A projectile is thrown from a point \(\mathrm{O}\) on the ground at an angle \(45^{\circ}\) from the vertical and with a speed \(5 \sqrt{2} \mathrm{~m} / \mathrm{s}\). The projectile at the highest point of its trajectory splits into two equal parts. One part falls vertically down to the ground, \(0.5 \mathrm{~s}\) after the splitting. The other part, \(t\) seconds after the splitting, falls to the ground at a distance \(x\) meters from the point O. The acceleration due to gravity \(g=10 \mathrm{~m} / \mathrm{s}^{2}\).
The value of $t$ is _________.

Three very large plates of same area are kept parallel and close to each other. They are considered as ideal black surfaces and have very high thermal conductivity. The first and third plates are maintained at temperatures \(2 T\) and \(3 T\) respectively. The temperature of the middle (i.e. second) plate under steady state condition is

As shown schematically in the figure, two vessels contain water solutions (at temperature $T$) of potassium permanganate $\left({\mathrm{KMnO}}_4\right)$ of different concentrations $n_1$ and $n_2\left(n_1>n_2\right)$ molecules per unit volume with $\mathrm{\Delta }n=\left(n_1-n_2\right)\ll n_1.$ When they are connected by a tube of small length $l$ and cross-sectional area $S,{\mathrm{KMnO}}_4$ starts to diffuse from the left to the right vessel through the tube. Consider the collection of molecules to behave as dilute ideal gases and the difference in their partial pressure in the two vessels causing the diffusion. The speed $v$ of the molecules is limited by the viscous force $-\beta v$ on each molecule, where $\beta$ is a constant. Neglecting all terms of the order ${\left(\mathrm{\Delta }n\right)}^2$. Which of the following is/are correct? ($k_B$ is the Boltzmann constant)

A rod of mass $m$ and length $L,$ pivoted at one of its ends, is hanging vertically. A bullet of the same mass moving at speed $v$ strikes the rod horizontally at a distance $x$ from its pivoted end and gets embedded in it. The combined system now rotates with an angular speed $\omega$ about the pivot. The maximum angular speed ${\omega }_{\mathrm{M}}$ is achieved for $x=x_{\mathrm{M}}$. Then

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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

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 rigid uniform bar $AB$ of length $L$ is slipping from its vertical position on a frictionless floor (as shown in the figure). At some instant of time, the angle made by the bar with the vertical is $\mathrm{\theta }$. Which of the following statements about its motion is correct?

The substituents \(R_{1}\) and \(R_{2}\) for nine peptides are listed in the table given below. How many of these peptides are positively charged at \(\mathrm{pH}=7.0\) ?

\(\begin{array}{|c|c|c|}\hline Peptide & \mathbf{R}_{\mathbf{1}} & \mathbf{R}_{\mathbf{2}} \\\hline I & \mathrm{H} & \mathrm{H} \\\hline II & \mathrm{H} & \mathrm{CH}_{3} \\\hline III & \mathrm{CH}_{2} \mathrm{COOH} & \mathrm{H} \\\hline IV & \mathrm{CH}_{2} \mathrm{CONH}_{2} & \left(\mathrm{CH}_{2}\right)_{4} \mathrm{NH}_{2} \\\hline V & \mathrm{CH}_{2} \mathrm{CONH}_{2} & \mathrm{CH}_{2} \mathrm{CONH}_{2} \\\hline VI & \left(\mathrm{CH}_{2}\right)_{4} \mathrm{NH}_{2} & \left(\mathrm{CH}_{2}\right)_{4} \mathrm{NH}_{2} \\\hline VII & \mathrm{CH}_{2} \mathrm{COOH} & \mathrm{CH}_{2} \mathrm{CONH}_{2} \\\hline VIII & \mathrm{CH}_{2} \mathrm{OH} & \left(\mathrm{CH}_{2}\right)_{4} \mathrm{NH}_{2} \\\hline IX & \left(\mathrm{CH}_{2}\right)_{4} \mathrm{NH}_{2} & \mathrm{CH}_{3} \\\hline\end{array}\)

The reaction of ${\mathrm{Xe}}_{}$ and ${\mathrm{O}}_2{\mathrm{F}}_2$ gives a $\mathrm{Xe}$ compound $\mathrm{P}$. The number of moles of $\mathrm{HF}$ produced by the complete hydrolysis of $1 \mathrm{mol}$ of $\mathrm{P}$ is _____ .

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A small block of mass \(M\) moves on a frictionless surface of an inclined plane, as shown in figure. The angle of the incline suddenly changes from \(60^{\circ}\) to \(30^{\circ}\) at point \(B\). The block is initially at rest at \(A\). Assume that collisions between the block and the incline are totally inelastic \(\left(g=10 \mathrm{~m} / \mathrm{s}^2\right)\)

Question:
The speed of the block at point \(B\) immediately after it strikes the second incline is

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The coordination number of \(\mathrm{Ni}^{2+}\) is 4 .
\(\mathrm{NiCl}_2+\mathrm{KCN}\) (excess) \(\rightarrow A\) (cyano complex)
\(\mathrm{NiCl}_2+\) conc. \(\mathrm{HCl}\) (excess) \(\rightarrow B\) (chloro complex)
Question:
The IUPAC name of \(A\) and \(B\) are

The correct option(s) regarding the complex ${\left[Co\left(en\right){\left(N{H}_3\right)}_3\left(H_{2}O\right)\right]}^{3+}$ :-
(en $= H_{2}NC{H}_{2}C{H}_{2}N{H}_2$ ) is (are)

For a radioactive material, its activity $\mathrm{A}$ and rate of change of its activity $\mathrm{R}$ are defined as $\mathrm{A}=-\frac{\mathrm{d}\mathrm{N}}{\mathrm{d}\mathrm{t}}$ and $\mathrm{R}=-\frac{\mathrm{d}\mathrm{A}}{\mathrm{d}\mathrm{t}}$ , where $\mathrm{N}\left(\mathrm{t}\right)$ is the number of nuclei at time $\mathrm{t}$ . Two radioactive sources $\mathrm{P}$ (mean life $\mathrm{\tau }$ ) and $\mathrm{Q}$ (mean life $2\mathrm{\tau }$ ) have the same activity at $\mathrm{t}=0$ . Their rates of change of activities at $\mathrm{t}=2\mathrm{\tau }$ are ${\mathrm{R}}_{\mathrm{P}}$ and ${\mathrm{R}}_{\mathrm{Q}}$ , respectively. If $\frac{{\mathrm{R}}_{\mathrm{P}}}{{\mathrm{R}}_{\mathrm{Q}}}=\frac{\mathrm{n}}{\mathrm{e}}$ , then the value of $\mathrm{n}$ is