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One main scale division of a Vernier calliper is equal to 1 mm and the number of divisions on the Vernier scale is 10. When both the jaws touch each other, the Vernier scale shifts to the left of zero of the main scale in such a way that 4th Vernier division coincides with a division of the main scale. If this Vernier calliper measures the length of the wire to be 1 cm, the actual length of the wire is:

A ray of light with wavelength $\lambda$ is incident on three different photoelectric cells namely 1, 2 and 3. The threshold wavelength of these photoelectric cells are $\lambda_{1}$, $\lambda_{2},$ and $\lambda_{3,}$ respectively and the magnitude of stopping potentials of these cells are $V_{1}$, $V_{2}$ and $V_{3},$ respectively. The relation between $\lambda$ and threshold wavelengths are $\lambda_{1}<\lambda$, $\lambda_{2}>\lambda$ and $\lambda_{3}>>\lambda$. The correct option is:

Consider that $\sigma_{S}$, $k_{B}$, b represents Stefan-Boltzmann constant, Boltzmann constant and Wien's displacement law constant, respectively. The dimension of $\sigma_{s}k_{B}^{-1}b$ is

In a solar system, the time-period of revolution of a planet tracing a circular orbit of radius R is proportional to:

Which of the following measurements require 'index correction'?

Consider the following nuclear reaction ${}^{238}U\longrightarrow{}^{234}Th+{}^{4}He$. Take masses of ${}^{238}U$, ${}^{234}Th$ and ${}^{4}He$ as 238.050 u, 234.043 u and 4.003 u, respectively. The Q value for the reaction, in keV, is: [Given: $1~u=931.5~MeV~c^{-2}$]

For sound waves, if the number of nodes for the $5^{th}$ harmonic of an open-ended pipe is n and that for the $9^{th}$ harmonic of the same pipe with one of its ends closed is m, the ratio $\frac{n}{m}$ is

A cylindrical cork of uniform density floats in a liquid of density $\rho_{1}$. If the cork is depressed slightly and released, it oscillates harmonically with time period T. If the same cork floats in another liquid of density $\rho_{2}$, then the similar oscillation has time period 2T. The value of $\rho_{2}/\rho_{1}$ is:

The mean free path of molecules in an ideal gas A is half that of another ideal gas B. The diameter of the spherical molecules of gas A is twice the diameter of the molecules of B. If number densities of the gases A and B are $n_{A}$ and $n_{B}$ respectively, the correct option is:

Consider that an electron is revolving in an excited state of Hydrogen atom with velocity $\sqrt{25.6}\times10^{5}ms^{-1}$. The radius of the orbit is $x\times10^{-9}$ m. The value of x is: [Take the mass of electron to the $9\times10^{-31}$ kg, charge of electron $=-1.6\times10^{-19}C$ and $\frac{1}{4\pi\epsilon_{0}}=9\times10^{9}Nm^{2}C^{-2}$]

A solid sphere A of radius R and mass M is attached at a point to a smaller solid sphere B of radius $r<R$ and mass $m<M.$ Assume that the line joining their centres lies along the horizontal. The moment of inertia of the system calculated about a vertical axis passing through the centre of A is $I_{A}$ and that calculated about a vertical axis passing through the centre of B is $I_{B}$. The difference $I_{A}-I_{B}$ is: [Image Placeholder]

The lens combination as shown in the figure, consists of two lenses, $L_{1}$ and $L_{2}$, of the focal lengths +10 cm and -10 cm, respectively. The position of the image formed is: [Image Placeholder]

Three identical p-n junction diodes $D_{1}$, $D_{2}$ and $D_{3}$ are connected across a battery as shown in the figure. If the width of the depletion regions of $D_{1},$ $D_{2}$ and $D_{3}$ are $W_{1},$ $W_{2}$ and $W_{3}$, respectively, then the correct option is: [Image Placeholder]

A frictionless circular wire of unit radius is fixed on the horizontal plane. Two-point particles of unit mass start moving simultaneously from point $A(\theta=\frac{\pi}{2})$ with identical uniform angular speeds in opposite directions, and meet again at point $B(\theta=-\frac{\pi}{2})$. During this time, which of the following figures schematically represent the magnitude of the total linear momentum P of the system, as a function of $\theta$? [Image Placeholder]

A car travels on a circular racetrack of radius 50 m, which is banked at an angle $\theta$. If the car travels at a speed $10~ms^{-1}$, then the wear and tear on its tyres is minimum. Taking the acceleration due to gravity to be $10~ms^{-2}$, the value of $\theta$ is:

Consider a fixed uniformly charged insulating sphere with radius R and total charge +Q. A point charge -q $(q<<Q)$ with mass m is released from rest at a distance of 3R from the centre of the charged sphere. When the point charge reaches the surface of the sphere, its speed is: ($\epsilon_0$ is the permittivity of vacuum, neglect gravitational forces).

Consider three media P, Q and R with refractive indices 1, 1.25, and 1.5 respectively. The medium Q having a thickness of 5 cm is placed between extended media P and R as shown in the figure. An object O is placed at the centre of medium Q. If viewed from medium P near the normal direction, the apparent depth of O is $h_1$. For similar observation from medium R, the apparent depth is $h_2$. The value of $|h_{1}-h_{2}|$, in cm, is: [Image Placeholder]

A particle of mass M moves along a horizontal x axis from $x=0$ to $x=L$. The coefficient of kinetic friction varies as a function of x as $\mu_{k}(x)=\mu_{0}-\alpha x,$ where $\mu_0$, $\alpha$ are constants of appropriate dimensions, so that $\mu_{k}(L)=0$. The total work done by the frictional force during the motion is $n\mu_0 MgL$, where g is the acceleration due to gravity. The value of n is:

An electromagnetic wave travelling in a lossless dielectric medium having a dielectric constant, $\epsilon_{r}=9$, has the electric field, $E_{x}=E_{0} \sin(kz-2\pi\times10^{6}t)Vm^{-1}$ where $E_{0}$ is the amplitude and k is the wave vector. Among the following options, the incorrect choice is

Bob B of mass m at rest is hanging vertically from the ceiling via a massless string of length 10 m, as shown in the figure. Point mass A of mass m travelling horizontally with speed $10~ms^{-1}$ hits bob B elastically. The bob B rises h meter after the collision. Taking the acceleration due to gravity $g=10~ms^{-2}$ and neglecting the size of the bob, the value of h is: [Image Placeholder]