A string of length $1 \mathrm{m}$ and mass $2\times {10}^{-5} \mathrm{kg}$ is under tension $T$. When the string vibrates, two successive harmonics are found to occur at frequencies $750 \mathrm{Hz}$ and $1000 \mathrm{Hz}$. The value of tension $T$ is _____ newton.

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A dense collection of equal number of electrons and positive ions is called neutral plasma. Certain solids containing fixed positive ions surrounded by free electrons can be treated as neutral plasma. Let \(N\) be the number density of free electrons, each of mass \(m\). When the electrons are subjected to an electric field, they are displaced relatively away from the heavy positive ions.If the electric field becomes zero, the electrons being to oscillate about the positive ions with a natural angular frequency \(\omega_p\), which is called the plasma frequency. To sustain the oscillations, a time varying electric field needs to be applied that has an angular frequency \(\omega\), where a part of the energy is absorbed and a part of it is reflected. As \(\omega\) approaches \(\omega_p\), all the free electrons are set to resonance together and all the energy is reflected. This is the explanation of high reflectivity of metals.
Question :
Taking the electronic charge as \(e\) and the permittivity as \(\varepsilon_0\), use dimensional analysis to determine the correct expression for \(\omega_p\).

Column I gives a list of possible set of parameters measured in some experiments. The variations of the parameters in the form of graphs are shown in Column II. Match the set of parameters given in Column I with the graphs given in Column II. Indicate your answer by darkening the appropriate bubble of the \(4 \times 4\) matrix given in the ORS.

Six point charges, each of the same magnitude \(q\), are arranged in different manners as shown in Column-II. In each case, a point \(M\) and a line \(P Q\) passing through \(M\) are shown. Let \(E\) be the electric field and \(V\) be the electric potential at \(M\) (potential at infinity is zero) due to the given charge distribution when it is at rest. Now, the whole system is set into rotation with a constant angular velocity about the line \(P Q\). Let \(B\) the magnetic field at \(M\) and \(\mu\) be the magnetic moment of the system in this condition. Assume each rotating charge to be equivalent to a steady current.

In a radioactive sample, ${\mathrm{}}_{19}^{40}\mathrm{K}$ nuclei either decay into stable ${\mathrm{}}_{20}^{40}\mathrm{C}\mathrm{a}$ nuclei with decay constant $4.5\times {10}^{-10}$ per year or into stable ${\mathrm{}}_{18}^{40}\mathrm{A}\mathrm{r}$ nuclei with decay constant $0.5\times {10}^{-10}$ per year. Given that in this sample all the stable ${\mathrm{}}_{20}^{40}\mathrm{C}\mathrm{a}$ and ${\mathrm{}}_{18}^{40}\mathrm{A}\mathrm{r}$ nuclei are produced by the ${\mathrm{}}_{19}^{40}\mathrm{K}$ nuclei only. In time $\mathrm{t}\times {10}^9$ years, if the ratio of the sum of stable ${\mathrm{}}_{20}^{40}\mathrm{C}\mathrm{a}$ and ${\mathrm{}}_{18}^{40}\mathrm{A}\mathrm{r}$ nuclei to the radioactive ${\mathrm{}}_{19}^{40}\mathrm{K}$ nuclei is $99,$ the value of t will be : [Given $\mathrm{l}\mathrm{n}\mathrm{}10=2.3$ ]

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\(\mathbf{P}_{17 \text { - } 19}\) : Paragraph for Questions Nos. 17 to 19 Two discs \(A\) and \(B\) are mounted coaxially on a vertical axle. The discs have moments of inertia I and 2I, respectively about the common axis. Disc \(A\) is imparted an initial angular velocity \(2 \omega\) using the entire potential energy of a spring compressed by a distance \(x_1\). Disc \(B\) is imparted an angular velocity \(\omega\) by a spring having the same spring constant and compressed by a distance \(x_2\). Both the discs rotate in the clockwise direction.
Question :
The loss of kinetic energy during the above process is

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When a particle is restricted to move along \(x\)-axis between \(x=0\) and \(x=a\), where \(a\) is of nanometer dimension, its energy can take only certain specific values. The allowed energies of the particle moving in such a restricted region, correspond to the formation of standing waves with nodes at its ends \(x=0\) and \(x=a\). The wavelength of this standing wave is related to the liner momentum \(p\) of the particle according to the de Broglie relation. The energy of the particle of mass \(m\) is related to its linear momentum as \(E=\frac{p^2}{2 m}\). Thus, the energy of the particle can be denoted by a quantum number \(n\) taking values \(1,2,3, \ldots(n=1\), called the ground state) corresponding to the number of loops in the standing wave.
Use the model described above to answer the following three questions for a particle moving in the line \(x=0\) to \(x=a\) [Take \(h=6.6 \times 10^{-34} \mathrm{Js}\) and \(e=1.6 \times 10^{-19} \mathrm{C}\) ]
Question:
If the mass of the particle is \(m=1.0 \times 10^{-30} \mathrm{~kg}\) and \(a=6.6 \mathrm{~nm}\), the energy of the particle in its ground state is closest to

For the following reaction scheme, percentage yields are given along the arrow:

\(\mathbf{x} g\) and \(\mathbf{y} g\) are mass of \(\mathbf{R}\) and \(\mathbf{U}\), respectively.
(Use: Molar mass (in \(\mathrm{g} \mathrm{mol}^{-1}\) ) of \(\mathrm{H}, \mathrm{C}\) and \(\mathrm{O}\) as 1,12 and 16 , respectively)
The value of$\mathrm{y}$ is ___.

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Let \(f:[0,1] \rightarrow \mathbb{R}\) (the set of all real numbers) be a function. Suppose the function \(f\) is twice differentiable, \(f(0)=f(1)=0\) and satisfies \(f^{\prime \prime}(x)-2 f^{\prime}(x)+f(x) \geq e^{x}, x \in[0,1] .\)


Question:

Which of the following is true for \(f(x)\) ?

The species present in solution when \(\mathrm{CO}_2\) is dissolved in water are

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\(\mathbf{P}_{17 \text { - } 19}\) : Paragraph for Questions Nos. 17 to 19 Two discs \(A\) and \(B\) are mounted coaxially on a vertical axle. The discs have moments of inertia I and 2I, respectively about the common axis. Disc \(A\) is imparted an initial angular velocity \(2 \omega\) using the entire potential energy of a spring compressed by a distance \(x_1\). Disc \(B\) is imparted an angular velocity \(\omega\) by a spring having the same spring constant and compressed by a distance \(x_2\). Both the discs rotate in the clockwise direction.Question:
The ratio \(\frac{x_1}{x_2}\) is

In the following reaction, compound $\mathrm{Q}$ is obtained from compound $\mathrm{P}$ via an ionic intermediate.

What is the degree of unsaturation of $\mathrm{Q}$?

Suppose that $\overrightarrow{p}, \overrightarrow{q} and \overrightarrow{r}$ are three non - coplanar vectors in ${\mathbb{R}}^3$ . Let the components of a vector $\overrightarrow{s}$ along \(p, q\) and \(r\) be 4,3 and 5 respectively. If the components of $\left(-\overrightarrow{p}+ \overrightarrow{q}+ \overrightarrow{r}\right), \left(\overrightarrow{p}- \overrightarrow{q}+ \overrightarrow{r}\right) and \left(- \overrightarrow{p}- \overrightarrow{q}+ \overrightarrow{r}\right)$ are x, y and z, respectively, then the value of $2x+y+z$ is