A hot air balloon is carrying some passengers, and a few sandbags of mass $1 \mathrm{kg}$ each so that its total mass is $480 \mathrm{kg}$. Its effective volume giving the balloon its buoyancy is $V$. The balloon is floating at an equilibrium height of $100 \mathrm{m}$. When $N$ number of sandbags are thrown out, the balloon rises to a new equilibrium height close to $150 \mathrm{m}$ with its volume $V$ remain unchanged. If the variation of the density of air with height $h$ from the ground is $\rho \left(h\right)={\rho }_0{e}^{-\frac{h}{h_0}},$ where ${\mathrm{\rho }}_0=1.25 \mathrm{kg} {\mathrm{m}}^{-3}$ and $h_0=6000 \mathrm{m},$ the value of $N$ is _________.

An electric dipole with dipole moment $\frac{{\mathrm{p}}_0}{\sqrt{2}}\left(\widehat{\mathrm{i}}+\widehat{\mathrm{j}}\right)$ is held fixed at the origin $\mathrm{O}$ in the presence of an uniform electric field of magnitude ${\mathrm{E}}_0.$ If the potential is constant on a circle of radius $\mathrm{R}$ centered at the origin as shown in figure, then the correct statement(s) is/are:
( ${\mathrm{\epsilon }}_0$ is permittivity of free space, $\mathrm{R}>>$ dipole size)

An electron in an excited state of $\mathrm{L}{\mathrm{i}}^{2+}$ ion has angular momentum $\frac{3\mathrm{h}}{2\mathrm{\pi }}$ . The de Broglie wavelength of the electron in this state is $\mathrm{p}\mathrm{\pi }{\mathrm{\alpha }}_0$ (where ${\mathrm{a}}_0$ is the Bohr radius). The value of $\mathrm{p}$ is

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A thermal power plant produces electric power of \(600 \mathrm{~kW}\) at \(4000 \mathrm{~V}\), which is to be transported to a place \(20 \mathrm{~km}\) away from the power plant for consumers' usage. It can be transported either directly with a cable of large current carrying capacity or by using a combination of step-up and step-down transformers at the two ends. The drawback of the direct transmission is the large energy dissipation. In the method using transformers, the dissipation is much smaller. In this method, a step-up transformer is used at the plant side so that the current is reduced to a smaller value. At the consumers' end, a step-down transformer is used to supply power to the consumers at the specified lower voltage. It is reasonable to assume that the power cable is purely resistive and the transformers are ideal with a power factor unity. All the currents and voltages mentioned are rms values.

Question :

In the method using the transformers, assume that the ratio of the number of turns in the primary to that in the secondary in the step-up transformer is \(1: 10\). If the power to the consumers has to be supplied at \(200 \mathrm{~V}\), the ratio of the number of turns in the primary to that in the secondary in the step-down transformer is

A train $\mathrm{S}1,$ moving with a uniform velocity of $108\mathrm{}\mathrm{k}\mathrm{m}/\mathrm{h},$ approaches another train $\mathrm{S}2$ standing on a platform. An observer $\mathrm{O}$ moves with a uniform velocity of $36\mathrm{}\mathrm{k}\mathrm{m}/\mathrm{h}$ towards $\mathrm{S}2,$ as shown in figure. Both the trains are blowing whistles of same frequency $120\mathrm{}\mathrm{H}\mathrm{z}.$ When $\mathrm{O}$ is $600\mathrm{}\mathrm{m}$ away from $\mathrm{S}2$ and distance between $\mathrm{S}1$ and $\mathrm{S}2$ is $800\mathrm{}\mathrm{m},$ the number of beats heard by $\mathrm{O}$ is____
[Speed of the sound $=330\mathrm{}\mathrm{m}/\mathrm{s}]$

A pulse of light of duration 100 ns is absorbed completely by a small object initially at rest. Power of the pulse is 30 mW and the speed of light is 3 ×10⁸ ms^-1. The final momentum of the object is

The radius of the orbit of an electron in a Hydrogen-like atom is \(4.5 a _0\), where \(a _0\) is the Bohr radius. Its orbital angular momentum is \(\frac{3 h}{2 \pi}\). It is given that \(h\) is Planck's constant and R is Rydberg constant. the possible wavelength \((s)\), when the atom de-excites, is (are)

Let $M$ be a $3\times 3$ invertible matrix with real entries and let $I$ denote the $3\times 3$ identity matrix. $M^{-1}=adj\left(adjM\right)$, then which of the following statements is/are ALWAYS TRUE?

A black body of temperature \(T\) is inside chamber of temperature \(T_0\). Now, the closed chamber is slightly opened to sun such that temperature of black body \((T)\) and chamber \(\left(T_0\right)\) remains constant.

Let $X$ be the set of all five digit numbers formed using $1,2,2,2,4,4,0$. For example, $22240$ is in $X$ while $02244$ and $44422$ are not in $X$. Suppose that each element of $X$ has an equal chance of being chosen. Let $p$ be the conditional probability that an element chosen at random is a multiple of $20$ given that it is a multiple of $5$. Then the value of $38 p$ is equal to

A stationary tuning fork is in resonance with an air column in a pipe. If the tuning fork is moved with a speed of $2 \mathrm{m} {\mathrm{s}}^{-1}$ in front of the open end of the pipe and parallel to it, the length of the pipe should be changed for the resonance to occur with the moving tuning fork. If the speed of sound in air is $320 \mathrm{m} {\mathrm{s}}^{-1},$ the smallest value of the percentage change required in the length of the pipe is ________.

Let \(f(x)=\frac{x^2-6 x+5}{x^2-5 x+6}\).

Two identical moving coil galvanometer have $10\mathrm{ }\mathrm{\Omega }$ resistance and full scale deflection at $2\mathrm{\mu }\mathrm{A}$ current. One of them is converted into a voltmeter of $100\mathrm{ }\mathrm{m}\mathrm{V}$ full scale reading and the other into an Ammeter of $1\mathrm{ }\mathrm{m}\mathrm{A}$ full scale current using appropriate resistors. These are then used to measure the voltage and current in the Ohm's law experiment with $\mathrm{R}=1000\mathrm{ }\mathrm{\Omega }$ resistor by using an ideal cell. Which of the following statement(s) is/are correct?