A monochromatic light wave is incident normally on a glass slab of thickness $d$, as shown in the figure. The refractive index of the slab increases linearly from $n_1$ to $n_2$ over the height $h$. Which of the following statement(s) is(are) true about the light wave emerging out of the slab?

The binding energy of nucleons in a nucleus can be affected by the pairwise Coulomb repulsion. Assume that all nucleons are uniformly distributed inside the nucleus. Let the binding energy of a proton be $E_b^p$ and the binding energy of a neutron be $E_b^n$ in the nucleus.
Which of the following statement(s) is(are) correct?

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Consider an evacuated cylindrical chamber of height \(h\) having rigid conducting plates at the ends and an insulating curved surface as shown in the figure. A number of spherical balls made of a light weight and soft material and coated with a conducting material are placed on the bottom plate. The balls have a radius \(r \ll h\). Now a high voltage source \((HV)\) is connected across the conducting plates such that the bottom plate is at \(+V_{0}\) and the top plate at \(-V_{0} .\) Due to their conducting surface, the balls will get charged, will become equipotential with the plate and are repelled by it. The balls will eventually collide with the top plate, where the coefficient of restitution can be taken to be zero due to the soft nature of the material of the balls. The electric field in the chamber can be considered to be that of a parallel plate capacitor. Assume that there are no collisions between the balls and the interaction between them is negligible. (Ignore gravity)



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The average current in the steady state registered by the ammeter in the circuit will be

A single slit diffraction experiment is performed to determine the slit width using the equation, \(\frac{b d}{D}=m \lambda\), where \(b\) is the slit width, \(D\) the shortest distance between the slit and the screen, \(d\) the distance between the \(m^{\text {th }}\) diffraction maximum and the central maximum, and \(\lambda\) is the wavelength. \(D\) and \(d\) are measured with scales of least count of 1 cm and 1 mm, respectively. The values of \(\lambda\) and \(m\) are known precisely to be 600 nm and 3, respectively. The maximum absolute error (in \(\mu \mathrm{m}\)) in the value of \(b\) estimated using the diffraction maximum that occurs for \(m=3\) with \(d=5 \mathrm{~mm}\) and \(D=1 \mathrm{~m}\) is ______.

A silver sphere of radius \(1 \mathrm{~cm}\) and work function \(4.7 \mathrm{eV}\) is suspended from an insulating thread in free-space. It is under continuous illumination of \(200 \mathrm{~nm}\) wavelength light. As photoelectrons are emitted, the sphere gets charged and acquires a potential. The maximum number of photoelectrons emitted from the sphere is \(A \times 10^Z\) (where \(1 < A < 10\) ). The value of \(Z\) is

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A wooden cylinder of diameter \(4 r\), height \(H\) and density \(\rho / 3\) is kept on a hole of diameter \(2 r\) of a tank, filled with liquid of density \(\rho\) as shown in the figure.

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Now level of the liquid starts decreasing slowly. When the level of liquid is at a height \(h_1\) above the cylinder, the block starts moving up. At what value of \(h_1\), will the block rise?

The focal length of a thin biconvex lens is \(20 \mathrm{~cm}\). When an object is moved from a distance of \(25 \mathrm{~cm}\) in front of it to \(50 \mathrm{~cm}\), the magnification of its image changes from \(m_{25}\) to \(m_{50}\). The ratio \(\frac{m_{25}}{m_{50}}\) is

The total number of stereoisomers that can exist for M is

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The reaction of compound \(\mathbf{P}\) with \(\mathrm{CH}_{3} \mathrm{MgBr}\) (excess) in \(\left(\mathrm{C}_{2} \mathrm{H}_{5}\right)_{2} \mathrm{O}\) followed by addition of \(\mathrm{H}_{2} \mathrm{O}\) gives \(\mathbf{Q}\). The compound \(\mathbf{Q}\) on treatment with \(\mathrm{H}_{2} \mathrm{SO}_{4}\) at \(0^{\circ} \mathrm{C}\) gives \(\mathbf{R}\). The reaction of \(\mathbf{R}\) with \(\mathrm{CH}_{3} \mathrm{COCl}\) in the presence of anhydrous \(\mathrm{AlCl}_{3}\) in \(\mathrm{CH}_{2} \mathrm{Cl}_{2}\) followed by treatment with \(\mathrm{H}_{2} \mathrm{O}\) produces compound \(\mathbf{S}\). [Et in compound \(\mathbf{P}\) is ethyl group]


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The reactions, \(\mathbf{Q}\) to \(\mathbf{R}\) and \(\mathbf{R}\) to \(\mathbf{S}\), are

The \(x-t\) graph of a particle undergoing simple harmonic motion is shown below. The acceleration of the particle at \(t=\frac{4}{3} \mathrm{~s}\) is

A particle of mass $m$ is initially at rest at the origin. It is subjected to a force and starts moving along the x - axis. Its kinetic energy changes with time as $dK/dt=\gamma t,$ where $\gamma$ is a positive constant of appropriate dimensions. Which of the following statements is (are) true?

A thermodynamic system is taken from an initial state $\mathrm{i}$ with internal energy ${\mathrm{U}}_{\mathrm{i}}\mathrm{}=100\mathrm{}\mathrm{J}$ to the final state f along two different paths iaf and ibf, as schematically shown in the figure. The work done by the system along the paths af, ib and bf are ${\mathrm{w}}_{\mathrm{a}\mathrm{f}}=\mathrm{}200\mathrm{}\mathrm{J},\mathrm{}{\mathrm{W}}_{\mathrm{i}\mathrm{b}}\mathrm{}=\mathrm{}50\mathrm{}\mathrm{J}$ and ${\mathrm{W}}_{\mathrm{b}\mathrm{f}}\mathrm{}=\mathrm{}100\mathrm{}\mathrm{J}$ respectively. The heat supplied to the system along the path iaf, ib and bf are iaf ib ${\mathrm{Q}}_{\mathrm{i}\mathrm{a}\mathrm{f}},\mathrm{}{\mathrm{Q}}_{\mathrm{i}\mathrm{b}}$ and ${\mathrm{Q}}_{\mathrm{b}\mathrm{f}}$ respectively. If the internal energy of the system in the state b is ${\mathrm{U}}_{\mathrm{b}}=200\mathrm{}\mathrm{J}$ and ${\mathrm{Q}}_{\mathrm{i}\mathrm{a}\mathrm{f}}=500\mathrm{}\mathrm{J}$ , the ratio $\frac{{\mathrm{Q}}_{\mathrm{b}\mathrm{f}}}{{\mathrm{Q}}_{\mathrm{i}\mathrm{b}}}\mathrm{}$ is

Let the functions $f:\left(-1,1\right)\to \mathrm{ℝ}$ and  $g:\left(-1,1\right)\to \left(-1,1\right)$ be defined by $f\left(x\right)= \left|2x-1\right|+\left|2x+1\right| \& g\left(x\right)=x-\left[x\right]$, where $\left[x\right]$ denotes the greatest integer less than or equal to$x$. Let $f\circ g:\left(-1,1\right)\to \mathrm{ℝ}$ be the composite function defined by $\left(f\circ g\right)\left(x\right)=f\left(g\left(x\right)\right)$. Suppose $c$ is the number of points in the interval $\left(-1,1\right)$ at which $f\circ g$ is not continuous, and suppose $d$ is the number of points in the interval $\left(-1, 1\right)$ at which  $f\circ g$ is not differentiable. Then the value of $c+d$ is _____________