A vibrating string of certain length \(l\) under a tension \(T\) resonates with a mode corresponding to the first overtone (third harmonic) of an air column of length \(75 \mathrm{~cm}\) inside a tube closed at one end. The string also generates 4 beat/s when excited along with a tuning fork of frequency \(n\). Now when the tension of the string is slightly increased the number of beats reduces to 2 per second. Assuming the velocity of sound in air to be \(340 \mathrm{~m} / \mathrm{s}\), the frequency \(n\) of the tuning fork (in \(\mathrm{Hz}\) ) is

In order to measure the internal resistance $r_1$ of a cell of emf $\epsilon ,$ a meter bridge of wire resistance $R_0=50 \mathrm{\Omega }$, a resistance $\frac{R_0}{2}$, another cell of emf $\frac{\epsilon }{2}$ (internal resistance $r$) and a galvanometer $G$ are used in a circuit, as shown in the figure. If the null point is found at $l=72 \mathrm{cm},$ then the value of $r_1=\_\_\_\_\mathrm{\Omega }.$

A plane polarized blue light ray is incident on a prism such that there is no reflection from the surface of the prism. The angle of deviation of the emergent ray is $\delta ={60}^{\mathrm{o}}$ (see Figure-$1$). The angle of minimum deviation for red light from the same prism is ${\delta }_{\min }={30}^{\mathrm{o}}$ (see Figure-$2$). The refractive index of the prism material for blue light is $\sqrt{3}$. Which of the following statement(s) is(are) correct?

A point source \(\mathrm{S}\) emits unpolarized light uniformly in all directions. At two points \(\mathrm{A}\) and \(\mathrm{B}\), the ratio \(r=I_A / I_B\) of the intensities of light is 2. If a set of two polaroids having \(45^{\circ}\) angle between their pass-axes is placed just before point \(\mathrm{B}\), then the new value of \(r\) will be ______

A light beam is travelling from Region I to Region IV (refer figure). The refractive index in Region I, II, III and IV are \(n_0, \frac{n_0}{2}, \frac{n_0}{6}\) and \(\frac{n_0}{8}\), respectively. The angle of incidence \(\theta\) for which the beam just misses entering Region IV is

A cylindrical furnace has height $\left(H\right)$ and diameter $\left(D\right)$ both $1\mathrm{m}$. It is maintained at temperature $360\mathrm{K}$. The air getsheated inside the furnace at constant pressure$P_a$and its temperature becomes $T=360\mathrm{K}$. The hot air with density $\rho$rises up a vertical chimney of diameter $d=0.1\mathrm{m}$ and height $h=9\mathrm{m}$ above the furnace and exits the chimney (seethe figure). As a result, atmospheric air of density${\rho }_a=1.2\mathrm{kg}{\mathrm{m}}^{-3}$,pressure $P_a$and temperature $T_a=300\mathrm{K}$ entersthe furnace. Assume air as an ideal gas, neglect the variations in$\rho$and $T$ inside the chimney and the furnace. Also ignore the viscous effects.
[Given: The acceleration due to gravity$g=10\mathrm{m}{\mathrm{s}}^{-2}$and$\pi =3.14$]


When the chimney is closed using a cap at the top, a pressure difference $∆P$develops between the top and the bottom surfaces of the cap. If the changes in the temperature and density of the hot air, due to the stoppage of air flow, are negligible then the value of $∆P$is _____ $\mathrm{N}{\mathrm{m}}^{-2}$.

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

In an experiment for determination of the focal length of a thin convex lens, the distance of the object from the lens is $10\pm 0.1 \mathrm{cm}$ and the distance of its real image from the lens is $20\pm 0.2 \mathrm{cm}$. The error in the determination of focal length of the lens is $n\%$. The value of $n$ is _____.

Paragraph:
A small spherical monoatomic ideal gas bubble \(\left(\gamma=\frac{5}{3}\right)\) is trapped inside a liquid of density \(\rho_1\) (see figure). Assume that the bubble does not exchange any heat with the liquid. The bubble contains \(n\) moles of gas. The temperature of the gas when the bubble is at the bottom is \(T_0\), the height of the liquid is \(H\) and the atmospheric pressure is \(p_0\) (Neglect surface tension).

Question:
When the gas bubble is at height \(y\) from the bottom, its temperature is

\[
\text { Match the conics in Column I with the statements/expressions in Column II. }
\]

Paragraph:
p-amino-N, N-dimethylaniline is added to a strongly acidic solution of \(X\). The resulting solution is treated with a few drops of aqueous solution of \(Y\) to yield blue colouration due to the formation of methylene blue. Treatment of the aqueous solution of \(Y\) with the reagent potassium hexacyanoferrate(II) leads to the formation of an intense blue precipitate. The precipitate dissolves an excess addition of the reagent. Similarly, treatment of the solution of \(Y\) with the solution of potassium hexacyanoferrate(III) leads to a brown colouration due to the formation of \(Z\).
Question:
The compound \(X\), is

The value of \(n\) in the molecular formula \(\mathrm{Be}_n \mathrm{Al}_2 \mathrm{Si}_6 \mathrm{O}_{18}\) is

A radioactive sample \(S_1\) having an activity of \(5 \mu \mathrm{Ci}\) has twice the number of nuclei as another sample \(S_2\) which has an activity of \(10 \mu \mathrm{Ci}\). The half lives of \(S_1\) and \(S_2\) can be