The positon vector \(\vec{r}\) of a particle of mass m is given by the following equation \(\vec{r}(t)=\alpha t^3 \hat{i} +\beta t^2 \hat{j}\), where \(\alpha=\frac{10}{3} m s^{-3}, \beta=5 m s^{-2}\) and \(m=0.1 kg\). At \(t=1 s\), which of the following statements (s) is (are) true about the particle?

The filament of a light bulb has surface area $64 {\mathrm{mm}}^2.$ The filament can be considered as a black body at temperature $2500 \mathrm{K}$ emitting radiation like a point source when viewed from far. At night the light bulb is observed from a distance of $100 \mathrm{m}$. Assume the pupil of the eyes of the observer to be circular with radius $3 \mathrm{mm}$. Then: (Take Stefan-Boltzmann constant $=5.67\times {10}^{-8} \mathrm{W} {\mathrm{m}}^{-2} {\mathrm{K}}^{-4},$ Wiens' displacement constant $=2.90\times {10}^{-3} \mathrm{m} \mathrm{K}$, Planck's constant $=6.60\times {10}^{-34} \mathrm{J} \mathrm{s}$, speed of light in vacuum $=3.00\times {10}^8 \mathrm{m}{\mathrm{s}}^{-1}$)

A disk of radius \(\frac{a}{4}\) having a uniformly distributed charge \(6 \mathrm{C}\) is placed in the \(x-y\) plane with its centre at \(\left(\frac{-a}{2}, 0,0\right)\). A rod of length a carrying a uniformly distributed charge \(8 \mathrm{C}\) is placed on the \(x\)-axis from \(x=\frac{a}{4}\) to \(x=\frac{5 a}{4}\). Two point charges \(-7 \mathrm{C}\) and \(3 \mathrm{C}\) are placed at \(\left(\frac{a}{4}, \frac{-a}{4}, 0\right)\) and \(\left(\frac{-3 a}{4}, \frac{3 a}{4}, 0\right)\), respectively. Consider a cubical surface formed by six surfaces \(x=\pm \frac{a}{2}, y=\pm \frac{a}{2}, z=\pm \frac{a}{2}\).
The electric flux through this cubical surface is

A long, hollow conducting cylinder is kept coaxially inside another long, hollow conducting cylinder of larger radius. Both the cylinders are initially electrically neutral.

An ideal gas is in thermodynamic equilibrium. The number of degrees of freedom of a molecule of the gas is $n$. The internal energy of one mole of the gas is $U_n$ and the speed of sound in the gas is $v_n$. At a fixed temperature and pressure, which of the following is the correct option?

A photoelectric material having work-function ${\varphi }_0$ is illuminated with a light of wavelength $\lambda \left(\lambda <\frac{hc}{{\varphi }_0}\right)$. The fastest photoelectron has a de Broglie wavelength ${\lambda }_d$. A change in wavelength of the incident light by $\mathrm{\Delta }\lambda$ results in a change $\mathrm{\Delta }{\lambda }_d$ in ${\lambda }_d$. The ratio $\frac{\mathrm{\Delta }{\lambda }_d}{\mathrm{\Delta }\lambda }$ is proportional to

In a scattering experiment, a particle of mass \(2 m\) collides with another particle of mass \(m\), which is initially at rest. Assuming the collision to be perfectly elastic, the maximum angular deviation \(\theta\) of the heavier particle, as shown in the figure, in radians is:

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

The freezing point (in \({ }^{\circ} \mathrm{C}\) ) of solution containing \(0.1 \mathrm{~g}\) of \(\mathrm{K}_3\left[\mathrm{Fe}(\mathrm{CN})_6\right]\) (molecular weight 329\()\) in \(100 \mathrm{~g}\) of water \(\left(K_f=1.86 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}\right)\) is

Let \(\overrightarrow{O P}=\frac{\alpha-1}{\alpha} \hat{i}+\hat{j}+\hat{k}, \overrightarrow{O Q}=\hat{i}+\frac{\beta-1}{\beta} \hat{j}+\hat{k}\) and \(\overrightarrow{O R}=\hat{i}+\hat{j}+\frac{1}{2} \hat{k}\) be three vectors, where \(\alpha, \beta \in \mathbb{R}-\{0\}\) and \(O\) denotes the origin. If \((\overrightarrow{O P} \times \overrightarrow{O Q}) \cdot \overrightarrow{O R}=0\) and the point \((\alpha, \beta, 2)\) lies on the plane \(3 x+3 y-z+l=0\), then the value of \(l\) is ____

The correct option(s) to distinguish nitrate salts of $M{n}^{2+} \mathrm{a}\mathrm{n}\mathrm{d} C{u}^{2+}$ taken separately is(are)

Two coherent monochromatic point sources $S_1$ and $S_2$ of wavelength $\lambda =600nm$ are placed symmetrically on either side of the center of the circle as shown. The sources are separated by a distance $d=1.8mm.$ This arrangement produces interference fringes visible as alternate bright and dark spots on the circumference of the circle. The angular separation between two consecutive bright spots is $\mathrm{\Delta }\theta$ . Which of the following options is/are correct?

For how many values of $p$, the circle $x^2+y^2+2x+4y-p=0$ and the coordinate axes have exactly three common points?