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

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

If $1^2\cdot \left({}^{15}{C}_1\right)+2^2\cdot \left({}^{15}{C}_2\right)+3^2\cdot \left({}^{15}{C}_3\right)+\dots +15^2\cdot \left({}^{15}{C}_{15}\right)=2^m\cdot 3^n\cdot 5^k$, where $m,n,k\in \mathbf{N}$, then $\mathrm{m}+\mathrm{n}+\mathrm{k}$ is equal to :

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 $\mathrm{P}\left(2n-1,n^2-4n\right)$ from the line $x+y=8$ is

In the expansion of ${\left(\sqrt[3]{2}+\frac{1}{\sqrt[3]{3}}\right)}^n,n\in \mathrm{N}$, 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 ${}^n{\mathrm{C}}_3$ is

If $\int {\mathrm{e}}^x\left(\frac{x{\sin }^{-1}x}{\sqrt{1-x^2}}+\frac{{\sin }^{-1}x}{{\left(1-x^2\right)}^{3/2}}+\frac{x}{1-x^2}\right)\mathrm{d}x=\mathrm{g}(x)+\mathrm{C}$, where C is the constant of integration, then $g\left(\frac{1}{2}\right)$ equals :

If $f(x)=\int \frac{1}{x^{1/4}\left(1+x^{1/4}\right)}\mathrm{d}x,f(0)=-6$, then $f(1)$ is equal to :

Let $\mathrm{I}(x)=\int \frac{dx}{(x-11{)}^{\frac{11}{13}}(x+15{)}^{\frac{15}{13}}}$. If $\mathrm{I}(37)-\mathrm{I}(24)=\frac{1}{4}\left(\frac{1}{{\mathrm{b}}^{\frac{1}{13}}}-\frac{1}{{\mathrm{c}}^{\frac{1}{13}}}\right),\mathrm{b},\mathrm{c}\in \mathcal{N}$, then $3(\mathrm{b}+\mathrm{c})$ is equal to

Let $\int x^3\sin x\mathrm{d}x=g(x)+C$, where $C$ is the constant of integration. If $8\left(g\left(\frac{\pi }{2}\right)+g^{\prime }\left(\frac{\pi }{2}\right)\right)=\alpha {\pi }^3+\beta {\pi }^2+\gamma ,\alpha ,\beta ,\gamma \in Z$, then $\alpha +\beta -\gamma$ equals :

If the domain of the function ${\log }_5(18x-x^2-77)$ is $(\alpha ,\beta )$ and the domain of the function ${\log }_{(x-1)}\left(\frac{2x^2+3x-2}{x^2-3x-4}\right)$ is $(\gamma ,\delta )$, then ${\alpha }^2+{\beta }^2+{\gamma }^2$ is equal to:

Let $\mathrm{A}=\{1,2,3,4\}$ and $\mathrm{B}=\{1,4,9,16\}$. Then the number of many-one functions $f:\mathrm{A}\to \mathrm{B}$ such that $1\in f(\mathrm{A})$ is equal to :

Let $f:[0,3]\to$ A be defined by $f(x)=2x^3-15x^2+36x+7$ and $g:[0,\infty )\to B$ be defined by $g(x)=\frac{x^{2025}}{x^{2025}+1}$, If both the functions are onto and $S=\{x\in Z;x\in A$ or $x\in B\}$, then $n(S)$ is equal to :

Let $f(x)={\log }_{\mathrm{e}}x$ and $g(x)=\frac{x^4-2x^3+3x^2-2x+2}{2x^2-2x+1}$. Then the domain of $f\circ g$ is

Let $f(x)=\frac{2^{x+2}+16}{2^{2x+1}+2^{x+4}+32}$. Then the value of $8\left(f\left(\frac{1}{15}\right)+f\left(\frac{2}{15}\right)+\dots +f\left(\frac{59}{15}\right)\right)$ is equal to

The function $f:(-\infty ,\infty )\to (-\infty ,1)$, defined by $f(x)=\frac{2^x-2^{-x}}{2^x+2^{-x}}$ is :

If $f(x)=\frac{2^x}{2^x+\sqrt{2}},\mathrm{x}\in \mathbb{R}$, then \sum_\limits{\mathrm{k}=1}^{81} f\left(\frac{\mathrm{k}}{82}\right) is equal to

Let $f:\mathbb{R}\to \mathbb{R}$ be a function defined by $f(x)=(2+3a)x^2+\left(\frac{a+2}{a-1}\right)x+b,a\ne 1$. If $f(x+y)=f(x)+f(\mathrm{y})+1-\frac{2}{7}x\mathrm{y}$, then the value of $28\sum _{i=1}^5|f(i)|$ is

If the range of the function $f(x)=\frac{5-x}{x^2-3x+2},x\ne 1,2,$ is $(-\infty ,\alpha ]\cup [\beta ,\infty )$, then ${\alpha }^2+{\beta }^2$ is equal to :

If the domain of the function $f(x)=\frac{1}{\sqrt{10+3x-x^2}}+\frac{1}{\sqrt{x+|x|}}$ is $(a, b)$, then $(1+a{)}^2+b^2$ is equal to :