Two identical uniform discs roll without slipping on two different surfaces AB and CD (see figure) starting at A and C with linear speeds \(v_1\) and \(v_2\), respectively, and always remain in contact with the surfaces. If they reach B and D with the same linear speed and \(v_1=3 m / s\), then \(v_2\) in \(m / s\) is \(\left( g =10 \frac{m}{ s ^2}\right)\)

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?

A projectile of mass 200 g is launched in a viscous medium at an angle \(60^{\circ}\) with the horizontal, with an initial velocity of \(270 \mathrm{~m} / \mathrm{s}\). It experiences a viscous drag force \(\vec{F}=-c \vec{v}\) where the drag coefficient \(c=0.1 \mathrm{~kg} / \mathrm{s}\) and \(\vec{v}\) is the instantaneous velocity of the projectile. The projectile hits a vertical wall after 2 s . Taking \(e=2.7\), the horizontal distance of the wall from the point of projection (in m ) is

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Light guidance in an optical fiber can be understood by considering a structure comprising of thin solid glass cylinder of refractive index \(n_{1}\) surrounded by a medium of lower refractive index \(n_{2}\). The light guidance in the structure takes place due to successive total internal reflections at the interface of the media \(n_{1}\) and \(n_{2}\) as shown in the figure. All rays with the angle of incidence \(i\) less than a particular value \(i_{m}\) are confined in the medium of refractive index \(n_{1}\). The numerical aperture (NA) of the structure is defined as \(\sin i_{m}\).



Question:

If two structures of same cross-sectional area, but different numerical apertures \(N A_{1}\) and \(N A_{2}\left(N A_{2} < N A_{1}\right)\) are joined longitudinally, the numerical aperture of the combined structure is

A solid sphere of radius \(R\) has a charge \(Q\) distributed in its volume with a charge density \(\rho=k r^a\), where \(k\) and \(a\) are constants and \(r\) is the distance from its centre. If the electric field at \(r=\frac{R}{2}\) is \(\frac{1}{8}\) times that at \(r=R\), find the value of \(a\).

List I describes thermodynamic processes in four different systems. List II gives the magnitudes (either exactly or as a close approximation) of possible changes in the internal energy of the system due to the process.

	List-I		List-II
(I)	${10}^{-3} \mathrm{kg}$ of water $100 °\mathrm{C}$ is converted to steam at the same temperature, at a pressure of ${10}^5\mathrm{Pa}$. The volume of the system changes from ${10}^{-6}{\mathrm{m}}^3$ to ${10}^{-3}{\mathrm{m}}^3$ in the process. Latent heat of water $=2250 \mathrm{kJ} {\mathrm{kg}}^{-1}$	(P)	$2 \mathrm{kJ}$
(II)	$0.2$ moles of a rigid diatomic ideal gas with volume $V$ at temperature $500 \mathrm{K}$ undergoes an isobaric expansion to volume $3 V$. Assume $8.0 \mathrm{J} {\mathrm{mol}}^{–1} {\mathrm{K}}^{–1}$	(Q)	$7 \mathrm{kJ}$
(III)	One mole of a monoatomic ideal gas is compressed adiabatically from volume $V=\frac{1}{3}{\mathrm{m}}^3$ and pressure $2 \mathrm{kPa}$ to volume $\frac{V}{8}$.	(R)	$4 \mathrm{kJ}$
(IV)	Three moles of a diatomic ideal gas whose molecules can vibrate, is given $9 \mathrm{kJ}$ of heat and undergoes isobaric expansion.	(S)	$5 \mathrm{kJ}$
		(T)	$3 \mathrm{kJ}$

Which one of the following options is correct?

A biconvex lens of focal length \(f\) forms a circular image of radius \(r\) of sun in focal plane. Then, which option is correct?

Let \(z\) be a complex number such that the imaginary part of \(z\) is non-zero and \(a=z^{2}+z+1\) is real. Then a cannot take the value

Consider a triangle $PQR$ having sides of lengths $p,q$ and $r$ opposite to the angles $P,Q$ and $R$, respectively. Then which of the following statements is (are) TRUE?

If \(|z|=1\) and \(z \neq \pm 1\), then all the values of \(\frac{z}{1-z^2}\) lie on

If \(\alpha=\int_{\frac{1}{2}}^2 \frac{\tan ^{-1} x}{2 x^2-3 x+2} d x\)
then the value of \(\sqrt{7} \tan \left(\frac{2 \alpha \sqrt{7}}{\pi}\right)\) is ____________.
(Here, the inverse trigonometric function \(\tan ^{-1} x\) assumes values in \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\).)

A pack contains n cards numbered from 1 to n. Two consecutive numbered cards are removed from the pack and the sum of the numbers on the remaining cards is 1224. If the smaller of the numbers on the removed cards is k, then $\text{k}-20=$

The Henry's law constant for the solubility of \(\mathrm{N}_2\) gas in water at \(298 \mathrm{~K}\) is \(1.0 \times 105 \mathrm{~atm}\). The mole fraction of \(\mathrm{N}_2\) in air is \(0.8\). The number of moles of \(\mathrm{N}_2\) from air dissolved in 10 moles of water of \(298 \mathrm{~K}\) and \(5 \mathrm{~atm}\) pressure is