A field line is shown in the figure. This field cannot represent

In a radioactive sample, ${\mathrm{}}_{19}^{40}\mathrm{K}$ nuclei either decay into stable ${\mathrm{}}_{20}^{40}\mathrm{C}\mathrm{a}$ nuclei with decay constant $4.5\times {10}^{-10}$ per year or into stable ${\mathrm{}}_{18}^{40}\mathrm{A}\mathrm{r}$ nuclei with decay constant $0.5\times {10}^{-10}$ per year. Given that in this sample all the stable ${\mathrm{}}_{20}^{40}\mathrm{C}\mathrm{a}$ and ${\mathrm{}}_{18}^{40}\mathrm{A}\mathrm{r}$ nuclei are produced by the ${\mathrm{}}_{19}^{40}\mathrm{K}$ nuclei only. In time $\mathrm{t}\times {10}^9$ years, if the ratio of the sum of stable ${\mathrm{}}_{20}^{40}\mathrm{C}\mathrm{a}$ and ${\mathrm{}}_{18}^{40}\mathrm{A}\mathrm{r}$ nuclei to the radioactive ${\mathrm{}}_{19}^{40}\mathrm{K}$ nuclei is $99,$ the value of t will be : [Given $\mathrm{l}\mathrm{n}\mathrm{}10=2.3$ ]

\(5.6 \mathrm{~L}\) of helium gas at STP is adiabatically compressed to \(0.7 \mathrm{~L}\). Taking the initial temperature to be \(T_1\), the work done in the process is

Electrons with de-Broglie wavelength \(\lambda\) fall on the target in an X-ray tube. The cut-off wavelength of the emitted X-rays is

A rod of length $2 \mathrm{cm}$ makes an angle $\frac{2\pi }{3}\mathrm{rad}$ with the principal axis of a thin convex lens. The lens has a focal length of $10 \mathrm{cm}$ and is placed at a distance of $\frac{40}{3} \mathrm{cm}$ from the object as shown in the figure. The height of the image is $\frac{30\sqrt{3}}{13} \mathrm{cm}$ and the angle made by it with respect to the principal axis is \(\alpha\) rad. The value of $\alpha$ is $\frac{\pi }{n}\mathrm{rad}$, where $n$ is

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

Two capacitors with capacitance values $C_1=\left(2000\pm 10\right) \mathrm{pF}$  and $C_2=\left(3000\pm 15\right) \mathrm{pF}$ are connected in series. The voltage applied across this combination is $V=\left(5.00\pm 0.02\right) \mathrm{V}$. The percentage error in the calculation of the energy stored in this combination of capacitors is __________.

Consider $4$ boxes, where each box contains $3$ red balls and $2$ blue balls. Assume that all $20$ balls are distinct. In how many different ways can $10$ balls be chosen from these $4$ boxes so that from each box at least one red ball and one blue ball are chosen?

Among the triatomic molecules/ions, $\mathrm{B}\mathrm{e}\mathrm{C}{\mathrm{l}}_2,\mathrm{ }{\mathrm{N}}_3^{-},\mathrm{ }{\mathrm{N}}_2\mathrm{O},\mathrm{ }{\mathrm{N}\mathrm{O}}_2^{+},\mathrm{ }{\mathrm{O}}_3,\mathrm{S}\mathrm{C}{\mathrm{l}}_2,\mathrm{ }{\mathrm{I}\mathrm{C}\mathrm{l}}_2^{-},\mathrm{ }{\mathrm{I}}_3^{-}$ and $\mathrm{X}\mathrm{e}{\mathrm{F}}_2$ , the total number of linear molecule(s)/ion(s) where the hybridization of the central atom does not have contribution from the d-orbital(s) is
[Atomic number: S = 16, Cl = 17, I = 53 and Xe = 54]

The reaction of ${\mathrm{HClO}}_3$ with $\mathrm{HCl}$ gives a paramagnetic gas, which upon reaction with ${\mathrm{O}}_3$ produces

Statement I The curve \(y=-\frac{x^2}{2}+x+1\) is symmetric with respect to the line \(x=1\)
Statement II A parabola is symmetric about its axis.

Paragraph:
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

Let $f:\left(0, \pi \right)\to \mathbb{R}$ be a twice differentiable function such that $\underset{t\to x}{\lim }\frac{f\left(x\right)\sin t-f\left(t\right)\sin x}{t-x}={\sin }^{2}x$ for all $x\in \left(0, \pi \right)$. If $f\left(\frac{\pi }{6}\right)=-\frac{\pi }{12}$, then which of the following statement(s) is (are) TRUE ?