A small object is placed 50 cm to the left of thin convex lens of focal length 30 cm. A convex spherical mirror of radius of curvature 100 cm is placed to the right of the lens at a distance of 50 cm. The mirror is tilted such that the axis of the mirror is at an angle $\theta ={30}^o$ to the axis of the lens, as shown in the figure. If the origin of the coordinate system is taken to be at the centre of the lens, the coordinates (in cm) of the point (x, y) at which the image is formed are:

In a Young's double slit experiment, the separation between the two slits is \(d\) and the wavelength of the light is \(\lambda\). The intensity of light falling on slit 1 is four times the intensity of light falling on slit 2 . Choose the correct choice(s),

A ball is dropped from a height of \(20 \mathrm{~m}\) above the surface of water in a lake. The refractive index of water is \(\frac{4}{3}\). A fish inside the lake, in the line of fall of the ball, is looking at the ball. At an instant, when the ball is \(12.8 \mathrm{~m}\) above the water surface, the fish sees the speed of ball as

Statement I A vertical iron rod has a coil of wire wound over it at the bottom end. An alternating current flows in the coil. The rod goes through a conducting ring as shown in the figure. The ring can float at a certain height above the coil.
Statement II In the above situation, a current is induced in the ring which interacts with the horizontal component of the magnetic field to produce an average force in the the upward direction.

Column I contains a list of processes involving expansion of an ideal gas. Match this with Column II describing the thermodynamic change during this process. Indicate your answer by darkening the appropriate bubbles of the \(4 \times 4\) matrix given in the ORS.

A person measures the depth of a well by measuring the time interval between dropping a stone and receiving the sound of impact with the bottom of the well. The error in his measurement of time is $\delta T=0.01$ seconds and he measures the depth of the well to be L = 20 meters. Take the acceleration due to gravity $g=10 m{s}^{-2}$ and the velocity of sound is $300m{s}^{-1}$ . Then the fractional error in the measurement, $\frac{\delta L}{L}$ , is closest to

Let us consider a system of units in which mass and angular momentum are dimensionless. If length has dimension of $\mathrm{L},$ which of the following statement (s) is/are correct?

Liquids A and B form ideal solution over the entire range of composition. At temperature $\mathrm{T}$, equimolar binary solution of liquids A and B has vapour pressure $45 \mathrm{T}\mathrm{o}\mathrm{r}\mathrm{r}$ . At the same temperature, a new solution of A and B having mole fractions ${\mathrm{x}}_{\mathrm{A}}\mathrm{ }\mathrm{and}\mathrm{ }{\mathrm{x}}_{\mathrm{B}}$, respectively, has vapour pressure of $22.5 \mathrm{T}\mathrm{o}\mathrm{r}\mathrm{r}$. The value of $\frac{{\mathrm{x}}_{\mathrm{A}}}{{\mathrm{x}}_{\mathrm{B}}}$, in the new solution is ____.
(Given that the vapour pressure of pure liquid A is $20 \mathrm{Torr}$ at temperature $\mathrm{T}$)

The value of $\sum _{k=1}^{13}\frac{1}{\sin \left(\frac{\pi }{4}+\frac{\left(k-1\right)\pi }{6}\right)\sin \left(\frac{\pi }{4}+\frac{k\pi }{6}\right)}$ is equal to

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When a particle of mass \(m\) moves on the \(x\)-axis in a potential of the form \(V(x)=k x^2\), it performs simple harmonic motion.

The corresponding time period is proportional to \(\sqrt{\frac{m}{k}}\), as can be seen easily using dimensional analysis. However, the motion of a particle can be periodic even when its potential energy increases on both sides of \(x=0\) in a way different from \(k x^2\) and its total energy is such that the particle does not escape to infinity. Consider a particle of mass \(\mathrm{m}\) moving on the \(x\)-axis. Its potential energy is \(V(x)=\alpha x^4(\alpha>0)\) for \(|x|\) near the origin and becomes a constant equal to \(V_0\) for \(|x| \geq X_0\) (see figure).
Question :
If the total energy of the particle is \(E\), it will perform periodic motion only if

A ray of light, travelling in the direction $\frac{1}{2}\left(\hat{\mathrm{i}}+\sqrt{3} \hat{\mathrm{j}}\right)$, is incident on a plane mirror. After reflection, it travels along the direction $\frac{1}{2}\left(\hat{\mathrm{i}}-\sqrt{3}\widehat{ \mathrm{j}}\right)$. The angle of incidence is

Two identical non-conducting solid spheres of same mass and charge are suspended in air from a common point by two non-conducting, massless strings of same length. At equilibrium, the angle between the strings is $\alpha$. The spheres are now immersed in a dielectric liquid of density $800 \mathrm{kg} {\mathrm{m}}^{-3}$ and dielectric constant $21$. If the angle between the strings remains the same after the immersion, then

In a one-litre flask, $6$ moles of $\mathrm{A}$ undergoes the reaction $\mathrm{A}( \mathrm{g})\rightleftharpoons \mathrm{P}( \mathrm{g})$. The progress of product formation at two temperatures (in Kelvin), ${\mathrm{T}}_1$ and ${\mathrm{T}}_2$, is shown in the figure: If ${\mathrm{T}}_1=2{\mathrm{T}}_2$ and$\left({\mathrm{\Delta G}}_2^{\mathrm{o}}-{\mathrm{\Delta G}}_1^{\mathrm{o}}\right)={\mathrm{RT}}_2\mathrm{lnx}$ , then the value of $\mathrm{x}$ is
[ ${\mathrm{\Delta G}}_1^{\mathrm{o}}$ and ${\mathrm{\Delta G}}_2^{\mathrm{o}}$ are standard Gibb's free energy change for the reaction at temperatures ${\mathrm{T}}_1$ and ${\mathrm{T}}_2$, respectively.]