One mole of an ideal gas undergoes two different cyclic processes $\mathrm{I}$ and $\mathrm{II}$, as shown in the $P-V$ diagrams below. In cycle $\mathrm{I}$, processes $a, b, c$ and $d$ are isobaric, isothermal, isobaric and isochoric, respectively. In cycle $\mathrm{II}$, processes $a\text{'}, b\text{'}, c\text{'}$ and $d\text{'}$ are isothermal, isochoric, isobaric and isochoric, respectively. The total work done during cycle $\mathrm{I}$ is $W_{\mathrm{I}}$ and that during cycle $\mathrm{II}$ is $W_{\mathrm{II}}$. The ratio $\frac{W_{\mathrm{I}}}{W_{\mathrm{II}}}$ is _____.

A train is moving along a straight line with a constant acceleration a. A boy standing in the train throws a ball forward with a speed of \(10 \mathrm{~m} / \mathrm{s}\), at an angle of \(60^{\circ}\) to the horizontal. The boy has to move forward by \(1.15 \mathrm{~m}\) inside the train to catch the ball back at the initial height. The acceleration of the train in \(\mathrm{m} / \mathrm{s}^2\), is

Young's double slit experiment is carried out by using green, red and blue light, one color at a time. The fringe widths recorded are \(\beta_{G}, \beta_{R}\) and \(\beta_{B}\), respectively. Then,

A small mass \(m\) is attached to a massless string whose other end is fixed at \(P\) as shown in the figure. The mass is undergoing circular motion in the \(x-y\) plane with centre at \(O\) and constant angular speed \(\omega\). If the angular momentum of the system, calculated about \(O\) and \(P\) are denoted by \(\vec{L}_{O}\) and \(\vec{L}_{P}\) respectively, then

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The figure shows a surface \(X Y\) separating two transparent media, medium-1 and medium-2. The lines \(a b\) and \(c d\) represent wavefronts of a light wave travelling in medium-1 and incident on \(X Y\). The lines ef and \(g h\) represent wavefronts of the light wave in medium-2 after refraction.
Question:
Speed of light is

A particle of mass $m$ is moving in the $xy$-plane such that its velocity at a point $(x, y)$ is given as $\overrightarrow{v}=\alpha \left(y\stackrel{\wedge }{x}+2x\stackrel{\wedge }{y}\right)$ where $\alpha$ is a non-zero constant. What is the force $\overrightarrow{F}$ acting on the particle?

In a Young's double slit experiment, the separation between the two slits is \(d\) and the wavelength of the light is \(\lambda\). The intensity of light falling on slit 1 is four times the intensity of light falling on slit 2 . Choose the correct choice(s),

A block of mass \(5 \mathrm{~kg}\) moves along the \(x\)-direction subject to the force \(F=(-20 x+10) \mathrm{N}\), with the value of \(x\) in metre. At time \(t=0 \mathrm{~s}\), it is at rest at position \(x=1 \mathrm{~m}\). The position and momentum of the block at \(t=(\pi / 4) \mathrm{s}\) are

For the given compound X, the total number of optically active stereoisomers is ____.

The number of real solutions of the equation ${\sin }^{-1}\left(\sum _{i=1}^{\infty }{x}^{i+1}-x\sum _{i=1}^{\infty }{\left(\frac{x}{2}\right)}^i\right)=\frac{\pi }{2}-{\cos }^{-1}\left(\sum _{i=1}^{\infty }{\left(-\frac{x}{2}\right)}^i-\sum _{i=1}^{\infty }{\left(-x\right)}^i\right)$ lying in the interval $\left(-\frac{1}{2},\frac{1}{2}\right)$ is____.
(Here, the inverse trigonometric functions ${\sin }^{-1}x and {\cos }^{-1}x$ assume values in $\left[-\frac{\pi }{2},\frac{\pi }{2}\right] \mathrm{a}\mathrm{n}\mathrm{d} \left[0,\pi \right]$
respectively.)

For the reaction, \(\mathbf{X}(s) \rightleftharpoons \mathbf{Y}(s)+\mathbf{Z}(g)\), the plot of \(\ln \frac{p_{\mathbf{Z}}}{p^{\theta}}\) versus \(\frac{10^{4}}{T}\) is given below (in solid line), where \(p_{\mathbf{Z}}\) is the pressure (in bar) of the gas \(\mathbf{Z}\) at temperature \(T\) and \(p^{\theta}=1\) bar.

(Given, \(\frac{\mathrm{d}(\ln K)}{\mathrm{d}\left(\frac{1}{T}\right)}=-\frac{\Delta H^{\theta}}{R}\), where the equilibrium constant, \(K=\frac{p_{z}}{p^{\theta}}\) and the gas constant, \(\mathrm{R}=8.314 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\) )
The value of $∆{\mathrm{S}}^{\mathrm{o}}$ (in $\mathrm{kJ} {\mathrm{mol}}^{-1}$) for the given reaction, at $1000 \mathrm{K}$ is ___.

Under hydrolytic conditions, the compounds used for the preparation of linear polymer and for chain termination, respectively, are

The increasing order of atomic radii of the following group $13$ elements is