A uniform circular disc of mass $\text{50 kg}$ and radius $\text{0.4 m}$ is rotating with an angular velocity of ${\text{10 rad s}}^{-1}$ about its own axis, which is vertical. Two uniform circular rings, each of mass $\text{6.25 kg}$ and radius $\text{0.2 m}$, are gently placed symmetrically on the disc in such a manner that they are touching each other along the axis of the disc and are horizontal. Assume that the friction is large enough such that the rings are at rest relative to the disc and the system rotates about the original axis. The new angular velocity $\left({in\; rad\; s}^{-1}\right)$ of the system is

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The figure shows a surface \(X Y\) separating two transparent media, medium-1 and medium-2. The lines \(a b\) and \(c d\) represent wavefronts of a light wave travelling in medium-1 and incident on \(X Y\). The lines ef and \(g h\) represent wavefronts of the light wave in medium-2 after refraction.
Question:
Speed of light is

System shown in figure is in equilibrium and at rest. The spring and string are massless, now the string is cut. The acceleration of mass \(2 m\) and \(m\) just after the string is cut will be

A block of base \(10 \mathrm{~cm} \times 10 \mathrm{~cm}\) and height \(15 \mathrm{~cm}\) is kept on an inclined plane. The coefficient of friction between them is \(\sqrt{3}\). The inclination \(\theta\) of this inclined plane from the horizontal plane is gradually increased from \(0^{\circ}\). Then,

You are given many resistances, capacitors and inductors. These are connected to a variable DC voltage source (the first two circuits) or an AC voltage source of \(50 \mathrm{~Hz}\) frequency (the next three circuits) in different ways as shown in Column II. When a current \(I\) (steady state for DC or rms for \(A C\) ) flows through the circuit, the corresponding voltage \(V_1\) and \(V_2\) (indicated in circuits) are related as shown in Column I. Match the two

A block with mass M is connected by a massless spring with stiffness constant k to a rigid wall and moves without friction on a horizontal surface. The block oscillates with small amplitude A about an equilibrium position $x_0$. Consider two cases : (i) when the block is at $x_0\text{;}$ and (ii) when the block is at $x=x_0+\text{A}\text{.}$ In both the cases, a particle with mass m $\left(<\text{M}\right)$ is softly placed on the block after which they stick to each other. Which of the following statement(s) is(are) true about the motion after the mass m is placed on the mass M?

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Phase space diagrams are useful tools in analyzing all kinds of dynamical problems. They are especially useful in studying the changes in motion as initial position and momentum are changed. Here we consider some simple dynamical systems in one-dimension. For such systems, phase space is a plane in which position is plotted along horizontal axis and momentum is plotted along vertical axis. The phase space diagram is \(x(t) v s p(t)\) curve in this plane. The arrow on the
curve indicates the time flow. For example, the phase space diagram for a particle moving with constant velocity is a straight line as shown in the figure. We use the sign convention in which position or momentum upwards (or to right) is positive and downwards (or to left) is negative.

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The phase space diagram for simple harmonic motion is a circle centered at the origin. In the figure, the two circles represent the same oscillator but for different initial conditions, and \(E_1\) and \(E_2\) are the total mechanical energies respectively. Then,

An organic compound \(\mathbf{P}\) having molecular formula \(\mathrm{C}_6 \mathrm{H}_6 \mathrm{O}_3\) gives ferric chloride test and does not have intramolecular hydrogen bond. The compound \(\mathbf{P}\) reacts with 3 equivalents of \(\mathrm{NH}_2 \mathrm{OH}\) to produce oxime \(\mathbf{Q}\). Treatment of \(\mathbf{P}\) with excess methyl iodide in the presence of \(\mathrm{KOH}\) produces compound \(\mathbf{R}\) as the major product. Reaction of \(\mathbf{R}\) with excess iso-butylmagnesium bromide followed by treatment with \(\mathrm{H}_3 \mathrm{O}^{+}\)gives compound \(\mathrm{S}\) as the major product.
The total number of methyl \(\left(-\mathrm{CH}_3\right)\) group(s) in compound \(\mathbf{S}\) is ________. (consider maximum value if there are two values)

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Two trains \(A\) and \(B\) are moving with speeds \(20 \mathrm{~m} / \mathrm{s}\) and \(30 \mathrm{~m} / \mathrm{s}\), respectively in the same direction on the same straight track, with \(B\) ahead of \(A\). The engines are at the front ends. The engine of train \(A\) blows a long whistle.
Assume that the sound of the whistle is composed of components varying in frequency from \(f_1=800 \mathrm{~Hz}\) to \(f_2=1120 \mathrm{~Hz}\), as shown in the figure. The spread in the frequency (highest frequency - lowest frequency) is thus \(320 \mathrm{~Hz}\). The speed of sound in still air is \(340 \mathrm{~m} / \mathrm{s}\).
Question:
The spread of frequency as observed by the passengers in train \(B\) is

A tin chloride $\mathrm{Q}$ undergoes the following reactions (not balanced)
$\mathrm{Q}+\mathrm{C}{\mathrm{l}}^{-}\to \mathrm{X}$
$\mathrm{Q}+\mathrm{M}{\mathrm{e}}_3\mathrm{N}\to \mathrm{Y}$
$\mathrm{Q}+\mathrm{C}\mathrm{u}\mathrm{C}{\mathrm{l}}_2\to \mathrm{Z}+\mathrm{C}\mathrm{u}\mathrm{C}\mathrm{l}$
$\mathrm{X}$ is a monoanion having pyramidal geometry. Both $\mathrm{Y}$ and $\mathrm{Z}$ are neutral compounds. Choose the correct option(s).

\(\frac{d^2 x}{d y^2}\) equals

An ideal monatomic gas of n moles is taken through a cycle \(W X Y Z W\) consisting of consecutive adiabatic and isobaric quasi-static processes, as shown in the schematic \(V-T\) diagram. The volume of the gas at \(W, X\) and \(Y\) points are, \(64 \mathrm{~cm}^3, 125 \mathrm{~cm}^3\) and \(250 \mathrm{~cm}^3\), respectively. If the absolute temperature of the gas \(T_W\) at the point \(W\) is such that \(n R T_W=1 \mathrm{~J}\) ( R is the universal gas constant), then the amount of heat absorbed (in J ) by the gas along the path \(X Y\) is \(\qquad\)

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Read the following passage and answer the questions.
Question:
The value of the elements of \(U^{-1}\) is