In the figure, potential difference between \(A\) and \(B\) is......\(V\)

In \(LC\) circuit the inductance \(\mathrm{L}=40\; \mathrm{mH}\) and capacitance \(\mathrm{C}=100\; \mu \mathrm{F} .\) If a voltage \(\mathrm{V}(\mathrm{t})=10 \sin (314 \mathrm{t})\) is applied to the circuit, the current in the circuit is given as 

A \(0.5\) kg mass is in contact against the inner wall of a cylindrical drum of radius \(4\) m rotating about its vertical axis. The minimum rotational speed of the drum to enable the mass to remain stuck to the wall (without falling) is \(5\) rad/s. The coefficient of friction between the drum's inner wall surface and mass is _______. (Take \(g = 10\) m/s\(^2\))

There is an air bubble of radius \(1.0\,mm\) in a liquid of surface tension \(0.075\,Nm ^{-1}\) and density \(1000\,kg\) \(m ^{-3}\) at a depth of \(10\,cm\) below the free surface. The amount by which the pressure inside the bubble is greater than the atmospheric pressure is \(....Pa \left( g =10\,ms ^{-2}\right)\)

Given below are two statements: one is labelled as Assertion \(A\) and the other is labelled as Reason \(R\). Assertion \(A:\) A pendulum clock when taken to Mount Everest becomes fast. Reason \(R:\) The value of \(g\) (acceleration due to gravity) is less at Mount Everest than its value on the surface of earth. In the light of the above statements, choose the most appropriate answer from the options given below

A person driving car at a constant speed of \(15\,m / s\) is approaching a vertical wall. The person notices a change of \(40\,Hz\) in the frequency of his car's horn upon reflection from the wall. The frequency of horn is ............ \(Hz\). (Given : Speed of sound : \(330\,m / s\) )

In a reflecting telescope, a secondary mirror is used to:

When light of frequency twice the threshold frequency is incident on the metal plate, the maximum velocity of emitted election is \(v _{1}\). When the frequency of incident radiation is increased to five times the threshold value, the maximum velocity of emitted electron becomes \(v _{2}\). If \(v _{2}= x\) \(v _{1}\), the value of \(x\) will be...........

Vectors \(a \hat{i}+b \hat{j}+\hat{k}\) and \(2 \hat{i}-3 \hat{j}+4 \hat{k}\) are perpendicular to each other when \(3 a+2 b=7\), the ratio of a to \(b\) is \(\frac{x}{2}\). The value of \(x\) is \(..............\)

If \(\int\left( e ^{2 x }+2 e ^{ x }- e ^{- x }-1\right) e ^{\left( e ^{ x }+ e ^{- x }\right)} d x\) \(=g(x) e^{\left(e^{x}+e^{-x}\right)}+c,\) where \(c\) is a constant of integration, then \(g (0)\) is equal to

Consider two vectors \(\overrightarrow{\mathrm{u}}=3 \hat{\mathrm{i}}-\hat{\mathrm{j}}\) and \(\overrightarrow{\mathrm{v}}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-\lambda \hat{\mathrm{k}}, \lambda \gt 0\). The angle between them is given by \(\cos ^{-1}\left(\frac{\sqrt{5}}{2 \sqrt{7}}\right)\). Let \(\overrightarrow{\mathrm{v}}=\overrightarrow{\mathrm{v}}_1+\overrightarrow{\mathrm{v}}_2\), where \(\overrightarrow{\mathrm{v}}_1\) is parallel to \(\overrightarrow{\mathrm{u}}\) and \(\overrightarrow{\mathrm{v}}_2\) is perpendicular to \(\overrightarrow{\mathrm{u}}\). Then the value \(\left|\overrightarrow{\mathrm{v}}_1\right|^2+\left|\overrightarrow{\mathrm{v}}_2\right|^2\) is equal to

Four particles each of mass \(1 \mathrm{~kg}\) are placed at four corners of a square of side \(2 \mathrm{~m}\). Moment of inertia of system about an axis perpendicular to its plane and passing through one of its vertex is_________ \(\mathrm{kgm}^2\).

Let a vector \(\vec{a}\) be coplanar with vectors \(\vec{b}=2 \hat{i}+\hat{j}+\hat{k}\) and \(\vec{c}=\hat{i}-\hat{j}+\hat{k} .\) If \(\vec{a}\) is perpendicular to \(\vec{d}=3 \vec{i}+2 \hat{j}+6 \hat{k}\), and \(|\vec{a}|=\sqrt{10} .\) Then a possible value of \([\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}]+[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \vec{d}]+[\overrightarrow{\mathrm{a}} \vec{c} \vec{d}]\) is equal to: