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

Two point-like objects of masses $20\mathrm{gm}$ and $30\mathrm{gm}$ are fixed at the two ends of a rigid massless rod of length $10\mathrm{cm}$. This system is suspended vertically from a rigid ceiling using a thin wire attached to its center of mass, as shown in the figure. The resulting torsional pendulum undergoes small oscillations. The torsional constant of the wire is $1.2\times {10}^{-8}\mathrm{N}\mathrm{m}{\mathrm{rad}}^{-1}$. The angular frequency of the oscillations in $n\times {10}^{-3}\mathrm{rad}{\mathrm{s}}^{-1}$. The value of $n$ is _____.

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

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A uniform thin cylindrical disk of mass \(M\) and radius \(R\) is attached to two identical massless springs of spring constant \(k\) which are fixed to the wall as shown in the figure. The springs are attached to the axle of the disk symmetrically on either side at a distance \(d\) from its centre. The axle is massless and both the springs and the axle are in a horizontal plane. The unstretched length of each spring is \(L\). The disk is initially at its equilibrium position with its centre of mass \((C M)\) at a distance Lfrom the wall. The disk rolls without slipping with velocity \(\mathbf{v}_0=v_0 \hat{\mathbf{i}}\) The coefficient of friction is \(\mu\).
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Question :
The centre of mass of the disk undergoes simple harmonic motion with angular frequency \(\omega\) equal to

Column I contains a list of processes involving expansion of an ideal gas. Match this with Column II describing the thermodynamic change during this process. Indicate your answer by darkening the appropriate bubbles of the \(4 \times 4\) matrix given in the ORS.

Airplanes A and B are flying with constant velocity in the same vertical plane at angles ${30}^{\mathrm{o}}$ and ${60}^{\mathrm{o}}$ with respect to the horizontal respectively as shown in the figure. The speed of A is $100\sqrt{3}\mathrm{}\mathrm{m}{\mathrm{s}}^{-1}$ . At time t = 0 s, an observer in A finds B at a distance of 500 m. This observer sees B moving with a constant velocity perpendicular to the line of motion of A. If at $\mathrm{t}={\mathrm{t}}_0$ , A just escapes being hit by B, t₀ in seconds is

A piece of wire is bent in the shape of a parabola \(y=k x^2\) (y-axis vertical) with a bead of mass \(m\) on it. The bead can slide on the wire without friction. It stays at the lowest point of the parabola when the wire is at rest. The wire is now accelerated parallel to the \(x\)-axis with the constant acceleration \(a\). The distance of the new equilibrium position of the bead, where the bead can stays at rest with respect to the wire, from the \(y\)-axis is

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Consider the lines: \(L_1: \frac{x+1}{3}=\frac{y+2}{1}=\frac{z+1}{2}, L_2: \frac{x-2}{1}=\frac{y+2}{2}=\frac{z-3}{3}\)Question:
The shortest distance between \(L_1\) and \(L_2\) is

A steady current \(I\) goes through a wire loop \(P Q R\) having shape of a right angle triangle with \(P Q=3 x, P R=4 x\) and \(Q R=5 x\). If the magnitude of the magnetic field at \(P\) due to this loop is \(k\left(\frac{\mu_0 I}{48 \pi x}\right)\), find the value of \(k\).

A crystalline solid of a pure substance has a face-centred cubic structure with a cell edge of $400\mathrm{ }\mathrm{pm}$ . If the density of the substance in the crystal is $8 \mathrm{g}\mathrm{ }{\mathrm{cm}}^{-3},\mathrm{ }$ then the number of atoms present in $256 \mathrm{g}$ of the crystal $\mathrm{is}\mathrm{ }\mathrm{N}\mathrm{ }\times \mathrm{ }{10}^{24}$ . The value of $\mathrm{N}$is

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Let \(A\) be the set of all \(3 \times 3\) symmetric matrices all of whose entries are either 0 or 1 . Five of these entries are 1 and four of them are 0 .Question:
The number of matrices \(A\) in \(A\) for which the system of linear equations \(A\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]\) is is inconsistent, is

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Let \(A\) be the set of all \(3 \times 3\) symmetric matrices all of whose entries are either 0 or 1 . Five of these entries are 1 and four of them are 0.
Question:
The number of matrices in \(A\) is

The number of $-{\mathrm{CH}}_{2-}$ (methylene) groups in the product formed from the following reaction sequence is _____ .