JEE Main Binomial Theorem Practice Questions With Solutions

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

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

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

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

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

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

Let $a$a be the sum of all coefficients in the expansion of ${(1-2x+2x^2)}^{2023}{(3-4x^2+2x^3)}^{2024}$\left(\left(\right. 1 - 2 x + 2 x^{2} \left.\right)\right)^{2023} \left(\left(\right. 3 - 4 x^{2} + 2 x^{3} \left.\right)\right)^{2024} and $b=\underset{x\to 0}{lim}\left(\frac{{\int }_0^x\frac{\log (1+t)}{t^{2024}+1}dt}{x^2}\right)$b = \underset{x \rightarrow 0}{lim} \left(\right. \frac{\int_{0}^{x} \frac{log ⁡ \left(\right. 1 + t \left.\right)}{t^{2024} + 1} d t}{x^{2}} \left.\right). If the equation $c{x}^2+dx+e=0$c x^{2} + d x + e = 0 and $2b{x}^2+ax+4=0$2 b x^{2} + a x + 4 = 0 have a common root, where $c,d,e\in \mathbb{R}$c , d , e \in \mathbb{R}, then $\mathrm{d}:\mathrm{c}:$d : c : e equals