An \(\alpha\)-particle and a proton are accelerated from rest by a potential difference of \(100 \mathrm{~V}\). After this, their de Broglie wavelengths are \(\lambda_\alpha\) and \(\lambda_p\) respectively. The ratio \(\frac{\lambda_p}{\lambda_\alpha}\), to the nearest integer, is

An electron and a proton are moving on straight parallel paths with same velocity. They enter a semi-infinite region of uniform magnetic field perpendicular to the velocity. Which of the following statement(s) is/are true?

A human body has a surface area of approximately $1 m^2$ . The normal body temperature is $10 K$ above the surrounding room temperature $T_0$ . Take the room temperature to be $T_0=300 K$ . For $T_0=300 K$ , the value of $\sigma T^4=460\frac{W}{m^2}$ (where $\sigma$ is the Stefan-Boltzmann constant). Which of the following options is/are correct?

A thermodynamic system is taken from an initial state $\mathrm{i}$ with internal energy ${\mathrm{U}}_{\mathrm{i}}\mathrm{}=100\mathrm{}\mathrm{J}$ to the final state f along two different paths iaf and ibf, as schematically shown in the figure. The work done by the system along the paths af, ib and bf are ${\mathrm{w}}_{\mathrm{a}\mathrm{f}}=\mathrm{}200\mathrm{}\mathrm{J},\mathrm{}{\mathrm{W}}_{\mathrm{i}\mathrm{b}}\mathrm{}=\mathrm{}50\mathrm{}\mathrm{J}$ and ${\mathrm{W}}_{\mathrm{b}\mathrm{f}}\mathrm{}=\mathrm{}100\mathrm{}\mathrm{J}$ respectively. The heat supplied to the system along the path iaf, ib and bf are iaf ib ${\mathrm{Q}}_{\mathrm{i}\mathrm{a}\mathrm{f}},\mathrm{}{\mathrm{Q}}_{\mathrm{i}\mathrm{b}}$ and ${\mathrm{Q}}_{\mathrm{b}\mathrm{f}}$ respectively. If the internal energy of the system in the state b is ${\mathrm{U}}_{\mathrm{b}}=200\mathrm{}\mathrm{J}$ and ${\mathrm{Q}}_{\mathrm{i}\mathrm{a}\mathrm{f}}=500\mathrm{}\mathrm{J}$ , the ratio $\frac{{\mathrm{Q}}_{\mathrm{b}\mathrm{f}}}{{\mathrm{Q}}_{\mathrm{i}\mathrm{b}}}\mathrm{}$ is

A point charge $q$ of mass $m$ is suspended vertically by a string of length $l.$ A point dipole of dipole moment $\overrightarrow{p}$ is now brought towards $q$ from infinity so that the charge moves away. The final equilibrium position of the system including the direction of the dipole, the angles and distances is shown in the figure below. If the work done in bringing the dipole to this position is $N\times \left(mgh\right)$, where $g$ is the acceleration due to gravity, then the value of $N$ is ____________ (Note that for three coplanar forces keeping a point mass in equilibrium, $\frac{\mathrm{F}}{\sin \theta }$ is the same for all forces, where $F$ is any one of the forces and $\theta$ is the angle between the other two forces)

A circular wire loop of radius \(R\) is placed in the \(x-y\) plane centered at the origin \(O\). A square loop of side \(a(a< < R)\) having two turns is placed with its centre at \(z=\sqrt{3} R\) along the axis of the circular wire loop, as shown in figure.

The plane of the square loop makes an angle of \(45^{\circ}\) with respect to the \(z\)-axis. If the mutual inductance between the loops is given by \(\frac{\mu_{0} a^{2}}{2^{p / 2} R}\), then the value of \(p\) is

Statement I If there is no external torque on a body about its centre of mass, then the velocity of the centre of mass remains constant.
Statement II The linear momentum of an isolated system remains constant.

Let \(f\) be a real-valued function defined on the interval \((-1,1)\) such that \(e^{-x} f(x)=2+\int_0^x \sqrt{t^4+1} d t, \quad\) for all \(x \in(-1,1)\) and let \(f^{-1}\) be the inverse function of \(f\). Then \(\left(f^{-1}\right)^{\prime}(2)\) is equal to

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The noble gases have closed-shell electronic configuration and are monoatomic gases under normal conditions. The low boiling points of the lighter noble gases are due to weak dispersion forces between the atoms and the absence of other interatomic interactions.
The direct reaction of xenon with fluorine leads to a series of compounds with oxidation numbers \(+2,+4\) and \(+6 . \mathrm{XeF}_4\) reacts violently with water to give \(\mathrm{XeO}_3\). The compounds of xenon exhibit rich stereochemistry and their geometries can be deduced considering the total number of electron pairs in the valence shell.
Question:
Argon is used in are welding because of its

Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a function such that \(f(x+y)=f(x)+f(y)\) for all \(x, y \in \mathbb{R}\), and \(g: \mathbb{R} \rightarrow(0, \infty)\) be a function such that \(g(x+y)=g(x) g(y)\) for all \(x, y \in \mathbb{R}\). If \(f\left(\frac{-3}{5}\right)=12\) and \(g\left(\frac{-1}{3}\right)=2\), then the value of \(\left(f\left(\frac{1}{4}\right)+g(-2)-8\right) g(0)\) is . ________.

Let \(\mathbb{R}\) denote the set of all real numbers. Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be defined by
\(f(x)= \begin{cases}\frac{6 x+\sin x}{2 x+\sin x} & \text { if } x \neq 0 \\ \frac{7}{3} & \text { if } x=0\end{cases}\)
Then which of the following statements is (are) TRUE?

A block $M$ hangs vertically at the bottom end of a uniform rope of constant mass per unit length. The top end of the rope is attached to a fixed rigid support at O. A transverse wave pulse (Pulse 1) of wavelength ${\lambda }_0$ is produced at point O on the rope. The pulse takes time $T_{OA}$ to reach point A. If the wave pulse of wavelength ${\lambda }_0$ is produced at point A (Pulse 2) without disturbing the position of M it takes time $T_{OA}$ to reach point O. Which of the following options is/are correct?

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Copper is the most noble of the first row transition metals and occurs in small deposits in several countries. Ores of copper include chalcanthite \(\left(\mathrm{CuSO}_4 \cdot 5 \mathrm{H}_2 \mathrm{O}\right)\), atacamite \(\left(\mathrm{Cu}_2 \mathrm{Cl}(\mathrm{OH})_3\right)\), cuprite \(\left(\mathrm{Cu}_2 \mathrm{O}\right)\), copper glance \(\left(\mathrm{Cu}_2 \mathrm{~S}\right)\) and malachite \(\left(\mathrm{Cu}_2(\mathrm{OH})_2 \mathrm{CO}_3\right)\). However, 80\% of the world copper production comes from the ore chalcopyrite \(\left(\mathrm{CuFeS}_2\right)\). The extraction of copper from chalcopyrite involves partial roasting, removal of iron and self-reduction.
Question:
Partial roasting of chalcopyrite produces