If the domain of the function $${\sin }^{-1}\left(\frac{3x-22}{2x-19}\right)+{\log }_{\mathrm{e}}\left(\frac{3x^2-8x+5}{x^2-3x-10}\right)$$ is $$(\alpha ,\beta ]$$, then $$3\alpha +10\beta$$ is equal to:

Given that the inverse trigonometric function assumes principal values only. Let $$x,y$$ be any two real numbers in $$[-1,1]$$ such that $${\cos }^{-1}x-{\sin }^{-1}y=\alpha ,\frac{-\pi }{2}\le \alpha \le \pi$$. Then, the minimum value of $$x^2+y^2+2xy\sin \alpha$$ is

The number of solutions of the equation $4{\sin }^{2}x-4{\cos }^{3}x+9-4\cos x=0;x\in [-2\pi ,2\pi ]$ is :

If $$2{\tan }^2\theta -5\sec \theta =1$$ has exactly 7 solutions in the interval $$\left[0,\frac{n\pi }{2}\right]$$, for the least value of $$n\in \mathbf{N}$$, then \sum_\limits{k=1}^n \frac{k}{2^k} is equal to:

If $$$\alpha$,-\frac{$\pi$}{2}

The sum of the solutions $$x\in \mathbb{R}$$ of the equation $$\frac{3\cos 2x+{\cos }^{3}2x}{{\cos }^{6}x-{\sin }^{6}x}=x^3-x^2+6$$ is

If $$2{\sin }^{3}x+\sin 2x\cos x+4\sin x-4=0$$ has exactly 3 solutions in the interval $$\left[0,\frac{\mathrm{n}\pi }{2}\right],\mathrm{n}\in \mathrm{N}$$, then the roots of the equation $$x^2+\mathrm{n}x+(\mathrm{n}-3)=0$$ belong to :

Let $$|\cos \theta \cos (60-\theta )\cos (60+\theta )|\le \frac{1}{8},\theta ϵ[0,2\pi ]$$. Then, the sum of all $$\theta \in [0,2\pi ]$$, where $$\cos 3\theta$$ attains its maximum value, is :

If $\mathrm{n}$ is the number of ways five different employees can sit into four indistinguishable offices where any office may have any number of persons including zero, then $\mathrm{n}$ is equal to :

Let $$\alpha =\frac{(4!)!}{(4!{)}^{3!}}$$ and $$\beta =\frac{(5!)!}{(5!{)}^{4!}}$$. Then :

The number of ways in which 21 identical apples can be distributed among three children such that each child gets at least 2 apples, is

If for some $$m,n;{}^6{C}_m+2\left({}^6{C}_{m+1}\right)+{}^6{C}_{m+2}>{}^8{C}_3$$ and $${}^{n-1}{P}_3:{}^n{P}_4=1:8$$, then $${}^n{P}_{m+1}+{}^{\mathrm{n}+1}{C}_m$$ is equal to

Number of ways of arranging 8 identical books into 4 identical shelves where any number of shelves may remain empty is equal to

There are 5 points $$P_1,P_2,P_3,P_4,P_5$$ on the side $$AB$$, excluding $$A$$ and $$B$$, of a triangle $$ABC$$. Similarly there are 6 points $${\mathrm{P}}_6,{\mathrm{P}}_7,\dots ,{\mathrm{P}}_{11}$$ on the side $$\mathrm{BC}$$ and 7 points $${\mathrm{P}}_{12},{\mathrm{P}}_{13},\dots ,{\mathrm{P}}_{18}$$ on the side $$\mathrm{CA}$$ of the triangle. The number of triangles, that can be formed using the points $${\mathrm{P}}_1,{\mathrm{P}}_2,\dots ,{\mathrm{P}}_{18}$$ as vertices, is:

The number of ways five alphabets can be chosen from the alphabets of the word MATHEMATICS, where the chosen alphabets are not necessarily distinct, is equal to:

Let $$[t]$$ be the greatest integer less than or equal to $$t$$. Let $$A$$ be the set of all prime factors of 2310 and $$f:A\to \mathbb{Z}$$ be the function $$f(x)=\left[{\log }_2\left(x^2+\left[\frac{x^3}{5}\right]\right)\right]$$. The number of one-to-one functions from $$A$$ to the range of $$f$$ is

Let the set $$S=\{2,4,8,16,\dots ,512\}$$ be partitioned into 3 sets $$A,B,C$$ with equal number of elements such that $$\mathrm{A}\cup \mathrm{B}\cup \mathrm{C}=\mathrm{S}$$ and $$\mathrm{A}\cap \mathrm{B}=\mathrm{B}\cap \mathrm{C}=\mathrm{A}\cap \mathrm{C}=\varphi$$. The maximum number of such possible partitions of $$S$$ is equal to:

60 words can be made using all the letters of the word $$\mathrm{BHBJO}$$, with or without meaning. If these words are written as in a dictionary, then the $$50^{\text{th }}$$ word is:

If all the words with or without meaning made using all the letters of the word "NAGPUR" are arranged as in a dictionary, then the word at $$315^{\text{th }}$$ position in this arrangement is :

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