A freshly prepared sample of a radioisotope of half-life 1386 s has activity \({10}^{3}\) disintegrations per second. Given that In 2 = 0.693, the fraction of the initial number of nuclei (expressed in nearest integer percentage) that will decay in the first 80 s after preparation of the sample is

A monochromatic light is travelling in a medium of refractive index $n=1.6$ . It enters a stack of glass layers from the bottom side at an angle $\theta =30°.$ The interfaces of the glass layers are parallel to each other. The refractive indices of different glass layers are monotonically decreasing as $n_m=n–m\mathrm{\Delta }n$ where $n_m$ is the refractive index of the $m^{th}$ slab and $\mathrm{\Delta }n=0.1$ (see the figure). The ray is refracted out parallel to the interface between the ${\left(m–1\right)}^{th}$ and $m^{th}$ slabs from the right side of the stack. What is the value of m?

As shown in the figures, a uniform rod \(O O^{\prime}\) of length \(l\) is hinged at the point \(O\) and held in place vertically between two walls using two massless springs of same spring constant. The springs are connected at the midpoint and at the top-end \(\left(O^{\prime}\right)\) of the rod, as shown in Fig. 1 and the rod is made to oscillate by a small angular displacement. The frequency of oscillation of the rod is \(f_1\). On the other hand, if both the springs are connected at the midpoint of the rod, as shown in Fig. 2 and the rod is made to oscillate by a small angular displacement, then the frequency of oscillation is \(f_2\). Ignoring gravity and assuming motion only in the plane of the diagram, the value of \(\frac{f_1}{f_2}\) is:

Consider regular polygons with number of sides $n=\mathrm{3,4},5\dots$ as shown in the figure. The center of mass of all the polygons is at height $h$ from the ground. They roll on a horizontal surface about the leading vertex without slipping and sliding as depicted. The maximum increase in height of the locus of the center of mass for each polygon is $\mathrm{\Delta }$ . Then, $\mathrm{\Delta }$ depends on $n$ and $h$ as

A block of mass ${\mathrm{m}}_1=1\mathrm{}\mathrm{k}\mathrm{g}$ another mass ${\mathrm{m}}_2=2\mathrm{}\mathrm{k}\mathrm{g}$ , are placed together (see figure) on an inclined plane with angle of inclination $\mathrm{\theta }$ . Various values of $\mathrm{\theta }$ are given in List I. The coefficient of friction between the block ${\mathrm{m}}_1$ and the plane is always zero. The coefficient of static and dynamic friction between the block ${\mathrm{m}}_2$ and the plane are equal to $\mathrm{\mu }=0.3$ . In List II expressions for the friction on the block ${\mathrm{m}}_2$ are given. Match the correct expression of the friction in List II with the angles given in List I, and choose the correct option. The acceleration due to gravity is denoted by g.
[Useful information: $\tan \left({5.5}^{\mathrm{o}}\right)\approx 0.1;\tan \left({11.5}^{\mathrm{o}}\right)\approx 0.2;\tan \left({16.5}^{\mathrm{o}}\right)\approx 0.3$ ]

	List I		List II
A.	$\mathrm{\theta }=5^{\mathrm{o}}$	$\mathrm{P}.$	${\mathrm{m}}_2\mathrm{g}\sin \mathrm{\theta }$
B.	$\mathrm{\theta }={10}^{\mathrm{o}}$	$\mathrm{Q}.$	$\left({\mathrm{m}}_1+{\mathrm{m}}_2\right)\mathrm{g}\sin \mathrm{\theta }$
C.	$\mathrm{\theta }={15}^{\mathrm{o}}$	$\mathrm{R}.$	$\mathrm{\mu }{\mathrm{m}}_2\mathrm{}\mathrm{g}\cos \mathrm{\theta }$
D.	$\mathrm{\theta }={20}^{\mathrm{o}}$	$\mathrm{S}.$	$\mathrm{\mu }\left({\mathrm{m}}_1+{\mathrm{m}}_2\right)\mathrm{g}\cos \mathrm{\theta }$

\(5.6 \mathrm{~L}\) of helium gas at STP is adiabatically compressed to \(0.7 \mathrm{~L}\). Taking the initial temperature to be \(T_1\), the work done in the process is

A ring of mass $\mathrm{M}$ and radius $\mathrm{R}$ is rotating with angular speed $\mathrm{\omega }$ about a fixed vertical axis passing through its centre O with two point masses each of mass $\frac{\mathrm{M}}{8}$ at rest at O. These masses can move radially outwards along two massless rods fixed on the ring as shown in the figure. At some instant the angular speed of the system is $\frac{8}{9}\mathrm{\omega }$ and one of the masses is at a distance of $\frac{3}{5}\mathrm{R}$ from O. At this instant the distance of the other mass from O is

Let $\mathrm{X}\mathrm{ }\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{ }\mathrm{Y}$ be two arbitrary, $3\mathrm{ }\times 3$ , non - zero, skew - symmetric matrices and Z be an arbitrary $3\mathrm{ }\times 3$ , non - zero, symmetric matrix. Then which of the following matrices is (are) skew symmetric ?

In the given reaction scheme, $\mathbf{P}$ is a phenyl alkyl ether, $\mathbf{Q}$  is an aromatic compound; $\mathbf{R}$ and $\mathbf{S}$ are the major products.

The correct statement about $\mathbf{S}$ is

Let $\mathrm{M}=\left[\begin{array}{ccc}0& 1& \mathrm{a}\\ 1& 2& 3\\ 3& b& 1\end{array}\right]$ and $\mathrm{a}\mathrm{d}\mathrm{j}\mathrm{M}=\left[\begin{array}{ccc}-1& 1& -1\\ 8& -6& 2\\ -5& 3& -1\end{array}\right]$ where $\mathrm{a}$ and $\mathrm{b}$ are real numbers. Which of the following options is/are correct?

For the first order reaction \(2 \mathrm{~N}_2 \mathrm{O}_5(\mathrm{~g}) \longrightarrow 4 \mathrm{NO}_2(g)+\mathrm{O}_2(g)\)

For any $y\in \mathrm{ℝ}$, let ${\cot }^{-1}\left(y\right)\in \left(0,\pi \right)$ and ${\tan }^{-1}\left(y\right)\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$. Then the sum of all the solutions of the equation ${\tan }^{-1}\left(\frac{6y}{9-y^2}\right)+{\cot }^{-1}\left(\frac{9-y^2}{6y}\right)=\frac{2\pi }{3}$ for $0<\left|y\right|<3$, is equal to

A solid sphere of radius \(R\) has moment of inertia \(J\) about its qeometrical axis. If it is melted into a disc of radius \(r\) and thickness \(t\). If its moment of inertia about the tangential axis (which is perpendicular to plane of the disc), is also equal to \(I\), then the value of \(r\) is equal to