Let $a_1=1,a_2,a_3,a_4,\dots .$. be consecutive natural numbers. Then ${\tan }^{-1}\left(\frac{1}{1+a_1{a}_2}\right)+{\tan }^{-1}\left(\frac{1}{1+a_2{a}_3}\right)+\dots ..+{\tan }^{-1}\left(\frac{1}{1+a_{2021}{a}_{2022}}\right)$ is equal to :

{\tan ^{ - 1}}\left( {{{1 + \sqrt 3 } \over {3 + \sqrt 3 }}} \right) + {\sec ^{ - 1}}\left( {\sqrt {{{8 + 4\sqrt 3 } \over {6 + 3\sqrt 3 }}} } \right) is equal to :

If the domain of the function $f(x)={\log }_e\left(4x^2+11x+6\right)+{\sin }^{-1}(4x+3)+{\cos }^{-1}\left(\frac{10x+6}{3}\right)$ is $(\alpha ,\beta ]$, then $36|\alpha +\beta |$ is equal to :

The number of elements in the set $$S=\left\{\theta \in [0,2\pi ]:3{\cos }^4\theta -5{\cos }^2\theta -2{\sin }^6\theta +2=0\right\}$$ is :

Let $$S=\left\{x\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right):9^{1-{\tan }^{2}x}+9^{{\tan }^{2}x}=10\right\}$$ and \beta=\sum_\limits{x \in S} \tan ^{2}\left(\frac{x}{3}\right), then $$\frac{1}{6}(\beta -14{)}^2$$ is equal to :

The value of $$\frac{1}{1!50!}+\frac{1}{3!48!}+\frac{1}{5!46!}+\dots .+\frac{1}{49!2!}+\frac{1}{51!1!}$$ is :

The number of ways of selecting two numbers $a$ and $b,a\in \{2,4,6,\dots .,100\}$ and $b\in \{1,3,5,\dots ..,99\}$ such that 2 is the remainder when $a+b$ is divided by 23 is :

The letters of the word OUGHT are written in all possible ways and these words are arranged as in a dictionary, in a series. Then the serial number of the word TOUGH is :

The number of 3 digit numbers, that are divisible by either 3 or 4 but not divisible by 48, is :

The number of numbers, strictly between 5000 and 10000 can be formed using the digits 1, 3, 5, 7, 9 without repetition, is :

$$\sum _{k=0}^6{}^{51-k}{C}_3$$ is equal to :

The number of integers, greater than 7000 that can be formed, using the digits 3, 5, 6, 7, 8 without repetition is :

The number of square matrices of order 5 with entries from the set {0, 1}, such that the sum of all the elements in each row is 1 and the sum of all the elements in each column is also 1, is :

The total number of three-digit numbers, divisible by 3, which can be formed using the digits $1,3,5,8$, if repetition of digits is allowed, is :

All words, with or without meaning, are made using all the letters of the word MONDAY. These words are written as in a dictionary with serial numbers. The serial number of the word MONDAY is :

The number of five digit numbers, greater than 40000 and divisible by 5 , which can be formed using the digits $$0,1,3,5,7$$ and 9 without repetition, is equal to :

If the letters of the word MATHS are permuted and all possible words so formed are arranged as in a dictionary with serial numbers, then the serial number of the word THAMS is :

The number of triplets $$(x,\mathrm{y},\mathrm{z})$$, where $$x,\mathrm{y},\mathrm{z}$$ are distinct non negative integers satisfying $$x+y+z=15$$, is :

Eight persons are to be transported from city A to city B in three cars of different makes. If each car can accommodate at most three persons, then the number of ways, in which they can be transported, is :

If the number of words, with or without meaning, which can be made using all the letters of the word MATHEMATICS in which $$\mathrm{C}$$ and $$\mathrm{S}$$ do not come together, is $$(6!)\mathrm{k}$$, then $$\mathrm{k}$$ is equal to :