left(1+x+x^2\right)^{2007}\). (22 Jan, Shift I, Memory Based)
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For some \(\mathrm{n}\neq 10,\) let the coefficients of the \({5}^{\mathrm{th}},{6}^{\mathrm{th}}\) 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:
[JEE Main 2025, 24 Jan (Shift 1)]
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If \(A\) and \(B\) are binomial coefficients of \(30^{\text {th }}\) and \(12^{\text {th }}\) term of binomial expansion \((1+x)^{2 n-1}\). If \(2 A\) \(=5 B\), then the value of \(n\) is (24 Jan, Shift II, Memory Based)
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Let \(\alpha, \beta, \gamma\) and \(\delta\) be the coefficients of \(x^7, x^5, x^3\) and \(x\) respectively in the expansion of \(\left(x+\sqrt{x^3-1}\right)^5+\left(x-\sqrt{x^3-1}\right)^5, x>1\). If u and v satisfy the equations
\(\begin{aligned}& \alpha u+\beta v=18 \\& \gamma u+\delta v=20\end{aligned}\)
then \(u+v\) equals :
[JEE Main 2025, 22 Jan (Shift 2)]
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The value of \(\frac{{}^{100}C_{50}}{51}+\frac{{}^{100}C_{51}}{52}+\ldots +\frac{{}^{100}C_{100}}{101}\) is:
[JEE Main 2026, 23 Jan (Shift 1)]
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If \(A\) and \(B\) are binomial coefficients of \(30^{\text {th }}\) and \(12^{\text {th }}\) term of binomial expansion \((1+x)^{2 n-1}\). If \(2 A\) \(=5 B\), then the value of \(n\) is (24 Jan, Shift II, Memory Based)
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Let the coefficients of three consecutive terms\(T_r, T_{r+1}\) and \(T_{r+2}\) 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 of all rational terms in the binomial expansion of \((\sqrt[4]{3}+\sqrt[3]{4})^{12}\). Then \(p+q\) is equal to :
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The number of integral terms in the expansion of \({\left({5}^{\frac{1}{2}}+{7}^{\frac{1}{8}}\right)}^{1016}\) is
[JEE Main 2025, 8 Apr (Shift 1)]
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If for \(3\leq r\leq 30\), \(\left({}^{30}C_{30-r}\right)+3\left({}^{30}C_{31-r}\right)+3\left({}^{30}C_{32-r}\right)+\left({}^{30}C_{33-r}\right)={}^{m}C_{r}\), then \(m\) equals:
[JEE Main 2026, 2 Apr (Shift 2)]
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If \(\sum _{r\text{ }=\text{ }1}^{9}\left(\frac{r+3}{{2}^{r}}\right)\text{ }.{\text{ }}^{9}{C}_{r}=\text{ }\alpha {\left(\frac{3}{2}\right)}^{9}−\text{ }\beta ,\alpha ,\beta \in \mathrm{N},\) then \((\alpha +\beta {)}^{2}\) is equal to
[JEE Main 2025, 3 Apr (Shift 1)]
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In the expansion of \({\left(9x-\frac{1}{3\sqrt{x}}\right)}^{18},x>0\), if the term independent of \(x\) is \((221)k\), then \(k\) is equal to:
[JEE Main 2026, 4 Apr (Shift 2)]
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The term independent of \(x\) in the expansion of \({\left(\frac{(\mathrm{x}+1)}{\left({\mathrm{x}}^{2/3}+1-{\mathrm{x}}^{1/3}\right)}-\frac{(\mathrm{x}-1)}{\left(\mathrm{x}-{\mathrm{x}}^{1/2}\right)}\right)}^{10},\mathrm{x}>1\) is:
[JEE Main 2025, 2 Apr (Shift 1)]
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The remainder, when \(7^{103}\) is divided by 23 , is equal to:
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In the expansion of \(\left(x+\sqrt{x^3-1}\right)^5+\left(x-\sqrt{x^3-1}\right)^5\), where \(\alpha, \beta, \gamma\) and \(\delta\) are the coefficient \(x^3, x^5\) \(x^7\) and \(x\) respectively. If \(\alpha u-\beta v=18\) and \(\gamma u+\delta v=20\), then \(u+v\) is equal to
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The number of integral terms in the expansion of \({\left({5}^{\frac{1}{2}}+{7}^{\frac{1}{8}}\right)}^{1016}\) is
[JEE Main 2025, 8 Apr (Shift 1)]
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The coefficient \(x^{48}\) in \(\left(1+x\right)+2{(1+x)}^{2}+3{(1+x)}^{3}+\ldots +100{(1+x)}^{100}\) is equal to
[JEE Main 2026, 22 Jan (Shift 1)]
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\(\text{ If }{3}^{107}\text{ is divided by }23\text{, then remainder is }\)
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The coefficient of \(x^2\) term in the binomial expansion of \(\left(\frac{1}{3} x^{\frac{1}{2}}+x^{-\frac{1}{4}}\right)^{10}\) is:
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The coefficient \(x^{48}\) in \(\left(1+x\right)+2{(1+x)}^{2}+3{(1+x)}^{3}+\ldots +100{(1+x)}^{100}\) is equal to
[JEE Main 2026, 22 Jan (Shift 1)]
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Suppose \(A\) and \(B\) are the coefficients of \(30^{\text {th }}\) and \(12^{\text {th }}\) terms respectively in the binomial expansion of \((1+x)^{2 n-1}\). If \(2 A=5 B\), then \(n\) is equal to:
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The least value of \(n\) for which the number of integral terms in the Binomial expansion of \((\sqrt[3]{7}+\sqrt[12]{11}{)}^{\mathrm{n}}\text{ is }183\text{, is : }\)
[JEE Main 2025, 29 Jan (Shift 1)]
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Let the coefficient of three consecutive terms \({T}_{r},{T}_{r+1},\&{T}_{r+2}\)in the binomial expansion of \({(a+b)}^{12}\) be in a A.P. and let p be the no. of all possible values of r, let q be the sum of all rational term in the binomial expansion \({(\sqrt[4]{3}+\sqrt[3]{4})}^{12}\). then p+q is equal to:
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The remainder, when \(7^{103}\) is divided by 23 , is equal to:
[JEE Main 2025, 29 Jan (Shift 2)]
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If in the expansion of \((1+x)^p(1-x)^q\), the coefficients of x and \(x^2\) are 1 and -2 , respectively, then \(\mathrm{p}^2+\mathrm{q}^2\) is equal to :
[JEE Main 2025, 23 Jan (Shift 2)]
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The sum of the coefficients of \({x}^{499}\) and \({x}^{500}\) in \({\left(1+x\right)}^{1000}+x{\left(1+x\right)}^{999}+{x}^{2}{\left(1+x\right)}^{998}+...+{x}^{1000}\) is:
[JEE Main 2026, 28 Jan (Shift 2)]
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Sum of all rational terms of \((1+(2^{1/3})+(3^{1/2}))^{6}\)
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If the \(5^{\text {th }}, 6^{\text {th }}\) and \(7^{\text {th }}\) term of the binomial expansion of \(\left(1+x^2\right)^{n+4}\) are in A.P. Then the greatest binomial coefficient in the expansion of \(\left(1+x^2\right)^{n+4}\) is(\(\mathrm{n}\neq 10\)) (24 Jan, Shift I, Memory Based)
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Let the smallest value of \(k\in N\) for which the coefficient of \({x}^{3}\) in \((1+x)^{3}+(1+x)^{4}+(1+x)^5+\ldots \ldots+(1+x)^{99}\)\(+(1+k x)^{100}, x \neq 0,\) is \(\left(43n+\frac{101}{4}\right)({}^{100}C_{3})\) for some \(n\in N\) be \(p\). Then the value of \(p+n\) is
[JEE Main 2026, 4 Apr (Shift 1)]
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Let \(S=\frac{1}{25!}+\frac{1}{3!23!}+\frac{1}{5!21!}+\ldots\) up to 13 terms. If \(13S=\frac{{2}^{k}}{n!},k\in ℕ,\) then \(n + k\) is equal to
[JEE Main 2026, 24 Jan (Shift 1)]
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Sum of all rational terms of \((1+(2^{1/3})+(3^{1/2}))^{6}\)
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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:
[JEE Main 2025, 4 Apr (Shift 1)]
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For some \(\mathrm{n}\neq 10,\) let the coefficients of the \({5}^{\mathrm{th}},{6}^{\mathrm{th}}\) 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:
[JEE Main 2025, 24 Jan (Shift 1)]
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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:
[JEE Main 2025, 4 Apr (Shift 1)]
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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 \(\left(1+x^2\right)^2(1+x)^n\), are in arithmetic progression is:
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If \({1}^{2}\cdot \left({}^{15}\mathrm{C}_{1}\right)+{2}^{2}\cdot \left({}^{15}\mathrm{C}_{2}\right)+{3}^{2}\cdot \left({}^{15}\mathrm{C}_{3}\right)+\ldots .+{15}^{2}\cdot \left({}^{15}\mathrm{C}_{15}\right)=\)\({2}^{\mathrm{m}}\cdot {3}^{\mathrm{n}}\cdot {5}^{\mathrm{k}}\), where \(\mathrm{m},\mathrm{n},\mathrm{k}\in \mathrm{N}\), then \(\mathrm{m}+\mathrm{n}+\mathrm{k}\) is equal to :-
[JEE Main 2025, 4 Apr (Shift 2)]
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The value of \({}^{49}{C}_{3}+{}^{48}{C}_{3}+{}^{47}{C}_{3}{+}^{46}{C}_{3}+{}^{45}{C}_{3}+{}^{45}{C}_{4}\) is
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Given below are 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:
[JEE Main 2026, 28 Jan (Shift 2)]
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In the expansion of \(\left(x+\sqrt{x^3-1}\right)^5+\left(x-\sqrt{x^3-1}\right)^5\), where \(\alpha, \beta, \gamma\) and \(\delta\) are the coefficient \(x^3, x^5\) \(x^7\) and \(x\) respectively. If \(\alpha u-\beta v=18\) and \(\gamma u+\delta v=20\), then \(u+v\) is equal to (22 Jan, Shift II, Memory Based)
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For some \(\mathrm{n}\neq 10,\) let the coefficients of the \({5}^{\mathrm{th}},{6}^{\mathrm{th}}\) 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:
[JEE Main 2025, 24 Jan (Shift 1)]
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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:EndFragment
[JEE Main 2024, 04 Apr (Shift 2)]
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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:EndFragment
[JEE Main 2024, 04 Apr (Shift 2)]
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If in the expansion of \(\ (1+x)^p(1-x)^q\) coefficient of \(x \\& x^2\) is \(1 \\&-2\) then find \(p^2+q^2\)
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\(\sum _{k=0}^{6}{}^{51-k}C_{3}\) is equal to
[JEE Main 2023, 25 Jan (Shift 2)]
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Let the number \((22{)}^{2022}+(2022{)}^{22}\) leave the remainder \(\alpha\) when divided by 3 and \(\beta\) when divided by 7 . Then \(\left({\alpha }^{2}+{\beta }^{2}\right)\) is equal to
[JEE Main 2023, 10 Apr (Shift 2)]
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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 these three terms, is ._____
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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
[JEE Main 2023, 11 Apr (Shift 2)]
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\(\sum _{k=0}^{20}{\left({}^{20}C_{k}\right)}^{2}\) is equal to.
[JEE Main 2021, 27 Aug (Shift 1)]
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If \(\frac{1}{n+1}{ }^n C_n+\frac{1}{n}{ }^n C_{n-1}+\ldots+\frac{1}{2}{ }^n C_1+{ }^n C_0=\frac{1023}{10}\) then \(n\) is equal to
[JEE Main 2023, 12 Apr (Shift 1)]
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A possible value of ' \(x\) ', for which the ninth term in the expansion of \(\left\{3^{\log _3 \sqrt{25^{x-1}+7}}+3^{\left(-\frac{1}{8}\right)^{\log _3\left(5^{x-1}+1\right)}}\right\}^{10}\) in the increasing powers of \(3^{\left(\frac{-1}{8}\right)^{\log _3\left(5^{x-1}+1\right)}}\) is equal to 180 , is:
[JEE Main 2021, 27 Jul (Shift 2)]
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\({50}^{th}\) root of a number \(x\) is 12 and \({50}^{th}\)root of another number \(y\) is 18 .Then the remainder obtained on dividing \((x+y)\) by 25 is____
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The sum, of the coefficients of the first 50 terms in the binomial expansion of \((1-x{)}^{100}\), is equal to
[JEE Main 2023, 12 Apr (Shift 1)]
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The sum of all those terms which are rational numbers in the expansion of \(\left(2^{\frac{1}{3}}+3^{\frac{1}{4}}\right)^{12}\) is:
[JEE Main 2021, 25 Jul (Shift 2)]
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The mean of the coefficients of \(x,{x}^{2},\ldots \ldots {x}^{7}\) in the binomial expansion of \((2+x{)}^{9}\)is_____
[JEE Main 2023, 11 Apr (Shift 1)]
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\({25}^{190}-{19}^{190}-{8}^{190}+{2}^{190}\) is divisible by
[JEE Main 2023, 8 Apr (Shift 2)]
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If the coefficients 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
[JEE Main 2023, 6 Apr (Shift 2)]
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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 \({\left(\alpha {x}^{1/9}+\beta {x}^{-1/6}\right)}^{10}\) is \(10k\), then \(k\) is equal to:
[JEE Main 2020, 2 Sep (Shift 1)]
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If \(\frac{1}{n+1}{}^{n}C_{n}+\frac{1}{n}{}^{n}C_{n-1}+\ldots +\frac{1}{2}{}^{n}C_{1}+{}^{n}C_{0}=\frac{1023}{10}\) then \(n\) is equal to:
[JEE Main 2023, 12 Apr (Shift 1)]
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The coefficient of \({x}^{256}\) in the expansion of \((1-x{)}^{101}{\left({x}^{2}+x+1\right)}^{100}\) is :
[JEE Main 2021, 20 Jul (Shift 1)]
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Let \(x=(8 \sqrt{3}+13)^{13}\) and \(y=(7 \sqrt{2}+9)^9\). If \([t]\) denotes the greatest integer \(\leq t\), then
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Coefficient of \(\ x^{13} \) in the expansion of \(\ (1-x)^5\left(1+x+x^2+x^3\right)^4 \) is
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The lowest integer which is greater than \(\left(1+\frac{1}{10^{100}}\right)^{10^{100}}\) is
[JEE Main 2021, 25 Jul (Shift 2)]
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If \(n \geq 2\) is a positive integer, then the sum of the series \({ }^{n+1} C_2+2\left({ }^2 C _2+{ }^3 C _2+{ }^4 C _2+\ldots+{ }^n C _2\right)\) is:
[JEE Main 2021, 24 Feb (Shift 2)]
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The remainder when \(3^{2022}\) is divided by 5 is:
[JEE Main 2022, 24 Jun (Shift 1)]
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The coefficient of \({x}^{256}\) in the expansion of \((1-x{)}^{101}\left({x}^{2}+\right.x+1{)}^{100}\) is:
[JEE Main 2021, 20 Jul (Shift 1)]
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The value \(\sum _{r=0}^{22}{}^{22}C_{r}{}^{23}C_{r}\) is:
[JEE Main 2023, 24 Jan (Shift 1)]
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If the greatest value of the term independent of '\( x \)' in the expansion of \( \left(x \sin \alpha+a \frac{\cos \alpha}{x}\right)^{10} \) is \( \frac{10 \text { ! }}{(5 !)^{2}} \), then the value of '\( a \)' is equal to
[JEE Main 2021, 25 Jul (Shift 2)]
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The number of integers, greater than 7000 that can be formed, using the digits \(3,5,6,7,8\) without repetition, is
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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 \(\frac{1}{a-b}+\frac{1}{a-2 b}+\frac{1}{a-3 b}+\ldots+\frac{1}{a-n b}=\alpha n+\beta n^2+\gamma n^3\), then the value of \(\gamma\) is.
[JEE Main 2021, 25 Jul (Shift 1)]
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The coefficient of \({x}^{5}\) in the expansion of \({\left(2{x}^{3}-\frac{1}{3{x}^{2}}\right)}^{5}\) is
[JEE Main 2023, 13 Apr (Shift 2)]
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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:
[JEE Main 2021, 16 Mar (Shift 1)]
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The value of \(\sum_{r=0}^6\left({ }^6 C_r \cdot{ }^6 C_{6-r}\right)\) is equal to:
[JEE Main 2021, 17 Mar (Shift 2)]
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The constant term in the expansion of \({\left(2x+\frac{1}{{x}^{7}}+3{x}^{2}\right)}^{5}\) is________
[JEE Main 2023, 25 Jan (Shift 1)]
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The absolute difference of the coefficients of \({x}^{10}\) and \({x}^{7}\) in the expansion of \({\left(2{x}^{2}+\frac{1}{2x}\right)}^{11}\)is equal to
[JEE Main 2023, 8 Apr (Shift 2)]
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If \(y=3 x+6 x^2+10 x^3+\ldots \ldots \ldots \infty\), then \(\frac{1}{3} y-\frac{1.4}{3^2 2!} y^2+\frac{1.4 .7}{3^2 3!} y^3-\ldots \ldots \infty\) is equal to
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If the coefficients of \({x}^{7}\) in \({\left({x}^{2}+\frac{1}{bx}\right)}^{11}\) and \({x}^{-7}\) in \({\left(x-\frac{1}{b{x}^{2}}\right)}^{11},b\neq 0\) are equal, then the value of \(b\) is equal to:
[JEE Main 2021, 27 Jul (Shift 1)]
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If \({\left({}^{30}C_{1}\right)}^{2}+2{\left({}^{30}C_{2}\right)}^{2}+3{\left({}^{30}C_{3}\right)}^{2}+\ldots ..+30{\left({}^{30}C_{30}\right)}^{2}\)\(=\frac{\alpha \cdot 60!}{(30!{)}^{2}}\),then \(\alpha\) is equal to
[JEE Main 2023, 24 Jan (Shift 2)]
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\(\sum_{k=0}^6{ }^{51-k} C_3\) is equal to
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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 \((l, n)\) is equal to:
[JEE Main 2023, 29 Jan (Shift 2)]
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If \(\frac{1}{n+1}{}^{n}C_{n}+\frac{1}{n}{}^{n}C_{n-1}+\ldots +\frac{1}{2}{}^{n}C_{1}+{}^{n}C_{0}=\frac{1023}{10}\) then n is equal to:
[JEE Main 2023, 12 Apr (Shift 1)]
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The ninth term in the expansion of \({\left\{ {{3^{{{\log }_3}\sqrt {{{25}^{x - 1}} + 7} }} + {3^{ - 1/8{{\log }_3}\left( {{5^{x - 1}} + 1} \right)}}} \right\}^{10}}\) is equal to \(180\), then \(x\) is: (x > 1)
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For the natural numbers \(m, n\), if \((1-y)^m(1+y)^n=1+a_1 y\) \(+a_2 y^2+\ldots a_{m+n} y^{m+n}\) and \(a_1=a_2=10\), then the value of \((m+n)\) is equal to:
[JEE Main 2021, 20 Jul (Shift 2)]
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Among the statements:
\(\left({S}_{1}\right):{2023}^{2022}-{1999}^{2022}\) is divisible by 8 .
\(\left({S}_{2}\right)\): \(13(13{)}^{n}-11n-13\) is divisible by 144 for infinitely many \(n\in N\) .
[JEE Main 2023, 6 Apr (Shift 2)]
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\left(1+x+2 x^2\right)^{20}=a_0+a_1 x+a_2 x^2+\ldots .+a_{40} x^{40} \text {. Then, } a_1+a_3+a_5+\ldots .+a_{37} \text { is equal to : }
[JEE Main 2021, 18 Mar (Shift 1)]
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Let \( [x] \) denote greatest integer less than or equal to \( x \). If for \( n \in \mathbb{N},\left(1-x+x^{3}\right)^{n}=\sum_{j=0}^{3 n} a_{j} x^{j} \), then \( \sum_{j=0}^{\left[\frac{3 n}{2}\right]} a_{2 j}+ \) \( 4 \sum_{j=0}^{\left[\frac{3 n-1}{2}\right]} a_{2 j+1} \) is equal to:
[JEE Main 2021, 16 Mar (Shift 1)]
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If \(y=3x+6{x}^{2}+10{x}^{3}+\ldots \ldots \ldots ∞\), then
\(\frac{1}{3}y−\frac{1\cdot 4}{{3}^{2}\cdot 2}{y}^{2}+\frac{1\cdot 4\cdot 7}{{3}^{2}\cdot 3}{y}^{3}−\ldots \ldots ∞\) is equal to
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The coefficient of \( x^{7} \) in the expression \( (1+x)^{10}+x(1+x)^{9}+x^{2}(1+x)^{8}+\ldots+x^{10} \) is
[JEE Main 2020, 7 Jan (Shift 2)]
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If the coefficients of \({x}^{7}\) in \({\left({x}^{2}+\frac{1}{bx}\right)}^{11}\) and \({x}^{-7}\) in \({\left(x-\frac{1}{b{x}^{2}}\right)}^{11},b\neq 0\) are equal, then the value of b is equal to:
[JEE Main 2021, 27 Jul (Shift 1)]
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Let \(x=(8\sqrt{3}+13{)}^{13}\) and \(y=(7\sqrt{2}+9{)}^{9}\) . If [t] denotes the greatest integer \(\leq\)t, then
[JEE Main 2023, 30 Jan (Shift 2)]
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If the coefficients of \( x^{3} \) and \( x^{4} \) in the expansion of \( \left(1+a x+b x^{2}\right)(1-2 x)^{18} \) in powers of \( x \) are both zero, then \( (a, b) \) is equal to
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The sum of the infinite series \(\frac{2^2}{2!}+\frac{2^4}{4!}+\frac{2^6}{6!}+\ldots\) is equal to
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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 \((l, n)\) is equal to:
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Let \([x]\) denote greatest integer less than or equal to \(x\). If for \(n \in N ,\left(1-x+x^3\right)^n=\sum_{j=0}^{3 n} a_j x^j\), then \(\sum_{j=0}^{\left[\frac{3 n}{2}\right]} a_{2 j}+4 \sum_{j=0}^{\left[\frac{3 n-1}{2}\right]} a_{2 j+1}\) is equal to:
[JEE Main 2021, 16 Mar (Shift 1)]
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If \({}^{20}\mathrm{C}_{\mathrm{r}}\) is the co-efficient of \({\mathrm{x}}^{\mathrm{r}}\) in the expansion of \((1+\mathrm{x}{)}^{20}\), then the value of \(\overset{20}{\underset{r=0}{\sum {r}^{2}}}{}^{20}C_{r}\) is equal to
[JEE Main 2021, 26 Aug (Shift 1)]
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If the ratio of the fifth term from the begining 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:
[JEE Main 2023, 6 Apr (Shift 1)]
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The absolute difference of the coefficients of \(x^{10}\) and \(x^7\) in the expansion of \(\left(2 x^2+\frac{1}{2 x}\right)^{11}\) is equal to
[JEE Main 2023, 08 Apr (Shift 2)]
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If the constant term, in binomial expansion of \({\left(2{x}^{r}+\frac{1}{{x}^{2}}\right)}^{10}\) is 180 , then \(r\) is equal to_____
[JEE Main 2021, 22 Jul (Shift 2)]
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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
[JEE Main 2023, 11 Apr (Shift 2)]
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If the fourth term in the expansion of \(\left(x+x^{\log _2 x}\right)^7\) is 4480 , then the value of \(x\) where \(x \in N\) is equal to:
[JEE Main 2021, 17 Mar (Shift 1)]
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The sum, of the coefficients of the first 50 terms in the binomial expansion of \((1-x)^{100}\), is equal to
[JEE Main 2023, 12 Apr (Shift 1)]
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If the coefficient of \({x}^{15}\) in the expansion of ,\({\left(a{x}^{3}+\frac{1}{b{x}^{\frac{1}{3}}}\right)}^{15}\) is equal to the coefficient of \({x}^{-15}\) in the expansion of \({\left(a{x}^{\frac{1}{3}}-\frac{1}{b{x}^{3}}\right)}^{15}\)where a and b are positive real numbers, then for each such ordered pair (a, b) :
[JEE Main 2023, 30 Jan (Shift 1)]
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The value of \(\frac{1}{1!50!}+\frac{1}{3!48!}+\frac{1}{5!46!}+\ldots +\frac{1}{49!2!}+\frac{1}{51!1!}\) is:
[JEE Main 2023, 1 Feb (Shift 1)]
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If the greatest value of the term independent of ‘x’ in the expansion of \({\left(x\sin \alpha +a\frac{\cos \alpha }{x}\right)}^{10}\text{ is }\) \(\frac{10!}{(5!{)}^{2}}\), then the value of ‘\(a\)’ is equal to :
[JEE Main 2021, 25 Jul (Shift 2)]
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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
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\(\sum_{k=0}^{20}\left({ }^{20} C_k\right)^2\) is equal to.
[JEE Main 2021, 27 Aug (Shift 1)]
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Let \( [x] \) denote greatest integer less than or equal to \( x \). If for \( n \in \mathbb{N},\left(1-x+x^{3}\right)^{n}=\sum_{j=0}^{3 n} a_{j} x^{j} \), then \( \sum_{j=0}^{\left[\frac{3 n}{2}\right]} a_{2 j}+ 4 \sum_{j=0}^{\left[\frac{3 n-1}{2}\right]} a_{2 j+1} \) is equal to:
[JEE Main 2021, 16 Mar (Shift 1)]
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The coefficient of \(x^7\) in \(\left(1-x+2 x^3\right)^{10}\) is \(\_\_\_\_\)
[JEE Main 2023, 10 Apr (Shift 1)]
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Fractional part of the number \(\frac{{4}^{2022}}{15}\) is equal to
[JEE Main 2023, 13 Apr (Shift 1)]
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The coefficient of \({x}^{301}\) in \((1+x{)}^{500}+x(1+x{)}^{499}+{x}^{2}(1+x{)}^{498}+\ldots +{x}^{500}\) is:
[JEE Main 2023, 30 Jan (Shift 1)]
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If the \(1011^{\text {th }}\) term from the end in the binomial expansion of \(\left(\frac{4 x}{5}-\frac{5}{2 x}\right)^{2022}\) is 1024 times \(1011^{\text {th }}\) term from the beginning, then \(|x|\) is equal to
[JEE Main 2023, 11 Apr (Shift 2)]
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The middle term in the expansion of \(\left(\frac{10}{x}+\frac{x}{10}\right)^{\mathbf{1 0}}\) is
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Among the statements:
\(\left({S}_{1}\right):{2023}^{2022}-{1999}^{2022}\) is divisible by 8.
\(\left({S}_{2}\right)\): \(13(13{)}^{n}-11n-13\) is divisible by 144 for infinitely many \(n\in N\).
[JEE Main 2023, 6 Apr (Shift 2)]
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The coefficient of \(x^5\) in the expansion of \(\left(2 x^3-\frac{1}{3 x^2}\right)^5\) is
[JEE Main 2023, 13 Apr (Shift 2)]
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Let \(\left(a+b x+c x^2\right)^{10}=\sum_{i=0}^{20} p_i x^i, a, b, c \in N\).If \(p_1=20\) and \(p_2=210\), then \(2(a+b+c)\) is equal to
[JEE Main 2023, 15 Apr (Shift 1)]
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Find the middle term of the expansion of \({\left(2x+\frac{1}{x}\right)}^{8}\)
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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
[JEE Main 2023, 11 Apr (Shift 2)]
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The coefficient of \({x}^{7}in{\left(1-x+2{x}^{3}\right)}^{10}\) is__________
[JEE Main 2023, 10 Apr (Shift 1)]
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If the coefficient of \( a^{7} b^{8} \) in the expansion of \( (a+2 b+4 a b)^{10} \) is \( K 2^{16} \), then \( K \) is equal to _________.
[JEE Main 2021, 31 Aug (Shift 2)]
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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, n) is equal to:
[JEE Main 2023, 29 Jan (Shift 2)]
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Let \({\left(a+bx+c{x}^{2}\right)}^{10}=\sum _{i=0}^{20}{p}_{i}{x}^{i},a,b,c\in N\). If \({p}_{1}=20and{p}_{2}=210\) , then \(2(a+b+c)\) is equal to
[JEE Main 2023, 15 Apr (Shift 1)]
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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
[JEE Main 2023, 10 Apr (Shift 2)]
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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
[JEE Main 2023, 11 Apr (Shift 2)]
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If the coefficient of \(x^{15}\) in the expansion of \(\left(a x^3+\frac{1}{b x^{\frac{1}{3}}}\right)^{15}\) is equal to the coefficient of \(x^{-15}\) in the expansion of \(\left(a x^{\frac{1}{3}}-\frac{1}{b x^3}\right)^{15}\), where \(a\) and \(b\) are positive real numbers, then for each such ordered pair \((a, b)\) :
[JEE Main 2023, 30 Jan (Shift 1)]
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Let \(\left(\begin{matrix}n \\ k\end{matrix}\right)\) denote \({}^{n}C_{k}\) and \(\left[\begin{matrix}n \\ k\end{matrix}\right]=\left\{\begin{matrix}\left(\begin{matrix}n \\ k\end{matrix}\right),\text{ if }0\leq k\leq n \\ 0,\text{ otherwise }\end{matrix}\right.\)
If \({A}_{k}=\sum _{i=0}^{9}\left(\begin{matrix}9 \\ i\end{matrix}\right)\left[\begin{matrix}12 \\ 12-k+i\end{matrix}\right]+\sum _{i=0}^{8}\left(\begin{matrix}8 \\ i\end{matrix}\right)\left[\begin{matrix}13 \\ 13-k+i\end{matrix}\right]\)
and \({A}_{4}-{A}_{3}=190p\), then \(p\) is equal to.........
[JEE Main 2021, 26 Aug (Shift 2)]
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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 \({\left(\alpha {x}^{1/9}+\beta {x}^{-1/6}\right)}^{10}\)is 10 k, then k is equal to:
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If \(\left({ }^{30} C _1\right)^2+2\left({ }^{30} C _2\right)^2+3\left({ }^{30} C _3\right)^2+\ldots . .+30\left({ }^{30} C _{30}\right)^2\) \(=\frac{\alpha \cdot 60 !}{(30 !)^2}\), then \(\alpha\) is equal to
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If the constant term, in binomial expansion of \({\left(2{x}^{r}+\frac{1}{{x}^{2}}\right)}^{10}\) is 180 , then r is equal to_____
[JEE Main 2021, 22 Jul (Shift 2)]
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