Two identical charged particles each having a mass \(10 \,g\) and charge \(2.0 \times 10^{-7}\,C\) area placed on a horizontal table with a separation of \(L\) between then such that they stay in limited equilibrium. If the coefficient of friction between each particle and the table is \(0.25\), find the value of \(L\).[Use \(g =10\,ms ^{-2}\) ]..........\(cm\)

A square loop of sides \(a=1 \mathrm{~m}\) is held normally in front of a point charge \(\mathrm{q}=1 \mathrm{C}\). The flux of the electric field through the shaded region is \(\frac{5}{\mathrm{p}} \times \frac{1}{\varepsilon_0} \frac{\mathrm{Nm}^2}{\mathrm{C}}\), where the value of p is _____.

The energy levels of an hydrogen atom are shown below. The transition corresponding to emission of shortest wavelength is

The amplitude of a particle executing \(SHM\) is \(3\,cm\). The displacement at which its kinetic energy will be \(25 \%\) more than the potential energy is: \(.............cm\).

The number of molecules in one litre of an ideal gas at \(300 \,{K}\) and \(2\) atmospheric pressure with mean kinetic energy \(2 \times 10^{-9}\, {J}\) per molecules is \(....\, \times 10^{11}\)

A bullet of mass \(200\,g\) having initial kinetic energy \(90\,J\) is shot inside a long swimming pool as shown in the figure. If it's kinetic energy reduces to \(40\,J\) within 1s, the minimum length of the pool, the bullet has a to travel so that it completely comes to rest is \(.....m\)

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

List \(I\)	List \(II\)
\((A)\) Torque	\((I)\) \(Nms^{-1}\)
\((B)\) Stress	\((II)\) \(J\,kg^{-1}\)
\((C)\) Latent Heat	\((III)\) \(Nm\)
\((D)\) Power	\((IV)\) \(Nm^{-2}\)

Choose the correct answer from the options below.

Let M denote the set of all real matrices of order \(3 \times 3\) and let \(\mathrm{S}=\{-3,-2,-1,1,2\}\). Let
\(\mathrm{S}_1=\left\{\mathrm{A}=\left[a_{\mathrm{ij}}\right] \in \mathrm{M}: \mathrm{A}=\mathrm{A}^{\mathrm{T}} \text { and } a_{\mathrm{ij}} \in \mathrm{~S}, \forall \mathrm{i}, \mathrm{j}\right\}, \)
\( \mathrm{S}_2=\left\{\mathrm{A}=\left[a_{\mathrm{ij}}\right] \in \mathrm{M}: \mathrm{A}=-\mathrm{A}^{\mathrm{T}} \text { and } a_{\mathrm{ij}} \in \mathrm{~S}, \forall \mathrm{i}, \mathrm{j}\right\}, \)
\( \mathrm{S}_3=\{\mathrm{A}=\left[a_{\mathrm{ij}}\right] \in \mathrm{M}: a_{11}+a_{22}+a_{33}=0\) and \(a_{\mathrm{ij}} \in \mathrm{~S}, \forall \mathrm{i}, \mathrm{j}\}\)
If \(n\left(\mathrm{~S}_1 \cup_2 \mathrm{US}_3\right)=125 \alpha\), then \(\alpha\) equals _______

Let \(y=y_{1}(x)\) and \(y=y_{2}(x)\) be two distinct solutions of the differential equation \(\frac{d y}{d x}=x+y\), with \(y _{1}(0)=0\) and \(y _{2}(0)=1\) respectively. Then, the number of points of intersection of \(y=y_{1}(x)\) and \(y=y_{2}(x)\) is.

The system of equations \(kx + y + z =1\) \(x + ky + z = k\) and \(x + y + zk = k ^{2}\) has no solution if \(k\) is equal to

A point \(\mathrm{z}\) moves in the complex plane such that \(\arg \left(\frac{\mathrm{z}-2}{\mathrm{z}+2}\right)=\frac{\pi}{4}\), then the minimum value of \(|z-9 \sqrt{2}-2 i|^{2}\) is equal to ..... .

Let the mean and variance of \(12\) observations be \(\frac{9}{2}\) and \(4\) respectively. Later on, it was observed that two observations were considered as \(9\) and \(10\) instead of \(7\) and \(14\) respectively. If the correct variance is \(\frac{m}{n}\), where \(m\) and \(n\) are co-prime, then \(m + n\) is equal to

When a ball is dropped into a lake from a height \(4.9 \,m\) above the water level, it hits the water with a velocity \(v\) and then sinks to the bottom with the constant velocity \(v\). It reaches the bottom of the lake \(4.0,s\) after it is dropped. The approximate depth of the lake is ............ \(m\)