A train $\mathrm{S}1,$ moving with a uniform velocity of $108\mathrm{}\mathrm{k}\mathrm{m}/\mathrm{h},$ approaches another train $\mathrm{S}2$ standing on a platform. An observer $\mathrm{O}$ moves with a uniform velocity of $36\mathrm{}\mathrm{k}\mathrm{m}/\mathrm{h}$ towards $\mathrm{S}2,$ as shown in figure. Both the trains are blowing whistles of same frequency $120\mathrm{}\mathrm{H}\mathrm{z}.$ When $\mathrm{O}$ is $600\mathrm{}\mathrm{m}$ away from $\mathrm{S}2$ and distance between $\mathrm{S}1$ and $\mathrm{S}2$ is $800\mathrm{}\mathrm{m},$ the number of beats heard by $\mathrm{O}$ is____
[Speed of the sound $=330\mathrm{}\mathrm{m}/\mathrm{s}]$

The work done on a particle of mass m by a force $\text{K}\left[\frac{\text{x}}{{\left({\text{x}}^2+{\text{y}}^2\right)}^{3/2}}\widehat{\text{i}}+\frac{\text{y}}{{\left({\text{x}}^2+{\text{y}}^2\right)}^{3/2}}\widehat{\text{j}}\right]$ (K being a constant of appropriate dimensions, when the particle is taken from the point (a,0) to the point (0,a) along a circular path of radius a about the origin in the x-y plane is

A few electric field lines for a system of two charges \(Q_1\) and \(Q_2\) fixed at two different points on the \(x\)-axis are shown in the figure. These lines suggest that

In the following circuit $C_1=12 \mathrm{\mu F},C_2=C_3=4 \mathrm{\mu F}$ and $C_4=C_5=2 \mathrm{\mu F}$. The charge stored in $C_3$ is $\mathrm{\mu C}.$

An ideal monoatomic gas is confined in a horizontal cylinder by a spring loaded piston (as shown in the figure).Initially the gas is at temperature ${\mathrm{T}}_1$ , pressure ${\mathrm{P}}_1$ and volume ${\mathrm{V}}_1$ and the spring is in its relaxed state. The gas is then heated very slowly to temperature ${\mathrm{T}}_2$ , pressure ${\mathrm{P}}_2$ and volume ${\mathrm{V}}_2$ . During this process the piston moves out by a distance $\mathrm{x}$ . Ignoring the friction between the piston the cylinder, the correct statement(s) is (are)

Put a uniform meter scale horizontally on your extended index fingers with the left one at $0.00 \mathrm{cm}$ and the right one at $90.00 \mathrm{cm}$. When you attempt to move both the fingers slowly towards the center, initially only the left finger slips with respect to the scale and the right finger does not. After some distance, the left finger stops and the right one starts slipping. Then the right finger stops at a distance $x_R$ from the center $(50.00 \mathrm{cm})$ of the scale and the left one starts slipping again. This happens because of the difference in the frictional forces on the two fingers. If the coefficients of static and dynamic friction between the fingers and the scale are $0.40$ and $0.32$, respectively, the value of $x_R$ (in cm) is___________.

Answer the following by appropriately matching the lists based on the information given in the paragraph.
A musical instrument is made using four different metal strings, $1,\mathrm{ }2,\mathrm{ }3$ and $4$ with mass per unit length $\mathrm{\mu },2\mathrm{\mu },3\mathrm{\mu }$ and $4\mathrm{\mu }$ respectively. The instrument is played by vibrating the strings by varying the free length in between the range ${\mathrm{L}}_0$ and $2{\mathrm{L}}_0.$ It is found that in string- $1\left(\mathrm{\mu }\right)$ at free length ${\mathrm{L}}_0$ and tension ${\mathrm{T}}_0$ the fundamental mode frequency is ${\mathrm{f}}_0.$
List-I gives the above four strings while list-II lists the magnitude of some quantity.

	List I		List II
$\left(\mathrm{a}\right)$	String $-1\left(\mathrm{\mu }\right)$	$\left(\mathrm{p}\right)$	$1$
$\left(\mathrm{b}\right)$	String $-2\left(2\mathrm{\mu }\right)$	$\left(\mathrm{q}\right)$	$\frac{1}{2}$
$\left(\mathrm{c}\right)$	String $-3\left(3\mathrm{\mu }\right)$	$\left(\mathrm{r}\right)$	$\frac{1}{\sqrt{2}}$
$\left(\mathrm{d}\right)$	String $-4\left(4\mathrm{\mu }\right)$	$\left(\mathrm{s}\right)$	$\frac{1}{\sqrt{3}}$
		$\left(\mathrm{t}\right)$	$\frac{3}{16}$
		$\left(\mathrm{u}\right)$	$\frac{1}{16}$

The length of the strings $1,\mathrm{ }2,\mathrm{ }3$ and $4$ are kept fixed at ${\mathrm{L}}_0,\frac{3{\mathrm{L}}_0}{2},\frac{5{\mathrm{L}}_0}{4}$ and $\frac{7{\mathrm{L}}_0}{4},$ respectively. Strings $1,\mathrm{ }2,\mathrm{ }3$ and $4$ are vibrated at their $1^{\mathrm{s}\mathrm{t}},\mathrm{ }{3}^{\mathrm{r}\mathrm{d}},\mathrm{ }{5}^{\mathrm{t}\mathrm{h}}$ and ${14}^{\mathrm{t}\mathrm{h}}$ harmonics, respectively such that all the strings have same frequency. The correct match for the tension in the four strings in the units of ${\mathrm{T}}_0$ will be.

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 ___.

For \(r=0,1, \ldots, 10\), let \(A_r, B_r\) and \(C_r\) denote, respectively, the coefficient of \(x^r\) in the expansions of \((1+x)^{10}\), \((1+x)^{20}\) and \((1+x)^{30}\). Then \(\sum_{r=1}^{10} A_r\left(B_{10} B_r-C_{10} A_r\right)\) is equal to

The reaction of white phosphorus with aqueous \(\mathrm{NaOH}\) gives phosphine along with another phosphorus containing compound. The reaction type; the oxidation states of phosphorus in phosphine and the other product are respectively

The acidic hydrolysis of ether (X) shown below is fastest when

Paragraph:

Schemes \(1\) and \(2\) describe sequential transformation of alkynes \(\mathbf{M}\) and \(\mathbf{N}\). Consider only the major products formed in each step for both the schemes.



Question:
The correct statement with respect to product \(\mathbf{Y}\) 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 $\mathrm{Q}$?