Let $a_{1}=1, a_{2}, a_{3}, a_{4}, \ldots .$. be consecutive natural numbers.

Then $\tan ^{-1}\left(\frac{1}{1+a_{1} a_{2}}\right)+\tan ^{-1}\left(\frac{1}{1+a_{2} a_{3}}\right)+\ldots . .+\tan ^{-1}\left(\frac{1}{1+a_{2021} a_{2022}}\right)$ is equal to :

For $\alpha, \beta \in \mathbb{R}$, suppose the system of linear equations

$$ \begin{aligned} & x-y+z=5 \\ & 2 x+2 y+\alpha z=8 \\ & 3 x-y+4 z=\beta \end{aligned} $$

has infinitely many solutions. Then $\alpha$ and $\beta$ are the roots of :

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

Let $S$ be the set of all values of $a_1$ for which the mean deviation about the mean of 100 consecutive positive integers $a_1, a_2, a_3, \ldots ., a_{100}$ is 25 . Then $S$ is :

Let $a, b, c>1, a^3, b^3$ and $c^3$ be in A.P., and $\log _a b, \log _c a$ and $\log _b c$ be in G.P. If the sum of first 20 terms of an A.P., whose first term is $\frac{a+4 b+c}{3}$ and the common difference is $\frac{a-8 b+c}{10}$ is $-444$, then $a b c$ is equal to :

Let $f, g$ and $h$ be the real valued functions defined on $\mathbb{R}$ as

$f(x)=\left\{\begin{array}{cc}\frac{x}{|x|}, & x \neq 0 \\ 1, & x=0\end{array}\right.$

$g(x)=\left\{\begin{array}{cc}\frac{\sin (x+1)}{(x+1)}, & x \neq-1 \\ 1, & x=-1\end{array}\right.$

and $h(x)=2[x]-f(x)$, where $[x]$ is the greatest integer $\leq x$. Then the

value of $\lim\limits_{x \rightarrow 1} g(h(x-1))$ is :

Let $q$ be the maximum integral value of $p$ in $[0,10]$ for which the roots of the equation $x^2-p x+\frac{5}{4} p=0$ are rational. Then the area of the region $\left\{(x, y): 0 \leq y \leq(x-q)^2, 0 \leq x \leq q\right\}$ is :

If the functions $f(x)=\frac{x^3}{3}+2 b x+\frac{a x^2}{2}$

and $g(x)=\frac{x^3}{3}+a x+b x^2, a \neq 2 b$

have a common extreme point, then $a+2 b+7$ is equal to :

The parabolas : $a x^2+2 b x+c y=0$ and $d x^2+2 e x+f y=0$ intersect on the line $y=1$. If $a, b, c, d, e, f$ are positive real numbers and $a, b, c$ are in G.P., then :

Let $\vec{a}$ and $\vec{b}$ be two vectors, Let $|\vec{a}|=1,|\vec{b}|=4$ and $\vec{a} \cdot \vec{b}=2$. If $\vec{c}=(2 \vec{a} \times \vec{b})-3 \vec{b}$, then the value of $\vec{b} \cdot \vec{c}$ is :

The range of the function $f(x)=\sqrt{3-x}+\sqrt{2+x}$ is :

The solution of the differential equation

$\frac{d y}{d x}=-\left(\frac{x^2+3 y^2}{3 x^2+y^2}\right), y(1)=0$ is :

Let $x=(8 \sqrt{3}+13)^{13}$ and $y=(7 \sqrt{2}+9)^9$. If $[t]$ denotes the greatest integer $\leq t$, then :

Let $A=\{1,2,3,5,8,9\}$. Then the number of possible functions $f: A \rightarrow A$ such that $f(m \cdot n)=f(m) \cdot f(n)$ for every $m, n \in A$ with $m \cdot n \in A$ is equal to ___________.

The number of seven digits odd numbers, that can be formed using all the

seven digits 1, 2, 2, 2, 3, 3, 5 is ____________.

Let $A$ be the area of the region

$\left\{(x, y): y \geq x^2, y \geq(1-x)^2, y \leq 2 x(1-x)\right\}$.

Then $540 \mathrm{~A}$ is equal to :

If the value of real number $a>0$ for which $x^2-5 a x+1=0$ and $x^2-a x-5=0$

have a common real root is $\frac{3}{\sqrt{2 \beta}}$ then $\beta$ is equal to ___________.

Let $P\left(a_1, b_1\right)$ and $Q\left(a_2, b_2\right)$ be two distinct points on a circle with center $C(\sqrt{2}, \sqrt{3})$. Let $\mathrm{O}$ be the origin and $\mathrm{OC}$ be perpendicular to both $\mathrm{CP}$ and $\mathrm{CQ}$. If the area of the triangle $\mathrm{OCP}$ is $\frac{\sqrt{35}}{2}$, then $a_1^2+a_2^2+b_1^2+b_2^2$ is equal to :

Let a line $L$ pass through the point $P(2,3,1)$ and be parallel to the line $x+3 y-2 z-2=0=x-y+2 z$. If the distance of $L$ from the point $(5,3,8)$ is $\alpha$, then $3 \alpha^2$ is equal to :

A bag contains six balls of different colours. Two balls are drawn in succession with replacement. The probability that both the balls are of the same colour is p. Next four balls are drawn in succession with replacement and the probability that exactly three balls are of the same colour is $q$. If $p: q=m: n$, where $m$ and $n$ are coprime, then $m+n$ is equal to :

If $\int \sqrt{\sec 2 x-1} d x=\alpha \log _e\left|\cos 2 x+\beta+\sqrt{\cos 2 x\left(1+\cos \frac{1}{\beta} x\right)}\right|+$ constant, then $\beta-\alpha$ is equal to ____________.

$50^{\text {th }}$ root of a number $x$ is 12 and $50^{\text {th }}$ root of another number $y$ is 18 . Then the remainder obtained on dividing $(x+y)$ by 25 is ____________.