A geostationary satellite above the equator is orbiting around the earth at a fixed distance \(r_1\) from the center of the earth. A second satellite is orbiting in the equatorial plane in the opposite direction to the earth's rotation, at a distance \(r_2\) from the center of the earth, such that \(r_1=1.21 r_2\). The time period of the second satellite as measured from the geostationary satellite is \(\frac{24}{p}\) hours. The value of \(p\) is

One mole of a monatomic ideal gas undergoes four thermodynamic processes as shown schematically in the PV - diagram below. Among these four processes, one is isobaric, one is isochoric, one is isothermal and one is adiabatic. Match the processes mentioned in List-1 with the corresponding statements in List-II.

	LIST -I		LIST -II
A)	In process $I$	P)	Work done by the gas is zero
B)	In process $II$	Q)	Temperature of the gas remains unchanged
C)	In process $III$	R)	No heat is exchanged between the gas and its surroundings
D)	In process $IV$	S)	Work done by the gas is $6P_0{V}_0$

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?

For a radioactive material, its activity $\mathrm{A}$ and rate of change of its activity $\mathrm{R}$ are defined as $\mathrm{A}=-\frac{\mathrm{d}\mathrm{N}}{\mathrm{d}\mathrm{t}}$ and $\mathrm{R}=-\frac{\mathrm{d}\mathrm{A}}{\mathrm{d}\mathrm{t}}$ , where $\mathrm{N}\left(\mathrm{t}\right)$ is the number of nuclei at time $\mathrm{t}$ . Two radioactive sources $\mathrm{P}$ (mean life $\mathrm{\tau }$ ) and $\mathrm{Q}$ (mean life $2\mathrm{\tau }$ ) have the same activity at $\mathrm{t}=0$ . Their rates of change of activities at $\mathrm{t}=2\mathrm{\tau }$ are ${\mathrm{R}}_{\mathrm{P}}$ and ${\mathrm{R}}_{\mathrm{Q}}$ , respectively. If $\frac{{\mathrm{R}}_{\mathrm{P}}}{{\mathrm{R}}_{\mathrm{Q}}}=\frac{\mathrm{n}}{\mathrm{e}}$ , then the value of $\mathrm{n}$ is

A circular disc with a groove along its diameter is placed horizontally. A block of mass \(1 \mathrm{~kg}\) is placed as shown. The coefficient of friction between the block and all surfaces of groove in contact is \(\mu=2 / 5\). The disc has an acceleration of \(25 \mathrm{~m} / \mathrm{s}^2\). Find the acceleration of the block with respect to disc.

A cylindrical furnace has height $\left(H\right)$ and diameter $\left(D\right)$ both $1 \mathrm{m}$. It is maintained at temperature $360 \mathrm{K}$. The air gets heated inside the furnace at constant pressure $P_a$ and its temperature becomes $T=360 \mathrm{K}$. The hot air with density $\rho$ rises up a vertical chimney of diameter $d=0.1 \mathrm{m}$ and height $h=9 \mathrm{m}$ above the furnace and exits the chimney (see the figure). As a result, atmospheric air of density ${\rho }_a=1.2 \mathrm{kg} {\mathrm{m}}^{-3}$, pressure $P_a$ and temperature $T_a=300 \mathrm{K}$ enters the furnace. Assume air as an ideal gas, neglect the variations in $\rho$ and $T$ inside the chimney and the furnace. Also ignore the viscous effects.
[Given: The acceleration due to gravity $g=10 \mathrm{m} {\mathrm{s}}^{-2}$ and $\pi =3.14$]

Considering the air flow to be streamline, the steady mass flow rate of air exiting the chimney is _____ $\mathrm{gm} {\mathrm{s}}^{-1}$.

The mass \(M\) shown in the figure oscillates in simple harmonic motion with amplitude \(A\). The amplitude of the point \(P\) is

Paragraph : If the measurement errors in all the independent quantities are known, then it is possible to determine the error in any dependent quantity. This is done by the use of series expansion and truncating the expansion at the first power of the error. For example, consider the relation $z =\frac{x}{y}$. If the errors in $x, y \mathrm{a}\mathrm{n}\mathrm{d} z \mathrm{a}\mathrm{r}\mathrm{e} ∆x, ∆y \mathrm{a}\mathrm{n}\mathrm{d} ∆z$ , respectively, then

$z\pm ∆z=\frac{x\pm ∆x}{y\pm ∆y}=\frac{x}{y}\left(1\pm \frac{∆x}{x}\right){\left(1\pm \frac{∆y}{y}\right)}^{-1}$

The series expansion for ${\left(1\pm \frac{∆y}{y}\right)}^{-1}$, to first power in $∆y/y, \mathrm{i}\mathrm{s} 1\mp \left(∆y/y\right)$. The relative errors in independent variables are always added. So the error in $z$ will be

$\text{Δ}z=z\left(\frac{\text{Δ}x}{x}+\frac{\text{Δ}y}{y}\right).$
The above derivation makes the assumption that \(\Delta r / x \ll 1, \Delta y / y < 1\). Therefore, the higher powers of these quantities are neglected.
Question : Consider the ratio \(r=\frac{(1-a)}{(1+a)}\) to be determined by measuring a dimensionless quantity a If the error in the measurement of a is \(\Delta a(\frac{\Delta a}{a}<<1)\), then what is the error \(\Delta r\) in determining \(r\) ?

Let \(L=\lim _{x \rightarrow 0} \frac{a-\sqrt{a^2-x^2}-\frac{x^2}{4}}{x^4}\), \(a>0\). If \(L\) is finite, then

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

For $\text{a}\in \text{R}$ (the set of all real numbers), $\text{a}\ne -1$, $\underset{\text{n}\to \infty }{\text{lim}}\frac{\left(1^{\text{a}}+2^{\text{a}}+\dots +{\text{n}}^{\text{a}}\right)}{{\left(\text{n}+1\right)}^{\text{a}-1}\left[\left(\text{na}+1\right)+\left(\text{na}+2\right)+\dots +\left(\text{na}+\text{n}\right)\right]}=\frac{1}{60}$ Then a =

In a conductometric titration, small volume of titrant of higher concentration is added stepwise to a larger volume of titrate of much lower concentration, and the conductance is measured after each addition.
The limiting ionic conductivity \(\left(\Lambda_0\right)\) values (in \(\mathrm{mS} \mathrm{m}^2 \mathrm{~mol}^{-1}\)) for different ions in aqueous solutions are given below:

For different combinations of titrates and titrants given in List-I, the graphs of 'conductance' versus 'volume of titrant' are given in List-II.
Match each entry in List-I with the appropriate entry in List-II and choose the correct option.

Let $f, g :\left[-1, 2\right]\to \mathbb{ }\mathbb{R}$   be continuous functions which are twice differentiable on the interval $\left(-1, 2\right)$. Let the values of $f$ and $g$ at the points $-1, 0$ and $2$ be as given in the following table:
 

	$x= -1$	$x = 0$	$x = 2$
$f\left(x\right)$	3	6	0
$g\left(x\right)$	0	1	-1

In each of the intervals $\left(-1, 0\right)$ and $\left(0, 2\right)$ the function $(f-3g)"$ never vanishes. Then the correct statement(s) is(are)