A student uses a simple pendulum of exactly \(1 \mathrm{~m}\) length to determine \(g\), the acceleration due to gravity. He uses a stop watch with the least count of \(1 \mathrm{~s}\) for this and records \(40 \mathrm{~s}\) for 20 oscillations. For this observation, which of the following statement(s) is/are true?

A large glass slab \(\left(\mu=\frac{5}{3}\right)\) of thickness 8 \(\mathrm{cm}\) is placed over a point source of light on a plane surface. It is seen that light emerges out of the top surface of the slab from a circular area of radius \(R \mathrm{~cm}\). What is the value of \(R\) ?

Two conducting cylinders of equal length but different radii are connected in series between two heat baths kept at temperatures $T_1=300\mathrm{K}\mathrm{a}\mathrm{n}\mathrm{d}{T}_2=100\mathrm{K}$as shown in the figure. The radius of the bigger cylinder is twice that of the smaller one and the thermal conductivities of the materials of the smaller and the larger cylinders are $K_1\mathrm{a}\mathrm{n}\mathrm{d}{K}_2$, respectively. If the temperature at the junction of the two cylinders in the steady state is $200\mathrm{}\mathrm{K}$, then $K_1/K_2$is

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\(P_{14-16}\) : Paragraph for Questions Nos. 14 to 16 A fixed thermally conducting cylinder has a radius \(R\) and height \(L_0\). The cylinder is open at its bottom and has a small hole at its top. A piston of mass \(M\) is held at a distance \(L\) from the top surface as shown in the figure. The atmospheric pressure is \(p_0\).

Question :
The piston is taken completely out of the cylinder. The hole at the top is sealed. A water tank is brought below the cylinder and put in a position so that the water surface in the tank is at the same level as the top of the cylinder as shown in the figure. The density of the water is \(\rho\). In equilibrium, the height \(H\) of the water column in the cylinder satisfies

A flat plate is moving normal to its plane through a gas under the action of a constant force F. The gas is kept at a very low pressure. The speed of the plate v I much less than the average speed u of the gas molecules. Which of the following options is true?

Two vectors $\overrightarrow{A} \mathrm{a}\mathrm{n}\mathrm{d} \overrightarrow{B}$ are defined as $\overrightarrow{A}=a\widehat{i} and \overrightarrow{B}=a\left(\cos \omega t \widehat{i}+\sin \omega t\widehat{j}\right)$, where a is a constant and $\omega =\frac{\pi }{6} rad s^{-1}.$ If $\left|\overrightarrow{A}+\overrightarrow{B}\right|=\sqrt{3}\left|\overrightarrow{A}-\overrightarrow{B}\right|$ at time $t=\tau$ for the first time, the value of $\tau$ in seconds, is_______

A small block is connected to one end of a massless spring of un-stretched length \(4.9 \mathrm{~m}\). The other end of the spring (see the figure) is fixed. The system lies on a horizontal frictionless surface. The block is stretched by \(0.2 \mathrm{~m}\) and released from rest at \(t=0\). It then executes simple harmonic motion with angular frequency \(\omega=\pi / 3 \mathrm{rad} / \mathrm{s}\). Simultaneously at \(t=0\), a small pebble is projected with speed \(v\) form point \(P\) at an angle of \(45^{\circ}\) as shown in the figure. Point \(P\) is at a horizontal distance of \(10 \mathrm{~m}\) from \(O\). If the pebble hits the block at \(t=1 \mathrm{~s}\), the value of \(v\) is (take \(g\) \(=10 \mathrm{~m} / \mathrm{s}^{2}\) )

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.

Let $f: \left(0, \infty \right)\to R$ be a differentiable function such that $f^{\text{'}}\left(x\right)=2-\frac{f\left(x\right)}{x}\forall x\in (0, \infty )$ and $f\left(1\right)\ne 1$. Then,

Adsorption of phenol from its aqueous solution on to fly ash obeys Freundlich isotherm. At a given temperature, from \(10 \mathrm{mg} \mathrm{g}^{-1}\) and \(16 \mathrm{mg} \mathrm{g}^{-1}\) aqueous phenol solutions, the concentrations of adsorbed phenol are measured to be \(4 \mathrm{mg} \mathrm{g}^{-1}\) and \(10 \mathrm{mg} \mathrm{g}^{-1}\), respectively. At this temperature, the concentration (in \(\mathrm{mg} \mathrm{g}^{-1}\) ) of adsorbed phenol from \(20 \mathrm{mg} \mathrm{g}^{-1}\) aqueous solution of phenol will be \(\qquad\) .
Use : \(\log _{10} 2=0.3\)

The correct statement(s) regarding,
(i) $\mathrm{H}\mathrm{C}\mathrm{l}\mathrm{O}$
(ii) $\mathrm{H}\mathrm{C}\mathrm{l}{\mathrm{O}}_2$
(iii) $\mathrm{H}\mathrm{C}\mathrm{l}{\mathrm{O}}_3$ and
(iv) $\mathrm{H}\mathrm{C}\mathrm{l}{\mathrm{O}}_4$ is /are

A monoatomic ideal gas undergoes a process in which the ratio of \(P\) to \(V\) at any instant is constant and equals to 1 . What is the molar heat capacity of the gas?

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 ___.