A monochromatic beam of light is incident at ${60}^{\mathrm{o}}$ on one face of an equilateral prism of refractive index $\mathrm{n}$ and emerges from the opposite face making an angle $\mathrm{\theta }\left(\mathrm{n}\right)$ with the normal (see the figure). For $\mathrm{n}=\sqrt{3}$ the value of $\mathrm{\theta }$ is ${60}^{\mathrm{o}}$ and $\frac{\mathrm{d}\mathrm{\theta }}{\mathrm{d}\mathrm{n}}=\mathrm{m}$ . The value of $\mathrm{m}\mathrm{}$ is

Three vectors $\overrightarrow{P},\overrightarrow{Q}$ and $\overrightarrow{R}$ are shown in the figure. Let, $S$ be any point on the vector $\overrightarrow{R}$ . The distance between the points $P$ and $S$ is $b\left|\overrightarrow{R}\right|$. The general relation among vectors $\overrightarrow{P},\overrightarrow{Q}$ and $\overrightarrow{S}$ are

The general motion of a rigid body can be considered to be a combination of (i) a motion of its centre of mass about an axis, and (ii) its motion about an instantaneous axis passing through the centre of mass.
These axes need not be stationary. Consider, for example, a thin uniform disc welded (rigidly fixed) horizontally at its rim to a massless, stick, as shown in the figure. When the disc-stick system is rotated about the origin on a horizontal frictionless plane with angular speed \(\omega\), the motion at any instant can be taken as a combination of (i) a rotation of the centre of mass of the disc about the \(z\)-axis and (ii) a rotation of the disc through an instantaneous vertical axis passing through its centre of mass (as is seen from the changed orientation of points \(P\) and \(Q\) ). Both these motions have the same angular speed \(\omega\) in this case

Now consider two similar systems as shown in the figure: Case
(a) the disc with its face vertical and parallel to \(x-z\) plane; Case
(b) the disc with its face making an angle of
\(45^{\circ}\) with \(x-y\) plane and its horizontal diameter parallel to \(x\)-axis. In both the cascs, the disc is welded at point \(\mathrm{P}\), and the systems are rotated with constant angular speed \(\omega\) about the \(z\) axis.

Question : Which of the following statements about the instantaneous axis (passing through the centre of mass) is correct?

The isotope ${}_5{}^{12}B$ having a mass 12.014 u undergoes $\beta$ - decay to ${}_6{}^{12}C . {}_6{}^{12}C$ has an excited state of the nucleus $\left({}_6{}^{12}{C}^{*}\right)$ at 4.041 MeV above its ground state. If ${}_5{}^{12}B$ decays to ${}_6{}^{12}{C}^{*},$ the maximum kinetic energy of the $\beta$ - particle in units of MeV is ( $1\mu =931.5 Me\frac{V}{c^2},$ where c is the speed of light in vacuum).

Suppose a ${}_{88}{}^{226}Ra$ nucleus at rest and in ground state undergoes $\mathrm{\alpha }$ -decay to a ${}_{86}{}^{222}Rn$ nucleus in its excited state. The kinetic energy of the emitted $\mathrm{\alpha }$ particle is found to be $4.44\mathrm{ }\mathrm{M}\mathrm{e}\mathrm{V}.{}_{86}{}^{222}Rn$ nucleus then goes to its ground state by $\mathrm{\gamma }$ -decay. The energy of the emitted $\mathrm{\gamma }$ -photon is _______ $\mathrm{k}\mathrm{e}\mathrm{V},$
[Given: atomic mass of ${\mathrm{ }}_{88}^{226}\mathrm{R}\mathrm{a}=226.005\mathrm{u},$ atomic mass of ${\mathrm{ }}_{86}^{222}\mathrm{R}\mathrm{n}=222.000\mathrm{u},$ atomic mass of $\mathrm{\alpha }$ particle $=4.000\mathrm{u},1\mathrm{u}=931\mathrm{M}\mathrm{e}\mathrm{V}/{\mathrm{c}}^2,\mathrm{c}$ is speed of the light]

In the circuit shown in the figure, there are two parallel plate capacitors each of capacitance C. The switch S₁ is pressed first to fully charge the capacitor C₁ and the released. The switch S₂ is then pressed to charge the capacitor C₂. After some time, S₂ is released and then is S₃ pressed. After some time,

Two beams of red and violet colours are made to pass separately through a prism (angle of the prism is \(60^{\circ}\) ). In the position of minimum deviation, the angle of refraction will be

A particle \(P\) starts from the point \(z_0=1+2 i\), where \(i=\sqrt{-1}\). It moves first horizontally away from origin by 5 units and then vertically away from origin by 3 units to reach a point \(z_1\). From \(z_1\) the particle moves \(\sqrt{2}\) units in the direction of the vector \(\hat{\mathbf{i}}+\hat{\mathbf{j}}\) and then it moves through an angle \(\frac{\pi}{2}\) in anti-clockwise direction on a circle with centre at origin, to reach a point \(z_2\). The point \(z_2\) is given by

A charged particle is introduced at the origin $x=0, y=0, z=0$ with a given initial velocity $\overrightarrow{v}$ . A uniform electric field $\overrightarrow{E}$ and a uniform magnetic field $\overrightarrow{B}$ exist everywhere. The velocity $\overrightarrow{v}$ , electric field $\overrightarrow{E}$ and a uniform magnetic field $\overrightarrow{B}$ are given in the columns below

Column 1	Column 2	Column 3
1. Electron with $\overrightarrow{v}=2\frac{E_0}{B_0}\widehat{x}$	(i) $\overrightarrow{E}=E_0\widehat{z}$	(P) $\overrightarrow{B}=-B_0\widehat{x}$
2. Electron with $\overrightarrow{v}=2\frac{E_0}{B_0}\widehat{y}$	(ii) $\overrightarrow{E}=-E_0\widehat{y}$	(Q) $\overrightarrow{B}=B_0\widehat{x}$
3.Proton with $\overrightarrow{v}$ =0	(iii) $\overrightarrow{E}={-E}_0\widehat{x}$	(R) $\overrightarrow{B}=-B_0\widehat{y}$
4. Proton with $\overrightarrow{v}=2\frac{E_0}{B_0}\widehat{x}$	(iv) $\overrightarrow{E}=E_0\widehat{x}$	(S) $\overrightarrow{B}=-B_0\widehat{z}$

In which case will the particle describe a helical path with axis along positive z direction?

A student performs an experiment to determine the Young's modulus of a wire, exactly \(2 \mathrm{~m}\) long, by Searle's method. In a particular reading, the student measures the extension in the length of the wire to be \(0.8 \mathrm{~mm}\) with an uncertainty of \(\pm 0.05 \mathrm{~mm}\) at a load of exactly \(1.0 \mathrm{~kg}\). The student also measures the diameter of the wire to be \(0.4 \mathrm{~mm}\) with an uncertainty of \(\pm 0.01 \mathrm{~mm}\). Take \(g=9.8 \mathrm{~m} / \mathrm{s}^2\) (exact). The Young's modulus obtained from the reading is

If the normal of the parabola ${\mathrm{y}}^2=4\mathrm{x}$ drawn at the end points of its latus rectum are tangents to the circle ${\left(\mathrm{x}-3\right)}^2+{\left(\mathrm{y}+2\right)}^2={\mathrm{r}}^2,$ then the value of ${\mathrm{r}}^2$ is

Let the functions $f:\left(-1,1\right)\to \mathrm{ℝ}$ and  $g:\left(-1,1\right)\to \left(-1,1\right)$ be defined by $f\left(x\right)= \left|2x-1\right|+\left|2x+1\right| \& g\left(x\right)=x-\left[x\right]$, where $\left[x\right]$ denotes the greatest integer less than or equal to$x$. Let $f\circ g:\left(-1,1\right)\to \mathrm{ℝ}$ be the composite function defined by $\left(f\circ g\right)\left(x\right)=f\left(g\left(x\right)\right)$. Suppose $c$ is the number of points in the interval $\left(-1,1\right)$ at which $f\circ g$ is not continuous, and suppose $d$ is the number of points in the interval $\left(-1, 1\right)$ at which  $f\circ g$ is not differentiable. Then the value of $c+d$ is _____________

The locus of the orthocentre of the triangle formed by the lines \((1+p) x-p y+p(1+p)=0\), \((1+q) x-q y+q(1+q)=0\) and \(y=0\), where \(p \neq q\), is