The value of  $I={\displaystyle \sum }_{k=1}^{98}\underset{k}{\overset{k+1}{{\displaystyle \int }}}\frac{k+1}{x(x+1)}dx,$ then

Let n be the number of ways in which 5 boys and 5 girls can stand in a queue in such a way that all the girls stand consecutively in the queue. Let m be the number of ways in which 5 boys and 5 girls can stand in queue in such a way that exactly four girls stand consecutively in the queue. Then the value of $\frac{\mathrm{m}}{\mathrm{n}}$ is

Let $\mathrm{M}=\left[\begin{array}{cc}{\sin }^4\mathrm{\theta }& -1-{\sin }^2\mathrm{\theta }\\ 1+{\cos }^2\mathrm{\theta }& {\cos }^4\mathrm{\theta }\end{array}\right]=\mathrm{\alpha }\mathrm{I}+\mathrm{\beta }{\mathrm{M}}^{-1},$ where $\mathrm{\alpha }=\mathrm{\alpha }\left(\mathrm{\theta }\right)$ and $\mathrm{\beta }=\mathrm{\beta }\left(\mathrm{\theta }\right)$ are real number, and $\mathrm{I}$ is the $2\times 2$ identity matrix. If ${\mathrm{\alpha }}^{\mathrm{*}}$ is the minimum of the set { $\mathrm{\alpha }\left(\mathrm{\theta }\right):\mathrm{\theta }\in \left[0,\mathrm{ }2\mathrm{\pi }\right)\}$ and ${\mathrm{\beta }}^{\mathrm{*}}$ is the minimum of the set $\left\{\mathrm{\beta }\left(\mathrm{\theta }\right):\mathrm{\theta }\in \left[0,\mathrm{ }2\mathrm{\pi }\right)\right\},$ then the value of ${\mathrm{\alpha }}^{\mathrm{*}}+{\mathrm{\beta }}^{\mathrm{*}}$ is

Let \(f(x)=x^4+a x^3+b x^2+c\) be a polynomial with real coefficients such that \(f(1)=-9\). Suppose that \(i \sqrt{3}\) is a root of the equation \(4 x^3+3 a x^2+2 b x=0\), where \(i=\sqrt{-1}\). If \(\alpha_1, \alpha_2, \alpha_3\), and \(\alpha_4\) are all the roots of the equation \(f(x)=0\), then \(\left|\alpha_1\right|^2+\left|\alpha_2\right|^2+\left|\alpha_3\right|^2+\left|\alpha_4\right|^2\) is equal to ________.

Let $\alpha$ be a positive real number. Let $f:\mathrm{ℝ}\to \mathrm{ℝ}$ and $g:\left(\alpha ,\infty \right)\to \mathrm{ℝ}$ be the functions defined by $f\left(x\right)=\sin \left(\frac{\pi x}{12}\right)$ and $g\left(x\right)=\frac{2{\log }_e\left(\sqrt{x}-\sqrt{\alpha }\right)}{{\log }_e\left(e^{\sqrt{x}}-e^{\sqrt{\alpha }}\right)}$.
Then the value of $\underset{x\to {\alpha }^{+}}{\lim }f\left(g\left(x\right)\right)$ is ______.

Let \(f:[-1,2] \rightarrow[0, \infty)\) be a continuous function such that \(f(x)=f(1-x)\) for all \(x \in[-1,2]\). Let \(R_1=\int_{-1}^2 x f(x) d x\) and \(R_2\) be the area of the region bounded by \(y=f(x), x=-1, x=2\) and the \(X\)-axis. Then,

The value of ${\left({\left({\log }_{2}9\right)}^2\right)}^{\frac{1}{{\log }_2\left({\log }_{2}9\right)}}\times {\left(\sqrt{7}\right)}^{\frac{1}{{\log }_{4}7}}$ is_____.

Consider the following reaction,
\(2 \mathrm{H}_2(\mathrm{~g})+2 \mathrm{NO}(\mathrm{g}) \rightarrow \mathrm{N}_2(\mathrm{~g})+2 \mathrm{H}_2 \mathrm{O}(\mathrm{g})\)
which follows the mechanism given below:
\(2 \mathrm{NO}(\mathrm{g}) \stackrel{k_1}{\underset{k_{-1}}{\rightleftharpoons}} \mathrm{N}_2 \mathrm{O}_2(\mathrm{~g})\) \(\quad\) (fast equlibrium)
\(\mathrm{N}_2 \mathrm{O}_2(\mathrm{~g})+\mathrm{H}_2(\mathrm{~g}) \xrightarrow{k_2} \mathrm{~N}_2 \mathrm{O}(\mathrm{g})+\mathrm{H}_2 \mathrm{O}(\mathrm{g})\) \(\quad\)(slow reaction)
\(\mathrm{N}_2 \mathrm{O}(\mathrm{g})+\mathrm{H}_2(\mathrm{~g}) \xrightarrow{k_3} \mathrm{~N}_2(\mathrm{~g})+\mathrm{H}_2 \mathrm{O}(\mathrm{g})\) \(\quad\) (fast reaction)
The order of the reaction is_____

In a particular system of units, a physical quantity can be expressed in terms of the electric charge $e$, electron mass $m_e$. Planck's constant $h$, and Coulomb's constant $k=\frac{1}{4\pi {\epsilon }_0}$, where ${\epsilon }_0$ is the permittivity of vacuum. In terms of these physical constants, the dimension of the magnetic field is $\left[B\right]={\left[e\right]}^{\alpha }{\left[m_e\right]}^{\beta }{\left[h\right]}^{\gamma }{\left[k\right]}^{\delta }$. The value of $\alpha +\beta +\gamma +\delta$ is _______.

Three vectors $\overrightarrow{P},\overrightarrow{Q}$ and $\overrightarrow{R}$ are shown in the figure. Let, $S$ be any point on the vector $\overrightarrow{R}$ . The distance between the points $P$ and $S$ is $b\left|\overrightarrow{R}\right|$. The general relation among vectors $\overrightarrow{P},\overrightarrow{Q}$ and $\overrightarrow{S}$ are

The common tangents to the circle ${\mathrm{x}}^2+{\mathrm{y}}^2=2$ and the parabola ${\mathrm{y}}^2=8\mathrm{x}$ touch the circle at the points P, and Q the parabola at the points R, S. Then the area of the quadrilateral PQRS is

For any two points \(M\) and \(N\) in the \(X Y\)-plane, let \(\overrightarrow{M N}\) denote the vector from \(M\) to \(N\), and \(\overrightarrow{0}\) denote the zero vector. Let \(P, Q\) and \(R\) be three distinct points in the \(X Y\)-plane. Let \(S\) be a point inside the triangle \(\triangle P Q R\) such that \(\overrightarrow{S P}+5 \overrightarrow{S Q}+6 \overrightarrow{S R}=\overrightarrow{0}\). Let \(E\) and \(F\) be the mid-points of the sides \(P R\) and \(Q R\), respectively. Then the value of \(\frac{\text { length of the line segment } E F}{\text { length of the line segment } E S}\) is ________

For the following reaction $2\mathrm{X}+\mathrm{Y}\stackrel{\mathrm{k}}{\to }\mathrm{P}$ the rate of reaction is $\frac{\mathrm{d}\left[\mathrm{P}\right]}{\mathrm{dt}}=\mathrm{k}\left[\mathrm{X}\right]$. Two moles of $\mathrm{X}$ are mixed with one mole of $\mathrm{Y}$ to make $1.0 \mathrm{L}$ of solution. At $50 \mathrm{s}, 0.5$ mole of $\mathrm{Y}$ is left in the reaction mixture. The correct statement(s) about the reaction is(are) $($Use: $\ln 2=0.693)$