A vessel contains \(16 \,g\) of hydrogen and \(128 \,g\) of oxygen at standard temperature and pressure. The volume of the vessel in \(cm ^{3}\) is

\(I _{ CM }\) is moment of inertia of a circular disc about an axis \(( CM )\) passing through its center and perpendicular to the plane of disc. \(I _{ AB }\) is it's moment of inertia about an axis \(A B\) perpendicular to plane and parallel to axis \(CM\) at a distance \(\frac{2}{3} R\) from center. Where \(R\) is the radius of the disc. The ratio of \(I _{ AB }\) and \(I _{ CM }\) is \(x: 9\). The value of \(x\) is \(........\)

In an octagon \(ABCDEFGH\) of equal side, what is the sum of \(\overrightarrow{ AB }+\overrightarrow{ AC }+\overrightarrow{ AD }+\overrightarrow{ AE }+\overrightarrow{ AF }+\overrightarrow{ AG }+\overrightarrow{ AH }\) if, \(\overrightarrow{ AO }=2 \hat{ i }+3 \hat{ j }-4 \hat{ k }\)

The current \(I\) in the given circuit will be \(......A\)

A resonance circuit having inductance and resistance \(2 \times 10^{-4} H\) and \(6.28 \Omega\) respectively oscillates at \(10\, MHz\) frequency. The value of quality factor of this resonator is ......... \([\pi=3.14]\)

In a series \(LCR\) resonant circuit, the quality factor is measured as \(100\) If the inductance is increased by two fold and resistance is decreased by two fold, then the quality factor after this change will be .........

The mass density of a planet of radius \(R\) varies with the distance \(r\) from its centre as \(\rho( r )=\rho_{0}\left(1-\frac{ r ^{2}}{ R ^{2}}\right) .\) Then the gravitational field is maximum at

The equation of the electric field of an electromagnetic wave propagating through free space is given by :
E \(=\sqrt{377} \sin \left(6.27 \times 10^3 t -2.09 \times 10^{-5} x \right)\) N/C
he average power of the electromagnetic wave is\(\left(\frac{1}{\alpha}\right) W / m ^2\). The value of \(\alpha\) is ___________
\(\left(\right.\) Take \(\sqrt{\frac{\mu_0}{\varepsilon_0}}=377\) in SI units \()\)

Let \(S=\{1,2,3,4,5,6,7,8,9,10\}\). Define \(f: S \rightarrow S\) as \(f(n)=\left\{\begin{array}{cc}2 n, & \text { if } n=1,2,3,4,5 \\ 2 n-11 & \text { if } n=6,7,8,9,10\end{array}\right.\). Let \(g : S \rightarrow S\) be a function such that \(f o g(n)=\left\{\begin{array}{ll}n+1 & \text {, if } n \text { is odd } \\ n-1 & \text {, if } n \text { is even }\end{array}\right.\), then \(g (10)(( g (1)+ g (2)+ g (3)+ g (4)+ g (5))\) is equal to

The shortest distance between the line \(\frac{x-3}{4}=\frac{y+7}{-11}=\frac{z-1}{5} \text { and } \frac{x-5}{3}=\frac{y-9}{-6}=\frac{z+2}{1}\) is :

Let \(( a , b ) \subset(0,2 \pi)\) be the largest interval for which \(\sin ^{-1}(\sin \theta)-\cos ^{-1}(\sin \theta) > 0, \theta \in(0,2 \pi)\) holds. If \(\alpha x^2+\beta x+\sin ^{-1}\left(x^2-6 x+10\right)+\cos ^{-1}\) \(\left(x^2-6 x+10\right)=0\) and \(\alpha-\beta=b-a\), then \(\alpha\) is equal to:

Let \(\left(2 x ^{2}+3 x +4\right)^{10}=\sum \limits_{ r =0}^{20} a _{ r } x ^{ r } \cdot\) Then \(\frac{ a _{7}}{ a _{13}}\) is equal to

Let \(L_1: \overrightarrow{\mathrm{r}}=(\hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}})+\lambda(\hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}}), \lambda \in \mathrm{R}\) \(L_2: \overrightarrow{\mathrm{r}}=(\hat{\mathrm{j}}-\hat{\mathrm{k}})+\mu(3 \hat{\mathrm{i}}+\hat{\mathrm{j}}+\mathrm{p} \hat{\mathrm{k}}), \mu \in \mathrm{R}\) and  \(L_3: \overrightarrow{\mathrm{r}}=\delta(\ell \hat{\mathrm{i}}+\mathrm{m} \hat{\mathrm{j}}+\mathrm{n} \hat{\mathrm{k}}) \delta \in \mathrm{R}\). Be three lines such that \(\mathrm{L}_1\) is perpendicular to \(\mathrm{L}_2\) and \(L_3\) is perpendicular to both \(L_1\) and \(L_2\). Then the point which lies on \(\mathrm{L}_3\) is