Choose the correct nuclear process from the below options.
[p: proton, n : neutron, \(\mathrm{e}^{-}\): electron, \(\mathrm{e}^{+}\): positron, \(v:\) neutrino, \(\bar{v}\) : antineutrino]

A particle is oscillating on the \(X-\) axis with an amplitude \(2\, cm\) about the point \(x_0 = 10\, cm\) with a frequency \(\omega \). A concave mirror of focal length \(5\,
cm\) is placed at the origin ( see figure) Identify the correct statements \((A)\) The image executes periodic motion \((B)\) The image executes non-periodic motion \((C)\) The turning points of the image are asymmetric w.r.t the image of the point at \(x =10\,cm\) \((D)\) The distance between the turning points of the oscillation of the image is \(\frac{{100}}{{21}}\)

A particle of mass \(m\) moving with velocity \(v\) collides with a stationary particle of mass \(2m\). After collision, they stick together and continue to move together with velocity ........

In an experiment with a convex lens. The plot of the image distance (\(v'\)) against the object distance \(\left(\mu^{\prime}\right)\) measured from the focus gives a curve \(v^{\prime} \mu^{\prime}=225\). If all the distances are measured in \(cm\). The magnitude of the focal length of the lens is \(cm\).

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

List\(-I\)	List\(-II\)
\((a)\) \({R}_{{H}}\) (Rydberg constant)	\((i)\) \({kg} {m}^{-1} {s}^{-1}\)
\((b)\) \(h\) (Planck's constant)	\((ii)\) \({kg} {m}^{2} {s}^{-1}\)
\((c)\) \(\mu_{{B}}\) (Magnetic field energy density)	\((iii)\) \({m}^{-1}\)
\((d)\) \(\eta\) (coefficient of viscocity)	\((iv)\) \({kg} {m}^{-1} {s}^{-2}\)

Choose the most appropriate answer from the options given below:

The momentum of an electron revolving in \(n ^{\text {th }}\) orbit is given by: (Symbols have their usual meanings)

The work functions of cesium (Cs) and lithium (Li) metals are 1.9 eV and 2.5 eV , respectively. If we incident a light of wavelength 550 nm on these two metal surfaces, then photo-electric effect is possible for the case of

For each \(x\,\in R,\) let \([x]\) be the greatest integer less than or equal to \(x.\) Then \(\mathop {\lim }\limits_{x \to {0^ + }} \frac{{x([x] + [x])\,\sin \,[x]}}{{\left| x \right|}}\) is equal to

If \(S=\frac{7}{5}+\frac{9}{5^{2}}+\frac{13}{5^{3}}+\frac{19}{5^{4}}+\ldots .\), then \(160 \mathrm{~S}\) is equal to....... .

A transverse wave is represented by \(y=2 \sin\) \((\omega t - kx ) cm\). The value of wavelength (in \(cm\) ) for which the wave velocity becomes equal to the maximum particle velocity, will be.

Which of the following are correct expression for torque acting on a body?
A. \(\vec{\tau}=\vec{r}\times\vec{L}\)
B. \(\vec{\tau}=\frac{\mathrm{d}}{\mathrm{dt}}(\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{p}})\)
C. \(\vec{\tau}=\overrightarrow{\mathrm{r}} \times \frac{\mathrm{d} \mathrm{p}}{\mathrm{dt}}\)
D. \(\vec{\tau}=\mathrm{I} \vec{\alpha}\)
E. \(\vec{\tau}=\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{F}}\)
( \(\overrightarrow{\mathrm{r}}=\) position vector; \(\overrightarrow{\mathrm{p}}=\) linear momentum;
\(\overrightarrow{\mathrm{L}}=\) angular momentum; \(\vec{\alpha}=\) angular acceleration;
\(I=\) moment of inertia; \(\vec{F}=\) force; \(t=\) time)
Choose the correct answer from the options given below :

If \(z_1, z_2\) are two distinct complex number such that \(\left|\frac{z_1-2 z_2}{\frac{1}{2}-z_1 \bar{z}_2}\right|=2\), then

Let \([t]\) denote the greatest integer \(\leq t .\) Then the equation in \(x ,[ x ]^{2}+2[ x +2]-7=0\) has