A uniform rod of length \(L\) and mass \(M\) is pivoted at the centre. Its two ends are attached to two springs of equal spring constants \(k\). The springs are fixed to rigid supports as shown in the figure, and rod is free to oscillate in the horizontal plane. The rod is gently pushed through a small angle \(\theta\) in one direction and released. The frequency of oscillation is

A solid sphere is in pure rolling motion on an inclined surface having inclination \(\theta\).

A ball is projected from the ground at an angle of ${45}^o$ with the horizontal surface. It reaches a maximum height of $120\mathrm{ m}$ and returns to the ground. Upon hitting the ground for the first time, it loses half of its kinetic energy. Immediately after the bounce, the velocity of the ball makes an angle of ${30}^o$ with the horizontal surface. The maximum height it reaches after the bounce (in meters) is _____________.

A satellite is moving with a constant speed \(v\) in a circular orbit about the earth. An object of mass \(m\) is ejected from the satellite such that it just escapes from the gravitational pull of the earth. At the time of its ejection, the kinetic energy of the object is

The positon vector \(\vec{r}\) of a particle of mass m is given by the following equation \(\vec{r}(t)=\alpha t^3 \hat{i} +\beta t^2 \hat{j}\), where \(\alpha=\frac{10}{3} m s^{-3}, \beta=5 m s^{-2}\) and \(m=0.1 kg\). At \(t=1 s\), which of the following statements (s) is (are) true about the particle?

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A point charge \(Q\) is moving in a circular orbit of radius \(R\) in the \(x-y\) plane with an angular velocity \(\omega .\) This can be considered as equivalent to a loop carrying a steady current \(\frac{Q \omega}{2 \pi}\). A uniform magnetic field along the positive \(z\)-axis is now switched on, which increases at a constant rate from 0 to \(B\) in one second. Assume that the radius of the orbit remains constant. The application of the magnetic field induces an emf in the orbit. The induced emf is defined as the work done by an induced electric field in moving a unit positive charge around a closed loop. It is known that, for an orbiting charge, the magnetic dipole moment is proportional to the angular momentum with a proportionality constant \(\gamma\).

Question:

The change in the magnetic dipole moment associated with the orbit, at the end of the time interval of the magnetic field change, is

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In a thin rectangular metallic strip a constant current \(I\) flows along the positive \(x\)-direction, as shown in the figure. The length, width and thickness of the strip are \(l, w\) and \(d\), respectively.

A uniform magnetic field \(\vec{B}\) is applied on the strip along the positive \(y\)-direction. Due to this, the charge carriers experience a net deflection along the \(z\)-direction. This results in accumulation of charge carriers on the surface \(P Q R S\) and appearance of equal and opposite charges on the face opposite to \(P Q R S\). A potential difference along the \(z\)-direction is thus developed. Charge accumulation continues until the magnetic force is balanced by the electric force. The current is assumed to be uniformly distributed on the cross section of the strip and carried by electrons.






Question:

Consider two different metallic strips (\(1\) and \(2\)) of same dimensions (length \(l\), width \(w\) and thickness \(d\) ) with carrier densities \(n_{1}\) and \(n_{2}\), respectively. Strip \(1\) is placed in magnetic field \(B_{1}\) and strip \(2\) is placed in magnetic field \(B_{2}\), both along positive \(y\)-directions. Then \(V_{1}\) and \(V_{2}\) are the potential differences developed between \(K\) and \(M\) in strips \(1\) and \(2\), respectively. Assuming that the current \(I\) is the same for both the strips, the correct option(s) is(are)

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Let \(p\) be an odd prime number and \(T_p\) be the following set of \(2 \times 2\) matrices
\[
T_p=\left\{A=\left[\begin{array}{ll}
a & b \\
c & a
\end{array}\right] ; a, b, c \in\{0,1,2, \ldots, p-1\}\right\}
\]Question:
The number of \(A\) in \(T_p\) such that the trace of \(A\) is not divisible by \(p\) but det \((A)\) is divisible by \(p\) is
[Note : The trace of a matrix is the sum of its diagonal entries.]

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In the figure a container is shown to have a movable (without friction) piston on top. The container and the piston are all made of perfectly insulating material allowing no heat transfer between outside and inside the container. The container is divided into two compartments by a rigid partition made of a thermally conducting material that allows slow transfer of heat. The lower compartment of the container is filled with \(2\) moles of an ideal monatomic gas at \(700 K\) and the upper compartment is filled with \(2\) moles of an ideal diatomic gas at \(400 K\). The heat capacities per mole of an ideal monatomic gas are \(C_{V}=\frac{3}{2} R, C_{P}=\frac{5}{2} R\), and those for an ideal diatomic gas are \(C_{V}=\frac{5}{2} R, C_{P}=\frac{7}{2} R\).


Question :
Consider the partition to be rigidly fixed so that it does not move. When equilibrium is achieved, the final temperature of the gases will be

A light source, which emits two wavelengths ${\mathrm{\lambda }}_1=400\mathrm{ }\mathrm{n}\mathrm{m}$ and ${\mathrm{\lambda }}_2=600\mathrm{ }\mathrm{n}\mathrm{m}$ , is used in a Young's double slit experiment. If recorded fringe widths for ${\mathrm{\lambda }}_1$ and ${\mathrm{\lambda }}_2$ are ${\mathrm{\beta }}_1$ and ${\mathrm{\beta }}_2$ and the number of fringes for them within a distance y on one side of the central maximum are ${\mathrm{m}}_1$ and ${\mathrm{m}}_2$ , respectively, then

The pair(s) of ions where BOTH the ions are precipitated upon passing ${\mathrm{H}}_2\mathrm{S}$ gas in presence of dilute HCl, is (are)

Consider the following molecules : ${\mathrm{Br}}_3{\mathrm{O}}_8, {\mathrm{F}}_2\mathrm{O},{\mathrm{H}}_2 {\mathrm{S}}_4{\mathrm{O}}_6,{\mathrm{H}}_2 {\mathrm{S}}_5{\mathrm{O}}_6$, and ${\mathrm{C}}_3{\mathrm{O}}_2$. Count the number of atoms existing in their zero oxidation state in each molecule. Their sum is

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Let \(A_1, G_1, H_1\) denote the arithmetic, geometric and harmonic means respectively, of two distinct positive numbers. For \(n \geq 2\), let \(A_{n-1}\) and \(H_{n-1}\) has arithmetic, geometric and harmonic means as \(A_n, G_n, H_n\), respectively.
Question:
Which of the following statements is correct?