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

Consider a rectangle \(ABCD\) having \(5,7,6,9\) points in the interior of the line segments \(AB,CD , BC , DA\) respectively. Let \(\alpha\) be the number of triangles having these points from different sides as vertices and \(\beta\) be the number of quadrilaterals having these points from different sides as vertices. Then \((\beta-\alpha)\) is equal to :

If \(m\) is a non-zero number and \(\int \frac{x^{5 m-1}+2 x^{4 m-1}}{\left(x^{2 m}+x^{m}+1\right)^{3}} d x=f(x)+c\) , then \(f(x)\) is

The mean and the standard deviation \((s.d.)\)  of five observations are \(9\) and \(0,\) respectively. If one of the observations is changed such that the mean of the new set of five observations becomes \(10,\)  then their \(s.d.\)  is?

Number of \(4-\)digit numbers (the repetition of digits is allowed) which are made using the digits \(1, 2, 3\) and \(5\) , and are divisible by \(15\) , is equal to \(............\).

Let \( |A|=6 \) where A is a \( 3 \times 3 \) matrix. If \( |adj(3adj(A^{2} \cdot adj(2A)))|=2^{m} \cdot 3^{n} \), \( m, n \in N \), then \( m+n \) is equal to:

For which of the following ordered pairs \((\mu, \delta)\) the system of linear equations  \(x+2 y+3 z=1\) ; \(3 x+4 y+5 z=\mu\) ; \(4 x+4 y+4 z=\delta\) is inconsistent?

Let \(\vec{c}\) be the projection vector of \(\vec{b}=\lambda \hat{i}+4 \hat{k}, \lambda\gt0\), on the vector \(\vec{a}=\hat{i}+2 \hat{j}+2 \hat{k}\). If \(|\vec{a}+\vec{c}|=7\), then the area of the parallelogram formed by the vectors \(\vec{b}\) and \(\vec{c}\) is ________

Let \(f : R \rightarrow R\) be a function defined by \(f ( x )=\) \(\log _{\sqrt{m}}\{\sqrt{2}(\sin x-\cos x)+m-2\}\), for some \(m\), such that the range of \(f\) is \([0,2]\). Then the value of \(m\) is \(............\)

Let \(k\) be a non-zero real number  If \(f(x) = {\rm{ }}\left\{ {\begin{array}{*{20}{c}}
{\frac{{\left( {{e^x} - 1} \right)^2}}{{\sin {\mkern 1mu} \left( {\frac{x}{k}} \right){\mkern 1mu} \log {\mkern 1mu} \left( {1 + \frac{x}{4}} \right)}}{\mkern 1mu} ,{\mkern 1mu} x \ne 0}\\
{{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} 12{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} ,x{\mkern 1mu} {\mkern 1mu}  = 0{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} }
\end{array}} \right.\)  is a continuous function then the value of \(k\) is

Let \(\vec{a}=4 \hat{i}+3 \hat{j}\) and \(\vec{b}=3 \hat{i}-4 \hat{j}+5 \hat{k}\) and \(\vec{c}\) is a vector such that \(\overrightarrow{ c } \cdot(\overrightarrow{ a } \times \overrightarrow{ b })+25=0, \overrightarrow{ c } \cdot(\hat{ i }+\hat{ j }+\hat{ k })=4\) and projection of \(\overrightarrow{ c }\) on \(\overrightarrow{ a }\) is \(1,\) then the projection of \(\overrightarrow{ c }\) on \(\overrightarrow{ b }\) equals:

A group of \(40\) students appeared in an examination of \(3\) subjects - Mathematics, Physics  Chemistry. It was found that all students passed in at least one of the subjects, \(20\) students passed in Mathematics, \(25\) students passed in Physics, \(16\) students passed in Chemistry, at most \(11\) students passed in both Mathematics and Physics, at most \(15\) students passed in both Physics and Chemistry, at most \(15\) students passed in both Mathematics and Chemistry. The maximum number of students passed in all the three subjects is ........... .

Let \(\mathrm{S}=\left\{\mathrm{m} \in \mathbf{Z}: \mathrm{A}^{\mathrm{m}^2}+\mathrm{A}^{\mathrm{m}}=3 \mathrm{I}-\mathrm{A}^{-6}\right\}\), where \(\mathrm{A}=\left[\begin{array}{cc}2 & -1 \\ 1 & 0\end{array}\right]\). Then \(\mathrm{n}(\mathrm{S})\) is equal to ______.