The \(P-V\) diagram of a diatomic ideal gas system going under cyclic process as shown in figure. The work done during an adiabatic process \(CD\) is (use \(\gamma=1.4\)) (in \(J\))

Statement \(I:\) If three forces \(\vec{F}_{1}, \vec{F}_{2}\) and \(\vec{F}_{3}\) are represented by three sides of a triangle and \(\overrightarrow{{F}}_{1}+\overrightarrow{{F}}_{2}=-\overrightarrow{{F}}_{3}\), then these three forces are concurrent forces and satisfy the condition for equilibrium. Statement \(II:\) A triangle made up of three forces \(\overrightarrow{{F}}_{1}, \overrightarrow{{F}}_{2}\) and \(\overrightarrow{{F}}_{3}\) as its sides taken in the same order, satisfy the condition for translatory equilibrium. In the light of the above statements, choose the most appropriate answer from the options given below:

The radius of gyration of a uniform rod of length \(l,\) about an axis passing through a point \(\frac{l}{4}\) away from the centre of the rod, and perpendicular to it, is

Match List \(I\) with List \(II\)

LIST\(-I\)	LIST\(-II\)
\((A)\) Surface tension	\((I)\) \(Kg m ^{-1} s ^{-1}\)
\((B)\) Pressure	\((II)\) \(Kg ms^{-1 }\)
\((C)\) Viscosity	\((III)\) \(Kg m ^{-1} s ^{-2}\)
\((D)\) Impulse	\((IV)\) \(Kg s ^{-2}\)

Choose the correct answer from the options given below :

A car is moving with speed of \(150\,km / h\) and after applying the brake it will move \(27\,m\) before it stops. If the same car is moving with a speed of one third the reported speed then it will stop after travelling \(....m\) distance.

As shown in the figure, a bob of mass \(\mathrm{m}\) is tied by a massless string whose other end portion is wound on a fly wheel (disc) of radius \(\mathrm{r}\) and mass \(m\). When released from rest the bob starts falling vertically. When it has covered a distance of \(h\). the angular speed of the wheel will be

As shown in the figure, a potentiometer wire of resistance \(20\,\Omega\) and length \(300\,cm\) is connected with resistance box (R.B.) and a standard cell of emf \(4\,V\). For a resistance ' \(R\) ' of resistance box introduced into the circuit, the null point for a cell of \(20\,mV\) is found to be \(60\,cm\). The value of ' \(R\) ' is \(.....\Omega\)

If the line \(y=m x\) bisects the area enclosed by the lines \(x=0, y=0, x=\frac{3}{2}\) and the curve \(\mathrm{y}=1+4 \mathrm{x}-\mathrm{x}^{2}\), then \(12 \mathrm{~m}\) is equal to ..... .

A straight wire carrying a current of \(14\,A\) is bent into a semicircular are of radius \(2.2\,cm\) as shown in the figure. The magnetic field produced by the current at the centre \((O)\) of the arc. is \(.........\,\times 10^{-4}\, T\)

If the point on the curve \(y^{2}=6 x\), nearest to the point \(\left(3, \frac{3}{2}\right)\) is \((\alpha, \beta)\), then \(2(\alpha+\beta)\) is equal to \(.....\)

Let \(f\) and \(g\) be two differentiable functions on \(R\) such that \(f'(x) > 0\) and \(g'(x) < 0\) for all \(x\in R\) .Then for all \(x\)

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

As shown in the figure, a metallic rod of linear density \(0.45\,kg\,m ^{-1}\) is lying horizontally on a smooth incline plane which makes an angle of \(45^{\circ}\) with the horizontal. The minimum current flowing in the rod required to keep it stationary, when \(0.15\,T\) magnetic field is acting on it in the vertical upward direction, will be \(....A\) \(\left\{\right.\) Use \(\left.g=10 m / s ^{2}\right\}\)