A person of height $1.6 \mathrm{m}$ is walking away from a lamp post of height $4 \mathrm{m}$ along a straight path on the flat ground. The lamp post and the person are always perpendicular to the ground. If the speed of the person is $60 \mathrm{cm} {\mathrm{s}}^{-1}$, the speed of the tip of the person’s shadow on the ground with respect to the person is _____ $\mathrm{cm} {\mathrm{s}}^{-1}$.

A point mass of \(1 \mathrm{~kg}\) collides elastically with a stationary point mass of \(5 \mathrm{~kg}\). After their collision, the \(1 \mathrm{~kg}\) mass reverses its direction and moves with a speed of \(2 \mathrm{~ms}^{-1}\). Which of the following statement(s) is/are correct for the system of these two masses?

The figure shows a circuit having eight resistances of $1 \mathrm{\Omega }$ each, labelled $R_1$ to $R_8$, and two ideal batteries with voltages ${\epsilon }_1=12 \mathrm{V}$ and ${\epsilon }_2=6 \mathrm{V}$.

Which of the following statement(s) is(are) correct?

A monochromatic light is incident from air on a refracting surface of a prism of angle ${75}^{\mathrm{o}}$ and refractive index ${\mathrm{n}}_0=\sqrt{3}.$ The other refracting surface of a prism is coated by a thin film of material of refractive index $\mathrm{n}$ as shown in figure. The light suffers total internal reflection at the coated prism surface for an incidence angle of $\mathrm{\theta }\le {60}^{\mathrm{o}}.$ The value of ${\mathrm{n}}^2$ is_______.

A heavy nucleus $N$, at rest, undergoes fission $N\to P+Q,$ where $P$ and $Q$ are two lighter nuclei. Let $\delta =M_N-M_P-M_Q$, where $M_P, M_Q$ and $M_N$ are the masses of $P,Q$ and $N$, respectively. $E_P$ and $E_Q$ are the kinetic energies of $P$ and $Q$, respectively. The speeds of $P$ and $Q$ are $v_P$ and $v_Q$, respectively. If $c$ is the speed of light, which of the following statement(s) is (are) correct?

A source, approaching with speed $u$ towards the open end of a stationary pipe of length $L$, is emitting a sound of frequency $f_s$. The farther end of the pipe is closed. The speed of sound in air is $v$ and $f_0$ is the fundamental frequency of the pipe. For which of the following combination(s) of $u$ and $f_s$, will the sound reaching the pipe lead to a resonance?

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

A monochromatic light is travelling in a medium of refractive index $n=1.6$ . It enters a stack of glass layers from the bottom side at an angle $\theta =30°.$ The interfaces of the glass layers are parallel to each other. The refractive indices of different glass layers are monotonically decreasing as $n_m=n–m\mathrm{\Delta }n$ where $n_m$ is the refractive index of the $m^{th}$ slab and $\mathrm{\Delta }n=0.1$ (see the figure). The ray is refracted out parallel to the interface between the ${\left(m–1\right)}^{th}$ and $m^{th}$ slabs from the right side of the stack. What is the value of m?

A student performed the experiment of determination of focal length of a concave mirror by \(u-v\) method using an optical bench of length \(1.5 \mathrm{~m}\). The focal length of the mirror used is 24 \(\mathrm{cm}\). The maximum error in the location of the image can be \(0.2 \mathrm{~cm}\).
The 5 sets of \((u, v)\) values recorded by the student (in \(\mathrm{cm})\) are : \((42,56),(48\), \(48),(60,40),(66,33),(78,39)\). The data set(s) that cannot come from experiment and is (are) incorrectly recorded, is(are)

For $x\in \mathrm{ℝ}$, let $y\left(x\right)$ be a solution of the differential equation $\left(x^2-5\right)\frac{dy}{dx}-2xy=-2x{\left(x^2-5\right)}^2$ such that $y\left(2\right)=7$. Then the maximum value of the function $y\left(x\right)$ is

For the following electrochemical cell at $\text{289 K,}$
\(\operatorname{Pt}( s ) \mid H _2(g, 1\) bar \() \mid H ^{+}( aq , 1 M ) \| M ^{4+}(\) aq. \(), M ^{2+}(\) aq. \() \mid \operatorname{Pt}( s )\)
${\mathrm{E}}_{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}=0.092\mathrm{ }\mathrm{V}$ when $\frac{\left[{\mathrm{M}}^{2+}\left(\mathrm{a}\mathrm{q}.\right)\right]}{\left[{\mathrm{M}}^{4+}\left(\mathrm{a}\mathrm{q}.\right)\right]}={10}^{\mathrm{x}}$
Given: ${\mathrm{E}}_{{\mathrm{M}}^{4+}/{\mathrm{M}}^{2+}}^0=0.151\mathrm{ }\mathrm{V}\mathrm{ };2.303\frac{\mathrm{R}\mathrm{T}}{\mathrm{F}}=0.059$
The value of $\text{x}$ is -

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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:
\(\mathrm{XeF}_4\) and \(\mathrm{XeF}_6\) are expected to be

For a solution formed by mixing liquids $\text{L}$ and $\text{M}$, the vapour pressure of $\text{L}$ plotted against the mole fraction of $\text{M}$ in solution is shown in the following figure, Here, ${\mathrm{X}}_{\mathrm{L}}$ and ${\mathrm{X}}_{\mathrm{M}}$ represent mole fractions of $\text{L}$ and $\text{M}$, respectively, in the solution. The correct statement(s) applicable to this system is(are) -