An $\alpha$-particle (mass $4$ amu) and a singly charged Sulphur ion (mass $32$ amu) are initially at rest. They are accelerated through a potential $V$ and then allowed to pass into a region of uniform magnetic field which is normal to the velocities of the particles. Within this region, the $\alpha$-particle and the sulphur ion move in circular orbits of radii $R_{\alpha }$ and $R_S$, respectively. The ratio $\left(\frac{R_S}{R_{\alpha }}\right)$ is __________ .

In a radioactive decay process, the activity is defined as $A=-\frac{dN}{dt}$, where $N\left(t\right)$ is the number of radioactive nuclei at time $\left(t\right)$. Two radioactive sources, $S_1$ and $S_2$ have same activity at time $t=0$. At a later time, the activities of $S_1$ and $S_2$ are $A_1$ and $A_2$, respectively. When $S_1$ and $S_2$ have just completed their $3^{\mathrm{rd}}$ and $7^{\mathrm{th}}$ half-lives, respectively, the ratio $\frac{A_1}{A_2}$ is _____.

A ray \(O P\) of monochromatic light is incident on the face \(A B\) of prism \(A B C D\) near vertex \(B\) at an incident angle of \(60^{\circ}\) (see figure). If the refractive index of the material of the prism is \(\sqrt{3}\), which of the following is (are) correct?

A circular wire loop of radius \(R\) is placed in the \(x-y\) plane centered at the origin \(O\). A square loop of side \(a(a< < R)\) having two turns is placed with its centre at \(z=\sqrt{3} R\) along the axis of the circular wire loop, as shown in figure.

The plane of the square loop makes an angle of \(45^{\circ}\) with respect to the \(z\)-axis. If the mutual inductance between the loops is given by \(\frac{\mu_{0} a^{2}}{2^{p / 2} R}\), then the value of \(p\) is

A ray of light travelling in water is incident on its surface open to air. The angle of incidence is \(\theta\), which is less than the critical angle. Then there will be

An infinitely long uniform line charge distribution of charge per unit length $\mathrm{\lambda }$ lies parallel to the $\mathrm{y}$ -axis in the $\mathrm{y}$ - $\mathrm{z}$ plane at $\mathrm{z}=\frac{\sqrt{3}}{2}\mathrm{\alpha }$ (see figure). If the magnitude of the flux of the electric field through the rectangular surface $\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$ lying in the $\mathrm{x}$ - $\mathrm{y}$ plane with its centre at the origin is $\frac{\mathrm{\lambda }\mathrm{L}}{\mathrm{n}{\mathrm{\epsilon }}_0}$ ( ${\mathrm{\epsilon }}_0=$ permittivity of free space), then the value of $\mathrm{n}$ is

A bullet is fired vertically upwards with velocity $\mathrm{\upsilon }$ from the surface of a spherical planet. When it reaches its maximum height, its acceleration due to the planet's gravity is $\frac{1}{4^{\mathrm{t}\mathrm{h}}}$ of its value at the surface of the planet. If the escape velocity from the planet is ${\mathrm{\upsilon }}_{\mathrm{e}\mathrm{s}\mathrm{c}}=\mathrm{\upsilon }\sqrt{\mathrm{N}}$ , then the value of N is (ignore energy loss due to atmosphere)

A sample \((5.6 \mathrm{~g})\) containing iron is completely dissolved in cold dilute \(\mathrm{HCl}\) to prepare a \(250 \mathrm{~mL}\) of solution. Titration of \(25.0 \mathrm{~mL}\) of this solution requires \(12.5 \mathrm{~mL}\) of \(0.03 \mathrm{M} \mathrm{KMnO}_{4}\) solution to reach the end point. Number of moles of \(\mathrm{Fe}^{2+}\) present in \(250 \mathrm{~mL}\) solution is \(\mathbf{x} \times 10^{-2}\) (consider complete dissolution of \(\mathrm{FeCl}_{2}\) ). The amount of iron present in the sample is \(\mathbf{y} \%\) by weight.
(Assume: \(\mathrm{KMnO}_{4}\) reacts only with \(\mathrm{Fe}^{2+}\) in the solution Use: Molar mass of iron as \(56 \mathrm{~g} \mathrm{~mol}^{-1}\) )
The value of $\mathrm{x}$ is________ .

The circle $C_1:x^2+y^2=3,$ with centre at O, intersects the parabola $x^2=2y$ at the point P in the first quadrant. Let the tangent to the circle $C_1$ at P touches other two circles $C_2$ and $C_3$ at $R_2$ and $R_3$ , respectively. Suppose $C_2$ and $C_3$ have equal radii $2\sqrt{3}$ and centres $Q_2$ and ${ Q}_3$ , respectively. If $Q_2$ and $Q_3$ lie on the y - axis, then

The number of distinct real roots of \(x^4-4 x^3+12 x^2+x-1=0\) is

Let \(p(x)\) be a polynomial of degree 4 having extremum at \(x=1,2\) and \(\lim _{x \rightarrow 0}\left[1+\frac{p(x)}{x^2}\right]=2\). Then, the value of \(p(2)\) is

In the Newman projection for 2,2-dimethylbutane

\(X\) and \(Y\) can respectively be

The smallest division on the main scale of a Vernier calipers is $0.1 \mathrm{cm}$. Ten divisions of the Vernier scale correspond to nine divisions of the main scale. The figure below on the left shows the reading of this calipers with no gap between its two jaws. The figure on the right shows the reading with a solid sphere held between the jaws. The correct diameter of the sphere is